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date	Tue, 10 Feb 2015 15:05:51 +0000
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wolffd@0	1 <HEAD>
wolffd@0	2 <TITLE>How to use the Bayes Net Toolbox</TITLE>
wolffd@0	3 </HEAD>
wolffd@0	4
wolffd@0	5 <BODY BGCOLOR="#FFFFFF">
wolffd@0	6 <!-- white background is better for the pictures and equations -->
wolffd@0	7
wolffd@0	8 <h1>How to use the Bayes Net Toolbox</h1>
wolffd@0	9
wolffd@0	10 This documentation was last updated on 13 November 2002.
wolffd@0	11 <br>
wolffd@0	12 Click <a href="changelog.html">here</a> for a list of changes made to
wolffd@0	13 BNT.
wolffd@0	14 <br>
wolffd@0	15 Click
wolffd@0	16 <a href="http://bnt.insa-rouen.fr/">here</a>
wolffd@0	17 for a French version of this documentation (which might not
wolffd@0	18 be up-to-date).
wolffd@0	19
wolffd@0	20
wolffd@0	21 <p>
wolffd@0	22
wolffd@0	23 <ul>
wolffd@0	24 <li> <a href="#install">Installation</a>
wolffd@0	25 <ul>
wolffd@0	26 <li> <a href="#installM">Installing the Matlab code</a>
wolffd@0	27 <li> <a href="#installC">Installing the C code</a>
wolffd@0	28 <li> <a href="../matlab_tips.html">Useful Matlab tips</a>.
wolffd@0	29 </ul>
wolffd@0	30
wolffd@0	31 <li> <a href="#basics">Creating your first Bayes net</a>
wolffd@0	32 <ul>
wolffd@0	33 <li> <a href="#basics">Creating a model by hand</a>
wolffd@0	34 <li> <a href="#file">Loading a model from a file</a>
wolffd@0	35 <li> <a href="http://bnt.insa-rouen.fr/ajouts.html">Creating a model using a GUI</a>
wolffd@0	36 </ul>
wolffd@0	37
wolffd@0	38 <li> <a href="#inference">Inference</a>
wolffd@0	39 <ul>
wolffd@0	40 <li> <a href="#marginal">Computing marginal distributions</a>
wolffd@0	41 <li> <a href="#joint">Computing joint distributions</a>
wolffd@0	42 <li> <a href="#soft">Soft/virtual evidence</a>
wolffd@0	43 <li> <a href="#mpe">Most probable explanation</a>
wolffd@0	44 </ul>
wolffd@0	45
wolffd@0	46 <li> <a href="#cpd">Conditional Probability Distributions</a>
wolffd@0	47 <ul>
wolffd@0	48 <li> <a href="#tabular">Tabular (multinomial) nodes</a>
wolffd@0	49 <li> <a href="#noisyor">Noisy-or nodes</a>
wolffd@0	50 <li> <a href="#deterministic">Other (noisy) deterministic nodes</a>
wolffd@0	51 <li> <a href="#softmax">Softmax (multinomial logit) nodes</a>
wolffd@0	52 <li> <a href="#mlp">Neural network nodes</a>
wolffd@0	53 <li> <a href="#root">Root nodes</a>
wolffd@0	54 <li> <a href="#gaussian">Gaussian nodes</a>
wolffd@0	55 <li> <a href="#glm">Generalized linear model nodes</a>
wolffd@0	56 <li> <a href="#dtree">Classification/regression tree nodes</a>
wolffd@0	57 <li> <a href="#nongauss">Other continuous distributions</a>
wolffd@0	58 <li> <a href="#cpd_summary">Summary of CPD types</a>
wolffd@0	59 </ul>
wolffd@0	60
wolffd@0	61 <li> <a href="#examples">Example models</a>
wolffd@0	62 <ul>
wolffd@0	63 <li> <a
wolffd@0	64 href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">
wolffd@0	65 Gaussian mixture models</a>
wolffd@0	66 <li> <a href="#pca">PCA, ICA, and all that</a>
wolffd@0	67 <li> <a href="#mixep">Mixtures of experts</a>
wolffd@0	68 <li> <a href="#hme">Hierarchical mixtures of experts</a>
wolffd@0	69 <li> <a href="#qmr">QMR</a>
wolffd@0	70 <li> <a href="#cg_model">Conditional Gaussian models</a>
wolffd@0	71 <li> <a href="#hybrid">Other hybrid models</a>
wolffd@0	72 </ul>
wolffd@0	73
wolffd@0	74 <li> <a href="#param_learning">Parameter learning</a>
wolffd@0	75 <ul>
wolffd@0	76 <li> <a href="#load_data">Loading data from a file</a>
wolffd@0	77 <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a>
wolffd@0	78 <li> <a href="#prior">Parameter priors</a>
wolffd@0	79 <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a>
wolffd@0	80 <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a>
wolffd@0	81 <li> <a href="#tying">Parameter tying</a>
wolffd@0	82 </ul>
wolffd@0	83
wolffd@0	84 <li> <a href="#structure_learning">Structure learning</a>
wolffd@0	85 <ul>
wolffd@0	86 <li> <a href="#enumerate">Exhaustive search</a>
wolffd@0	87 <li> <a href="#K2">K2</a>
wolffd@0	88 <li> <a href="#hill_climb">Hill-climbing</a>
wolffd@0	89 <li> <a href="#mcmc">MCMC</a>
wolffd@0	90 <li> <a href="#active">Active learning</a>
wolffd@0	91 <li> <a href="#struct_em">Structural EM</a>
wolffd@0	92 <li> <a href="#graphdraw">Visualizing the learned graph structure</a>
wolffd@0	93 <li> <a href="#constraint">Constraint-based methods</a>
wolffd@0	94 </ul>
wolffd@0	95
wolffd@0	96
wolffd@0	97 <li> <a href="#engines">Inference engines</a>
wolffd@0	98 <ul>
wolffd@0	99 <li> <a href="#jtree">Junction tree</a>
wolffd@0	100 <li> <a href="#varelim">Variable elimination</a>
wolffd@0	101 <li> <a href="#global">Global inference methods</a>
wolffd@0	102 <li> <a href="#quickscore">Quickscore</a>
wolffd@0	103 <li> <a href="#belprop">Belief propagation</a>
wolffd@0	104 <li> <a href="#sampling">Sampling (Monte Carlo)</a>
wolffd@0	105 <li> <a href="#engine_summary">Summary of inference engines</a>
wolffd@0	106 </ul>
wolffd@0	107
wolffd@0	108
wolffd@0	109 <li> <a href="#influence">Influence diagrams/ decision making</a>
wolffd@0	110
wolffd@0	111
wolffd@0	112 <li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a>
wolffd@0	113 </ul>
wolffd@0	114
wolffd@0	115 </ul>
wolffd@0	116
wolffd@0	117
wolffd@0	118
wolffd@0	119
wolffd@0	120 <h1><a name="install">Installation</h1>
wolffd@0	121
wolffd@0	122 <h2><a name="installM">Installing the Matlab code</h2>
wolffd@0	123
wolffd@0	124 <ul>
wolffd@0	125 <li> <a href="bnt_download.html">Download</a> the BNT.zip file.
wolffd@0	126
wolffd@0	127 <p>
wolffd@0	128 <li> Unpack the file. In Unix, type
wolffd@0	129 <!--"tar xvf BNT.tar".-->
wolffd@0	130 "unzip FullBNT.zip".
wolffd@0	131 In Windows, use
wolffd@0	132 a program like <a href="http://www.winzip.com">Winzip</a>. This will
wolffd@0	133 create a directory called FullBNT, which contains BNT and other libraries.
wolffd@0	134
wolffd@0	135 <p>
wolffd@0	136 <li> Read the file <tt>BNT/README</tt> to make sure the date
wolffd@0	137 matches the one on the top of <a href=bnt.html>the BNT home page</a>.
wolffd@0	138 If not, you may need to press 'refresh' on your browser, and download
wolffd@0	139 again, to get the most recent version.
wolffd@0	140
wolffd@0	141 <p>
wolffd@0	142 <li> <b>Edit the file "BNT/add_BNT_to_path.m"</b> so it contains the correct
wolffd@0	143 pathname.
wolffd@0	144 For example, in Windows,
wolffd@0	145 I download FullBNT.zip into C:\kpmurphy\matlab, and
wolffd@0	146 then comment out the second line (with the % character), and uncomment
wolffd@0	147 the third line, which reads
wolffd@0	148 <pre>
wolffd@0	149 BNT_HOME = 'C:\kpmurphy\matlab\FullBNT';
wolffd@0	150 </pre>
wolffd@0	151
wolffd@0	152 <p>
wolffd@0	153 <li> Start up Matlab.
wolffd@0	154
wolffd@0	155 <p>
wolffd@0	156 <li> Type "ver" at the Matlab prompt (">>").
wolffd@0	157 <b>You need Matlab version 5.2 or newer to run BNT</b>.
wolffd@0	158 (Versions 5.0 and 5.1 have a memory leak which seems to sometimes
wolffd@0	159 crash BNT.)
wolffd@0	160
wolffd@0	161 <p>
wolffd@0	162 <li> Move to the BNT directory.
wolffd@0	163 For example, in Windows, I type
wolffd@0	164 <pre>
wolffd@0	165 >> cd C:\kpmurphy\matlab\FullBNT\BNT
wolffd@0	166 </pre>
wolffd@0	167
wolffd@0	168 <p>
wolffd@0	169 <li> Type "add_BNT_to_path".
wolffd@0	170 This executes the command
wolffd@0	171 <tt>addpath(genpath(BNT_HOME))</tt>,
wolffd@0	172 which adds all directories below FullBNT to the matlab path.
wolffd@0	173
wolffd@0	174 <p>
wolffd@0	175 <li> Type "test_BNT".
wolffd@0	176 <b>If all goes well, this will just produce a bunch of numbers and pictures.</b>
wolffd@0	177 It may produce some
wolffd@0	178 warning messages (which you can ignore), but should not produce any error messages.
wolffd@0	179 (The warnings should only be of the form
wolffd@0	180 "Warning: Maximum number of iterations has been exceeded", and are
wolffd@0	181 produced by Netlab.)
wolffd@0	182
wolffd@0	183 <p>
wolffd@0	184 <li> <a href="http://groups.yahoo.com/group/BayesNetToolbox/join">
wolffd@0	185 Join the BNT email list</a>
wolffd@0	186
wolffd@0	187 </ul>
wolffd@0	188
wolffd@0	189
wolffd@0	190 If you are new to Matlab, you might like to check out
wolffd@0	191 <a href="matlab_tips.html">some useful Matlab tips</a>.
wolffd@0	192 For instance, this explains how to create a startup file, which can be
wolffd@0	193 used to set your path variable automatically, so you can avoid having
wolffd@0	194 to type the above commands every time.
wolffd@0	195
wolffd@0	196
wolffd@0	197
wolffd@0	198
wolffd@0	199 <h2><a name="installC">Installing the C code</h2>
wolffd@0	200
wolffd@0	201 Some BNT functions also have C implementations.
wolffd@0	202 <b>It is not necessary to install the C code</b>, but it can result in a speedup
wolffd@0	203 of a factor of 2-5 of certain simple functions.
wolffd@0	204 To install all the C code,
wolffd@0	205 edit installC_BNT.m so it contains the right path,
wolffd@0	206 then type <tt>installC_BNT</tt>.
wolffd@0	207 To uninstall all the C code,
wolffd@0	208 edit uninstallC_BNT.m so it contains the right path,
wolffd@0	209 then type <tt>uninstallC_BNT</tt>.
wolffd@0	210 For an up-to-date list of the files which have C implementations, see
wolffd@0	211 BNT/installC_BNT.m.
wolffd@0	212
wolffd@0	213 <p>
wolffd@0	214 mex is a script that lets you call C code from Matlab - it does not compile matlab to
wolffd@0	215 C (see mcc below).
wolffd@0	216 If your C/C++ compiler is set up correctly, mex should work out of
wolffd@0	217 the box. (Matlab 6 now ships with its own minimal C compiler.)
wolffd@0	218 If not, you might need to type
wolffd@0	219 <p>
wolffd@0	220 <tt> mex -setup</tt>
wolffd@0	221 <p>
wolffd@0	222 before calling installC.
wolffd@0	223 <p>
wolffd@0	224 To make mex call gcc on Windows,
wolffd@0	225 you must install <a
wolffd@0	226 href="http://www.mrc-cbu.cam.ac.uk/Imaging/gnumex20.html">gnumex</a>.
wolffd@0	227 You can use the <a href="http://www.mingw.org/">minimalist GNU for
wolffd@0	228 Windows</a> version of gcc, or
wolffd@0	229 the <a href="http://sources.redhat.com/cygwin/">cygwin</a> version.
wolffd@0	230 <p>
wolffd@0	231 In general, typing
wolffd@0	232 'mex foo.c' from inside Matlab creates a file called
wolffd@0	233 'foo.mexglx' or 'foo.dll' (the exact file
wolffd@0	234 extension is system dependent - on Linux it is 'mexglx', on Windows it is '.dll').
wolffd@0	235 The resulting file will hide the original 'foo.m' (if it existed), i.e.,
wolffd@0	236 typing 'foo' at the prompt will call the compiled C version.
wolffd@0	237 To reveal the original matlab version, just delete foo.mexglx (this is
wolffd@0	238 what uninstallC does).
wolffd@0	239 <p>
wolffd@0	240 Sometimes it takes time for Matlab to realize that the file has
wolffd@0	241 changed from matlab to C or vice versa; try typing 'clear all' or
wolffd@0	242 restarting Matlab to refresh it.
wolffd@0	243 To find out which version of a file you are running, type
wolffd@0	244 'which foo'.
wolffd@0	245 <p>
wolffd@0	246 <a href="http://www.mathworks.com/products/compiler">mcc</a>, the
wolffd@0	247 Matlab to C compiler, is a separate product,
wolffd@0	248 and is quite different from mex. It does not yet support
wolffd@0	249 objects/classes, which is why we can't compile all of BNT to C automatically.
wolffd@0	250 Also, hand-written C code is usually much
wolffd@0	251 better than the C code generated by mcc.
wolffd@0	252
wolffd@0	253
wolffd@0	254 <p>
wolffd@0	255 Acknowledgements:
wolffd@0	256 Although I wrote some of the C code, most of
wolffd@0	257 the C code (e.g., for jtree and dpot) was written by Wei Hu;
wolffd@0	258 the triangulation C code was written by Ilya Shpitser.
wolffd@0	259
wolffd@0	260
wolffd@0	261 <h1><a name="basics">Creating your first Bayes net</h1>
wolffd@0	262
wolffd@0	263 To define a Bayes net, you must specify the graph structure and then
wolffd@0	264 the parameters. We look at each in turn, using a simple example
wolffd@0	265 (adapted from Russell and
wolffd@0	266 Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall,
wolffd@0	267 1995, p454).
wolffd@0	268
wolffd@0	269
wolffd@0	270 <h2>Graph structure</h2>
wolffd@0	271
wolffd@0	272
wolffd@0	273 Consider the following network.
wolffd@0	274
wolffd@0	275 <p>
wolffd@0	276 <center>
wolffd@0	277 <IMG SRC="Figures/sprinkler.gif">
wolffd@0	278 </center>
wolffd@0	279 <p>
wolffd@0	280
wolffd@0	281 <P>
wolffd@0	282 To specify this directed acyclic graph (dag), we create an adjacency matrix:
wolffd@0	283 <PRE>
wolffd@0	284 N = 4;
wolffd@0	285 dag = zeros(N,N);
wolffd@0	286 C = 1; S = 2; R = 3; W = 4;
wolffd@0	287 dag(C,[R S]) = 1;
wolffd@0	288 dag(R,W) = 1;
wolffd@0	289 dag(S,W)=1;
wolffd@0	290 </PRE>
wolffd@0	291 <P>
wolffd@0	292 We have numbered the nodes as follows:
wolffd@0	293 Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4.
wolffd@0	294 <b>The nodes must always be numbered in topological order, i.e.,
wolffd@0	295 ancestors before descendants.</b>
wolffd@0	296 For a more complicated graph, this is a little inconvenient: we will
wolffd@0	297 see how to get around this <a href="usage_dbn.html#bat">below</a>.
wolffd@0	298 <p>
wolffd@0	299 <!--
wolffd@0	300 In Matlab 6, you can use logical arrays instead of double arrays,
wolffd@0	301 which are 4 times smaller:
wolffd@0	302 <pre>
wolffd@0	303 dag = false(N,N);
wolffd@0	304 dag(C,[R S]) = true;
wolffd@0	305 ...
wolffd@0	306 </pre>
wolffd@0	307 --
wolffd@0	308 <p>
wolffd@0	309 A preliminary attempt to make a <b>GUI</b>
wolffd@0	310 has been writte by Philippe LeRay and can be downloaded
wolffd@0	311 from <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>.
wolffd@0	312 <p>
wolffd@0	313 You can visualize the resulting graph structure using
wolffd@0	314 the methods discussed <a href="#graphdraw">below</a>.
wolffd@0	315
wolffd@0	316 <h2>Creating the Bayes net shell</h2>
wolffd@0	317
wolffd@0	318 In addition to specifying the graph structure,
wolffd@0	319 we must specify the size and type of each node.
wolffd@0	320 If a node is discrete, its size is the
wolffd@0	321 number of possible values
wolffd@0	322 each node can take on; if a node is continuous,
wolffd@0	323 it can be a vector, and its size is the length of this vector.
wolffd@0	324 In this case, we will assume all nodes are discrete and binary.
wolffd@0	325 <PRE>
wolffd@0	326 discrete_nodes = 1:N;
wolffd@0	327 node_sizes = 2*ones(1,N);
wolffd@0	328 </pre>
wolffd@0	329 If the nodes were not binary, you could type e.g.,
wolffd@0	330 <pre>
wolffd@0	331 node_sizes = [4 2 3 5];
wolffd@0	332 </pre>
wolffd@0	333 meaning that Cloudy has 4 possible values,
wolffd@0	334 Sprinkler has 2 possible values, etc.
wolffd@0	335 Note that these are cardinal values, not ordinal, i.e.,
wolffd@0	336 they are not ordered in any way, like 'low', 'medium', 'high'.
wolffd@0	337 <p>
wolffd@0	338 We are now ready to make the Bayes net:
wolffd@0	339 <pre>
wolffd@0	340 bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes);
wolffd@0	341 </PRE>
wolffd@0	342 By default, all nodes are assumed to be discrete, so we can also just
wolffd@0	343 write
wolffd@0	344 <pre>
wolffd@0	345 bnet = mk_bnet(dag, node_sizes);
wolffd@0	346 </PRE>
wolffd@0	347 You may also specify which nodes will be observed.
wolffd@0	348 If you don't know, or if this not fixed in advance,
wolffd@0	349 just use the empty list (the default).
wolffd@0	350 <pre>
wolffd@0	351 onodes = [];
wolffd@0	352 bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes);
wolffd@0	353 </PRE>
wolffd@0	354 Note that optional arguments are specified using a name/value syntax.
wolffd@0	355 This is common for many BNT functions.
wolffd@0	356 In general, to find out more about a function (e.g., which optional
wolffd@0	357 arguments it takes), please see its
wolffd@0	358 documentation string by typing
wolffd@0	359 <pre>
wolffd@0	360 help mk_bnet
wolffd@0	361 </pre>
wolffd@0	362 See also other <a href="matlab_tips.html">useful Matlab tips</a>.
wolffd@0	363 <p>
wolffd@0	364 It is possible to associate names with nodes, as follows:
wolffd@0	365 <pre>
wolffd@0	366 bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4);
wolffd@0	367 </pre>
wolffd@0	368 You can then refer to a node by its name:
wolffd@0	369 <pre>
wolffd@0	370 C = bnet.names{'cloudy'}; % bnet.names is an associative array
wolffd@0	371 bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]);
wolffd@0	372 </pre>
wolffd@0	373
wolffd@0	374
wolffd@0	375 <h2><a name="cpt">Parameters</h2>
wolffd@0	376
wolffd@0	377 A model consists of the graph structure and the parameters.
wolffd@0	378 The parameters are represented by CPD objects (CPD = Conditional
wolffd@0	379 Probability Distribution), which define the probability distribution
wolffd@0	380 of a node given its parents.
wolffd@0	381 (We will use the terms "node" and "random variable" interchangeably.)
wolffd@0	382 The simplest kind of CPD is a table (multi-dimensional array), which
wolffd@0	383 is suitable when all the nodes are discrete-valued. Note that the discrete
wolffd@0	384 values are not assumed to be ordered in any way; that is, they
wolffd@0	385 represent categorical quantities, like male and female, rather than
wolffd@0	386 ordinal quantities, like low, medium and high.
wolffd@0	387 (We will discuss CPDs in more detail <a href="#cpd">below</a>.)
wolffd@0	388 <p>
wolffd@0	389 Tabular CPDs, also called CPTs (conditional probability tables),
wolffd@0	390 are stored as multidimensional arrays, where the dimensions
wolffd@0	391 are arranged in the same order as the nodes, e.g., the CPT for node 4
wolffd@0	392 (WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself.
wolffd@0	393 Hence the child is always the last dimension.
wolffd@0	394 If a node has no parents, its CPT is a column vector representing its
wolffd@0	395 prior.
wolffd@0	396 Note that in Matlab (unlike C), arrays are indexed
wolffd@0	397 from 1, and are layed out in memory such that the first index toggles
wolffd@0	398 fastest, e.g., the CPT for node 4 (WetGrass) is as follows
wolffd@0	399 <P>
wolffd@0	400 <P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P>
wolffd@0	401 <P>
wolffd@0	402 where we have used the convention that false==1, true==2.
wolffd@0	403 We can create this CPT in Matlab as follows
wolffd@0	404 <PRE>
wolffd@0	405 CPT = zeros(2,2,2);
wolffd@0	406 CPT(1,1,1) = 1.0;
wolffd@0	407 CPT(2,1,1) = 0.1;
wolffd@0	408 ...
wolffd@0	409 </PRE>
wolffd@0	410 Here is an easier way:
wolffd@0	411 <PRE>
wolffd@0	412 CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]);
wolffd@0	413 </PRE>
wolffd@0	414 In fact, we don't need to reshape the array, since the CPD constructor
wolffd@0	415 will do that for us. So we can just write
wolffd@0	416 <pre>
wolffd@0	417 bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]);
wolffd@0	418 </pre>
wolffd@0	419 The other nodes are created similarly (using the old syntax for
wolffd@0	420 optional parameters)
wolffd@0	421 <PRE>
wolffd@0	422 bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]);
wolffd@0	423 bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]);
wolffd@0	424 bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]);
wolffd@0	425 bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]);
wolffd@0	426 </PRE>
wolffd@0	427
wolffd@0	428
wolffd@0	429 <h2><a name="rnd_cpt">Random Parameters</h2>
wolffd@0	430
wolffd@0	431 If we do not specify the CPT, random parameters will be
wolffd@0	432 created, i.e., each "row" of the CPT will be drawn from the uniform distribution.
wolffd@0	433 To ensure repeatable results, use
wolffd@0	434 <pre>
wolffd@0	435 rand('state', seed);
wolffd@0	436 randn('state', seed);
wolffd@0	437 </pre>
wolffd@0	438 To control the degree of randomness (entropy),
wolffd@0	439 you can sample each row of the CPT from a Dirichlet(p,p,...) distribution.
wolffd@0	440 If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0).
wolffd@0	441 If p = 1, each entry is drawn from U[0,1].
wolffd@0	442 If p >> 1, the entries will all be near 1/k, where k is the arity of
wolffd@0	443 this node, i.e., each row will be nearly uniform.
wolffd@0	444 You can do this as follows, assuming this node
wolffd@0	445 is number i, and ns is the node_sizes.
wolffd@0	446 <pre>
wolffd@0	447 k = ns(i);
wolffd@0	448 ps = parents(dag, i);
wolffd@0	449 psz = prod(ns(ps));
wolffd@0	450 CPT = sample_dirichlet(p*ones(1,k), psz);
wolffd@0	451 bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT);
wolffd@0	452 </pre>
wolffd@0	453
wolffd@0	454
wolffd@0	455 <h2><a name="file">Loading a network from a file</h2>
wolffd@0	456
wolffd@0	457 If you already have a Bayes net represented in the XML-based
wolffd@0	458 <a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/">
wolffd@0	459 Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the
wolffd@0	460 <a
wolffd@0	461 href="http://www.cs.huji.ac.il/labs/compbio/Repository">
wolffd@0	462 Bayes Net repository</a>),
wolffd@0	463 you can convert it to BNT format using
wolffd@0	464 the
wolffd@0	465 <a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java
wolffd@0	466 program</a> written by Ken Shan.
wolffd@0	467 (This is not necessarily up-to-date.)
wolffd@0	468
wolffd@0	469
wolffd@0	470 <h2>Creating a model using a GUI</h2>
wolffd@0	471
wolffd@0	472 Click <a href="http://bnt.insa-rouen.fr/ajouts.html">here</a>.
wolffd@0	473
wolffd@0	474
wolffd@0	475
wolffd@0	476 <h1><a name="inference">Inference</h1>
wolffd@0	477
wolffd@0	478 Having created the BN, we can now use it for inference.
wolffd@0	479 There are many different algorithms for doing inference in Bayes nets,
wolffd@0	480 that make different tradeoffs between speed,
wolffd@0	481 complexity, generality, and accuracy.
wolffd@0	482 BNT therefore offers a variety of different inference
wolffd@0	483 "engines". We will discuss these
wolffd@0	484 in more detail <a href="#engines">below</a>.
wolffd@0	485 For now, we will use the junction tree
wolffd@0	486 engine, which is the mother of all exact inference algorithms.
wolffd@0	487 This can be created as follows.
wolffd@0	488 <pre>
wolffd@0	489 engine = jtree_inf_engine(bnet);
wolffd@0	490 </pre>
wolffd@0	491 The other engines have similar constructors, but might take
wolffd@0	492 additional, algorithm-specific parameters.
wolffd@0	493 All engines are used in the same way, once they have been created.
wolffd@0	494 We illustrate this in the following sections.
wolffd@0	495
wolffd@0	496
wolffd@0	497 <h2><a name="marginal">Computing marginal distributions</h2>
wolffd@0	498
wolffd@0	499 Suppose we want to compute the probability that the sprinker was on
wolffd@0	500 given that the grass is wet.
wolffd@0	501 The evidence consists of the fact that W=2. All the other nodes
wolffd@0	502 are hidden (unobserved). We can specify this as follows.
wolffd@0	503 <pre>
wolffd@0	504 evidence = cell(1,N);
wolffd@0	505 evidence{W} = 2;
wolffd@0	506 </pre>
wolffd@0	507 We use a 1D cell array instead of a vector to
wolffd@0	508 cope with the fact that nodes can be vectors of different lengths.
wolffd@0	509 In addition, the value [] can be used
wolffd@0	510 to denote 'no evidence', instead of having to specify the observation
wolffd@0	511 pattern as a separate argument.
wolffd@0	512 (Click <a href="cellarray.html">here</a> for a quick tutorial on cell
wolffd@0	513 arrays in matlab.)
wolffd@0	514 <p>
wolffd@0	515 We are now ready to add the evidence to the engine.
wolffd@0	516 <pre>
wolffd@0	517 [engine, loglik] = enter_evidence(engine, evidence);
wolffd@0	518 </pre>
wolffd@0	519 The behavior of this function is algorithm-specific, and is discussed
wolffd@0	520 in more detail <a href="#engines">below</a>.
wolffd@0	521 In the case of the jtree engine,
wolffd@0	522 enter_evidence implements a two-pass message-passing scheme.
wolffd@0	523 The first return argument contains the modified engine, which
wolffd@0	524 incorporates the evidence. The second return argument contains the
wolffd@0	525 log-likelihood of the evidence. (Not all engines are capable of
wolffd@0	526 computing the log-likelihood.)
wolffd@0	527 <p>
wolffd@0	528 Finally, we can compute p=P(S=2\|W=2) as follows.
wolffd@0	529 <PRE>
wolffd@0	530 marg = marginal_nodes(engine, S);
wolffd@0	531 marg.T
wolffd@0	532 ans =
wolffd@0	533 0.57024
wolffd@0	534 0.42976
wolffd@0	535 p = marg.T(2);
wolffd@0	536 </PRE>
wolffd@0	537 We see that p = 0.4298.
wolffd@0	538 <p>
wolffd@0	539 Now let us add the evidence that it was raining, and see what
wolffd@0	540 difference it makes.
wolffd@0	541 <PRE>
wolffd@0	542 evidence{R} = 2;
wolffd@0	543 [engine, loglik] = enter_evidence(engine, evidence);
wolffd@0	544 marg = marginal_nodes(engine, S);
wolffd@0	545 p = marg.T(2);
wolffd@0	546 </PRE>
wolffd@0	547 We find that p = P(S=2\|W=2,R=2) = 0.1945,
wolffd@0	548 which is lower than
wolffd@0	549 before, because the rain can ``explain away'' the
wolffd@0	550 fact that the grass is wet.
wolffd@0	551 <p>
wolffd@0	552 You can plot a marginal distribution over a discrete variable
wolffd@0	553 as a barchart using the built 'bar' function:
wolffd@0	554 <pre>
wolffd@0	555 bar(marg.T)
wolffd@0	556 </pre>
wolffd@0	557 This is what it looks like
wolffd@0	558
wolffd@0	559 <p>
wolffd@0	560 <center>
wolffd@0	561 <IMG SRC="Figures/sprinkler_bar.gif">
wolffd@0	562 </center>
wolffd@0	563 <p>
wolffd@0	564
wolffd@0	565 <h2><a name="observed">Observed nodes</h2>
wolffd@0	566
wolffd@0	567 What happens if we ask for the marginal on an observed node, e.g. P(W\|W=2)?
wolffd@0	568 An observed discrete node effectively only has 1 value (the observed
wolffd@0	569 one) --- all other values would result in 0 probability.
wolffd@0	570 For efficiency, BNT treats observed (discrete) nodes as if they were
wolffd@0	571 set to 1, as we see below:
wolffd@0	572 <pre>
wolffd@0	573 evidence = cell(1,N);
wolffd@0	574 evidence{W} = 2;
wolffd@0	575 engine = enter_evidence(engine, evidence);
wolffd@0	576 m = marginal_nodes(engine, W);
wolffd@0	577 m.T
wolffd@0	578 ans =
wolffd@0	579 1
wolffd@0	580 </pre>
wolffd@0	581 This can get a little confusing, since we assigned W=2.
wolffd@0	582 So we can ask BNT to add the evidence back in by passing in an optional argument:
wolffd@0	583 <pre>
wolffd@0	584 m = marginal_nodes(engine, W, 1);
wolffd@0	585 m.T
wolffd@0	586 ans =
wolffd@0	587 0
wolffd@0	588 1
wolffd@0	589 </pre>
wolffd@0	590 This shows that P(W=1\|W=2) = 0 and P(W=2\|W=2) = 1.
wolffd@0	591
wolffd@0	592
wolffd@0	593
wolffd@0	594 <h2><a name="joint">Computing joint distributions</h2>
wolffd@0	595
wolffd@0	596 We can compute the joint probability on a set of nodes as in the
wolffd@0	597 following example.
wolffd@0	598 <pre>
wolffd@0	599 evidence = cell(1,N);
wolffd@0	600 [engine, ll] = enter_evidence(engine, evidence);
wolffd@0	601 m = marginal_nodes(engine, [S R W]);
wolffd@0	602 </pre>
wolffd@0	603 m is a structure. The 'T' field is a multi-dimensional array (in
wolffd@0	604 this case, 3-dimensional) that contains the joint probability
wolffd@0	605 distribution on the specified nodes.
wolffd@0	606 <pre>
wolffd@0	607 >> m.T
wolffd@0	608 ans(:,:,1) =
wolffd@0	609 0.2900 0.0410
wolffd@0	610 0.0210 0.0009
wolffd@0	611 ans(:,:,2) =
wolffd@0	612 0 0.3690
wolffd@0	613 0.1890 0.0891
wolffd@0	614 </pre>
wolffd@0	615 We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass
wolffd@0	616 to be wet if both the rain and sprinkler are off.
wolffd@0	617 <p>
wolffd@0	618 Let us now add some evidence to R.
wolffd@0	619 <pre>
wolffd@0	620 evidence{R} = 2;
wolffd@0	621 [engine, ll] = enter_evidence(engine, evidence);
wolffd@0	622 m = marginal_nodes(engine, [S R W])
wolffd@0	623 m =
wolffd@0	624 domain: [2 3 4]
wolffd@0	625 T: [2x1x2 double]
wolffd@0	626 >> m.T
wolffd@0	627 m.T
wolffd@0	628 ans(:,:,1) =
wolffd@0	629 0.0820
wolffd@0	630 0.0018
wolffd@0	631 ans(:,:,2) =
wolffd@0	632 0.7380
wolffd@0	633 0.1782
wolffd@0	634 </pre>
wolffd@0	635 The joint T(i,j,k) = P(S=i,R=j,W=k\|evidence)
wolffd@0	636 should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible
wolffd@0	637 with the evidence that R=2.
wolffd@0	638 Instead of creating large tables with many 0s, BNT sets the effective
wolffd@0	639 size of observed (discrete) nodes to 1, as explained above.
wolffd@0	640 This is why m.T has size 2x1x2.
wolffd@0	641 To get a 2x2x2 table, type
wolffd@0	642 <pre>
wolffd@0	643 m = marginal_nodes(engine, [S R W], 1)
wolffd@0	644 m =
wolffd@0	645 domain: [2 3 4]
wolffd@0	646 T: [2x2x2 double]
wolffd@0	647 >> m.T
wolffd@0	648 m.T
wolffd@0	649 ans(:,:,1) =
wolffd@0	650 0 0.082
wolffd@0	651 0 0.0018
wolffd@0	652 ans(:,:,2) =
wolffd@0	653 0 0.738
wolffd@0	654 0 0.1782
wolffd@0	655 </pre>
wolffd@0	656
wolffd@0	657 <p>
wolffd@0	658 Note: It is not always possible to compute the joint on arbitrary
wolffd@0	659 sets of nodes: it depends on which inference engine you use, as discussed
wolffd@0	660 in more detail <a href="#engines">below</a>.
wolffd@0	661
wolffd@0	662
wolffd@0	663 <h2><a name="soft">Soft/virtual evidence</h2>
wolffd@0	664
wolffd@0	665 Sometimes a node is not observed, but we have some distribution over
wolffd@0	666 its possible values; this is often called "soft" or "virtual"
wolffd@0	667 evidence.
wolffd@0	668 One can use this as follows
wolffd@0	669 <pre>
wolffd@0	670 [engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence);
wolffd@0	671 </pre>
wolffd@0	672 where soft_evidence{i} is either [] (if node i has no soft evidence)
wolffd@0	673 or is a vector representing the probability distribution over i's
wolffd@0	674 possible values.
wolffd@0	675 For example, if we don't know i's exact value, but we know its
wolffd@0	676 likelihood ratio is 60/40, we can write evidence{i} = [] and
wolffd@0	677 soft_evidence{i} = [0.6 0.4].
wolffd@0	678 <p>
wolffd@0	679 Currently only jtree_inf_engine supports this option.
wolffd@0	680 It assumes that all hidden nodes, and all nodes for
wolffd@0	681 which we have soft evidence, are discrete.
wolffd@0	682 For a longer example, see BNT/examples/static/softev1.m.
wolffd@0	683
wolffd@0	684
wolffd@0	685 <h2><a name="mpe">Most probable explanation</h2>
wolffd@0	686
wolffd@0	687 To compute the most probable explanation (MPE) of the evidence (i.e.,
wolffd@0	688 the most probable assignment, or a mode of the joint), use
wolffd@0	689 <pre>
wolffd@0	690 [mpe, ll] = calc_mpe(engine, evidence);
wolffd@0	691 </pre>
wolffd@0	692 mpe{i} is the most likely value of node i.
wolffd@0	693 This calls enter_evidence with the 'maximize' flag set to 1, which
wolffd@0	694 causes the engine to do max-product instead of sum-product.
wolffd@0	695 The resulting max-marginals are then thresholded.
wolffd@0	696 If there is more than one maximum probability assignment, we must take
wolffd@0	697 care to break ties in a consistent manner (thresholding the
wolffd@0	698 max-marginals may give the wrong result). To force this behavior,
wolffd@0	699 type
wolffd@0	700 <pre>
wolffd@0	701 [mpe, ll] = calc_mpe(engine, evidence, 1);
wolffd@0	702 </pre>
wolffd@0	703 Note that computing the MPE is someties called abductive reasoning.
wolffd@0	704
wolffd@0	705 <p>
wolffd@0	706 You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar,
wolffd@0	707 that does a forwards max-product pass, and then a backwards traceback
wolffd@0	708 pass, which is how Viterbi is traditionally implemented.
wolffd@0	709
wolffd@0	710
wolffd@0	711
wolffd@0	712 <h1><a name="cpd">Conditional Probability Distributions</h1>
wolffd@0	713
wolffd@0	714 A Conditional Probability Distributions (CPD)
wolffd@0	715 defines P(X(i) \| X(Pa(i))), where X(i) is the i'th node, and X(Pa(i))
wolffd@0	716 are the parents of node i. There are many ways to represent this
wolffd@0	717 distribution, which depend in part on whether X(i) and X(Pa(i)) are
wolffd@0	718 discrete, continuous, or a combination.
wolffd@0	719 We will discuss various representations below.
wolffd@0	720
wolffd@0	721
wolffd@0	722 <h2><a name="tabular">Tabular nodes</h2>
wolffd@0	723
wolffd@0	724 If the CPD is represented as a table (i.e., if it is a multinomial
wolffd@0	725 distribution), it has a number of parameters that is exponential in
wolffd@0	726 the number of parents. See the example <a href="#cpt">above</a>.
wolffd@0	727
wolffd@0	728
wolffd@0	729 <h2><a name="noisyor">Noisy-or nodes</h2>
wolffd@0	730
wolffd@0	731 A noisy-OR node is like a regular logical OR gate except that
wolffd@0	732 sometimes the effects of parents that are on get inhibited.
wolffd@0	733 Let the prob. that parent i gets inhibited be q(i).
wolffd@0	734 Then a node, C, with 2 parents, A and B, has the following CPD, where
wolffd@0	735 we use F and T to represent off and on (1 and 2 in BNT).
wolffd@0	736 <pre>
wolffd@0	737 A B P(C=off) P(C=on)
wolffd@0	738 ---------------------------
wolffd@0	739 F F 1.0 0.0
wolffd@0	740 T F q(A) 1-q(A)
wolffd@0	741 F T q(B) 1-q(B)
wolffd@0	742 T T q(A)q(B) q-q(A)q(B)
wolffd@0	743 </pre>
wolffd@0	744 Thus we see that the causes get inhibited independently.
wolffd@0	745 It is common to associate a "leak" node with a noisy-or CPD, which is
wolffd@0	746 like a parent that is always on. This can account for all other unmodelled
wolffd@0	747 causes which might turn the node on.
wolffd@0	748 <p>
wolffd@0	749 The noisy-or distribution is similar to the logistic distribution.
wolffd@0	750 To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j)
wolffd@0	751 be the prob. that j inhibits i. Then
wolffd@0	752 <pre>
wolffd@0	753 Pr(S(i)=1 \| parents(S(i))) = 1 - prod_{j} q(i,j)^S(j)
wolffd@0	754 </pre>
wolffd@0	755 Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then
wolffd@0	756 <pre>
wolffd@0	757 Pr(S(i)=1 \| parents(S(i))) = rho(sum_j w(i,j) S(j))
wolffd@0	758 </pre>
wolffd@0	759 For a sigmoid node, we have
wolffd@0	760 <pre>
wolffd@0	761 Pr(S(i)=1 \| parents(S(i))) = sigma(-sum_j w(i,j) S(j))
wolffd@0	762 </pre>
wolffd@0	763 where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of
wolffd@0	764 the activation function (although both are monotonically increasing).
wolffd@0	765 In addition, in the case of a noisy-or, the weights are constrained to be
wolffd@0	766 positive, since they derive from probabilities q(i,j).
wolffd@0	767 In both cases, the number of parameters is <em>linear</em> in the
wolffd@0	768 number of parents, unlike the case of a multinomial distribution,
wolffd@0	769 where the number of parameters is exponential in the number of parents.
wolffd@0	770 We will see an example of noisy-OR nodes <a href="#qmr">below</a>.
wolffd@0	771
wolffd@0	772
wolffd@0	773 <h2><a name="deterministic">Other (noisy) deterministic nodes</h2>
wolffd@0	774
wolffd@0	775 Deterministic CPDs for discrete random variables can be created using
wolffd@0	776 the deterministic_CPD class. It is also possible to 'flip' the output
wolffd@0	777 of the function with some probability, to simulate noise.
wolffd@0	778 The boolean_CPD class is just a special case of a
wolffd@0	779 deterministic CPD, where the parents and child are all binary.
wolffd@0	780 <p>
wolffd@0	781 Both of these classes are just "syntactic sugar" for the tabular_CPD
wolffd@0	782 class.
wolffd@0	783
wolffd@0	784
wolffd@0	785
wolffd@0	786 <h2><a name="softmax">Softmax nodes</h2>
wolffd@0	787
wolffd@0	788 If we have a discrete node with a continuous parent,
wolffd@0	789 we can define its CPD using a softmax function
wolffd@0	790 (also known as the multinomial logit function).
wolffd@0	791 This acts like a soft thresholding operator, and is defined as follows:
wolffd@0	792 <pre>
wolffd@0	793 exp(w(:,i)'*x + b(i))
wolffd@0	794 Pr(Q=i \| X=x) = -----------------------------
wolffd@0	795 sum_j exp(w(:,j)'*x + b(j))
wolffd@0	796
wolffd@0	797 </pre>
wolffd@0	798 The parameters of a softmax node, w(:,i) and b(i), i=1..\|Q\|, have the
wolffd@0	799 following interpretation: w(:,i)-w(:,j) is the normal vector to the
wolffd@0	800 decision boundary between classes i and j,
wolffd@0	801 and b(i)-b(j) is its offset (bias). For example, suppose
wolffd@0	802 X is a 2-vector, and Q is binary. Then
wolffd@0	803 <pre>
wolffd@0	804 w = [1 -1;
wolffd@0	805 0 0];
wolffd@0	806
wolffd@0	807 b = [0 0];
wolffd@0	808 </pre>
wolffd@0	809 means class 1 are points in the 2D plane with positive x coordinate,
wolffd@0	810 and class 2 are points in the 2D plane with negative x coordinate.
wolffd@0	811 If w has large magnitude, the decision boundary is sharp, otherwise it
wolffd@0	812 is soft.
wolffd@0	813 In the special case that Q is binary (0/1), the softmax function reduces to the logistic
wolffd@0	814 (sigmoid) function.
wolffd@0	815 <p>
wolffd@0	816 Fitting a softmax function can be done using the iteratively reweighted
wolffd@0	817 least squares (IRLS) algorithm.
wolffd@0	818 We use the implementation from
wolffd@0	819 <a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>.
wolffd@0	820 Note that since
wolffd@0	821 the softmax distribution is not in the exponential family, it does not
wolffd@0	822 have finite sufficient statistics, and hence we must store all the
wolffd@0	823 training data in uncompressed form.
wolffd@0	824 If this takes too much space, one should use online (stochastic) gradient
wolffd@0	825 descent (not implemented in BNT).
wolffd@0	826 <p>
wolffd@0	827 If a softmax node also has discrete parents,
wolffd@0	828 we use a different set of w/b parameters for each combination of
wolffd@0	829 parent values, as in the <a href="#gaussian">conditional linear
wolffd@0	830 Gaussian CPD</a>.
wolffd@0	831 This feature was implemented by Pierpaolo Brutti.
wolffd@0	832 He is currently extending it so that discrete parents can be treated
wolffd@0	833 as if they were continuous, by adding indicator variables to the X
wolffd@0	834 vector.
wolffd@0	835 <p>
wolffd@0	836 We will see an example of softmax nodes <a href="#mixexp">below</a>.
wolffd@0	837
wolffd@0	838
wolffd@0	839 <h2><a name="mlp">Neural network nodes</h2>
wolffd@0	840
wolffd@0	841 Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron
wolffd@0	842 to implement a mapping from continuous parents to discrete children,
wolffd@0	843 similar to the softmax function.
wolffd@0	844 (If there are also discrete parents, it creates a mixture of MLPs.)
wolffd@0	845 It uses code from <a
wolffd@0	846 href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>.
wolffd@0	847 This is work in progress.
wolffd@0	848
wolffd@0	849 <h2><a name="root">Root nodes</h2>
wolffd@0	850
wolffd@0	851 A root node has no parents and no parameters; it can be used to model
wolffd@0	852 an observed, exogeneous input variable, i.e., one which is "outside"
wolffd@0	853 the model.
wolffd@0	854 This is useful for conditional density models.
wolffd@0	855 We will see an example of root nodes <a href="#mixexp">below</a>.
wolffd@0	856
wolffd@0	857
wolffd@0	858 <h2><a name="gaussian">Gaussian nodes</h2>
wolffd@0	859
wolffd@0	860 We now consider a distribution suitable for the continuous-valued nodes.
wolffd@0	861 Suppose the node is called Y, its continuous parents (if any) are
wolffd@0	862 called X, and its discrete parents (if any) are called Q.
wolffd@0	863 The distribution on Y is defined as follows:
wolffd@0	864 <pre>
wolffd@0	865 - no parents: Y ~ N(mu, Sigma)
wolffd@0	866 - cts parents : Y\|X=x ~ N(mu + W x, Sigma)
wolffd@0	867 - discrete parents: Y\|Q=i ~ N(mu(:,i), Sigma(:,:,i))
wolffd@0	868 - cts and discrete parents: Y\|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i))
wolffd@0	869 </pre>
wolffd@0	870 where N(mu, Sigma) denotes a Normal distribution with mean mu and
wolffd@0	871 covariance Sigma. Let \|X\|, \|Y\| and \|Q\| denote the sizes of X, Y and Q
wolffd@0	872 respectively.
wolffd@0	873 If there are no discrete parents, \|Q\|=1; if there is
wolffd@0	874 more than one, then \|Q\| = a vector of the sizes of each discrete parent.
wolffd@0	875 If there are no continuous parents, \|X\|=0; if there is more than one,
wolffd@0	876 then \|X\| = the sum of their sizes.
wolffd@0	877 Then mu is a \|Y\|\|Q\| vector, Sigma is a \|Y\|\|Y\|*\|Q\| positive
wolffd@0	878 semi-definite matrix, and W is a \|Y\|\|X\|\|Q\| regression (weight)
wolffd@0	879 matrix.
wolffd@0	880 <p>
wolffd@0	881 We can create a Gaussian node with random parameters as follows.
wolffd@0	882 <pre>
wolffd@0	883 bnet.CPD{i} = gaussian_CPD(bnet, i);
wolffd@0	884 </pre>
wolffd@0	885 We can specify the value of one or more of the parameters as in the
wolffd@0	886 following example, in which \|Y\|=2, and \|Q\|=1.
wolffd@0	887 <pre>
wolffd@0	888 bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y));
wolffd@0	889 </pre>
wolffd@0	890 <p>
wolffd@0	891 We will see an example of conditional linear Gaussian nodes <a
wolffd@0	892 href="#cg_model">below</a>.
wolffd@0	893 <p>
wolffd@0	894 When fitting a Gaussian using EM, you might encounter some numerical
wolffd@0	895 problems. See <a
wolffd@0	896 href="http://www.ai.mit.edu/~murphyk/Software/HMM/hmm.html#loglikpos">here</a>
wolffd@0	897 for a discussion of this in the context of HMMs.
wolffd@0	898 (Note: BNT does not currently support spherical covariances, although it
wolffd@0	899 would be easy to add, since KPMstats/clg_Mstep supports this option;
wolffd@0	900 you would just need to modify gaussian_CPD/update_ess to accumulate
wolffd@0	901 weighted inner products.)
wolffd@0	902
wolffd@0	903
wolffd@0	904 <h2><a name="nongauss">Other continuous distributions</h2>
wolffd@0	905
wolffd@0	906 Currently BNT does not support any CPDs for continuous nodes other
wolffd@0	907 than the Gaussian.
wolffd@0	908 However, you can use a mixture of Gaussians to
wolffd@0	909 approximate other continuous distributions. We will see some an example
wolffd@0	910 of this with the IFA model <a href="#pca">below</a>.
wolffd@0	911
wolffd@0	912
wolffd@0	913 <h2><a name="glm">Generalized linear model nodes</h2>
wolffd@0	914
wolffd@0	915 In the future, we may incorporate some of the functionality of
wolffd@0	916 <a href =
wolffd@0	917 "http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a>
wolffd@0	918 into BNT.
wolffd@0	919
wolffd@0	920
wolffd@0	921 <h2><a name="dtree">Classification/regression tree nodes</h2>
wolffd@0	922
wolffd@0	923 We plan to add classification and regression trees to define CPDs for
wolffd@0	924 discrete and continuous nodes, respectively.
wolffd@0	925 Trees have many advantages: they are easy to interpret, they can do
wolffd@0	926 feature selection, they can
wolffd@0	927 handle discrete and continuous inputs, they do not make strong
wolffd@0	928 assumptions about the form of the distribution, the number of
wolffd@0	929 parameters can grow in a data-dependent way (i.e., they are
wolffd@0	930 semi-parametric), they can handle missing data, etc.
wolffd@0	931 However, they are not yet implemented.
wolffd@0	932 <!--
wolffd@0	933 Yimin Zhang is currently (Feb '02) implementing this.
wolffd@0	934 -->
wolffd@0	935
wolffd@0	936
wolffd@0	937 <h2><a name="cpd_summary">Summary of CPD types</h2>
wolffd@0	938
wolffd@0	939 We list all the different types of CPDs supported by BNT.
wolffd@0	940 For each CPD, we specify if the child and parents can be discrete (D) or
wolffd@0	941 continuous (C) (Binary (B) nodes are a special case).
wolffd@0	942 We also specify which methods each class supports.
wolffd@0	943 If a method is inherited, the name of the parent class is mentioned.
wolffd@0	944 If a parent class calls a child method, this is mentioned.
wolffd@0	945 <p>
wolffd@0	946 The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this
wolffd@0	947 requires that the child and all parents are discrete.
wolffd@0	948 The CPT might be exponentially big...
wolffd@0	949 <tt>convert_to_table</tt> evaluates a CPD with evidence, and
wolffd@0	950 represents the the resulting potential as an array.
wolffd@0	951 This requires that the child is discrete, and any continuous parents
wolffd@0	952 are observed.
wolffd@0	953 <tt>convert_to_pot</tt> evaluates a CPD with evidence, and
wolffd@0	954 represents the resulting potential as a dpot, gpot, cgpot or upot, as
wolffd@0	955 requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u =
wolffd@0	956 utility).
wolffd@0	957
wolffd@0	958 <p>
wolffd@0	959 When we sample a node, all the parents are observed.
wolffd@0	960 When we compute the (log) probability of a node, all the parents and
wolffd@0	961 the child are observed.
wolffd@0	962 <p>
wolffd@0	963 We also specify if the parameters are learnable.
wolffd@0	964 For learning with EM, we require
wolffd@0	965 the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and
wolffd@0	966 <tt>maximize_params</tt>.
wolffd@0	967 For learning from fully observed data, we require
wolffd@0	968 the method <tt>learn_params</tt>.
wolffd@0	969 By default, all classes inherit this from generic_CPD, which simply
wolffd@0	970 calls <tt>update_ess</tt> N times, once for each data case, followed
wolffd@0	971 by <tt>maximize_params</tt>, i.e., it is like EM, without the E step.
wolffd@0	972 Some classes implement a batch formula, which is quicker.
wolffd@0	973 <p>
wolffd@0	974 Bayesian learning means computing a posterior over the parameters
wolffd@0	975 given fully observed data.
wolffd@0	976 <p>
wolffd@0	977 Pearl means we implement the methods <tt>compute_pi</tt> and
wolffd@0	978 <tt>compute_lambda_msg</tt>, used by
wolffd@0	979 <tt>pearl_inf_engine</tt>, which runs on directed graphs.
wolffd@0	980 <tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H
wolffd@0	981 The pearl methods can exploit special properties of the CPDs for
wolffd@0	982 computing the messages efficiently, whereas belprop does not.
wolffd@0	983 <p>
wolffd@0	984 The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>,
wolffd@0	985 which is not shown, since it is not very interesting.
wolffd@0	986
wolffd@0	987
wolffd@0	988 <p>
wolffd@0	989
wolffd@0	990
wolffd@0	991 <table>
wolffd@0	992 <table border units = pixels><tr>
wolffd@0	993 <td align=center>Name
wolffd@0	994 <td align=center>Child
wolffd@0	995 <td align=center>Parents
wolffd@0	996 <td align=center>Comments
wolffd@0	997 <td align=center>CPD_to_CPT
wolffd@0	998 <td align=center>conv_to_table
wolffd@0	999 <td align=center>conv_to_pot
wolffd@0	1000 <td align=center>sample
wolffd@0	1001 <td align=center>prob
wolffd@0	1002 <td align=center>learn
wolffd@0	1003 <td align=center>Bayes
wolffd@0	1004 <td align=center>Pearl
wolffd@0	1005
wolffd@0	1006
wolffd@0	1007 <tr>
wolffd@0	1008 <!-- Name--><td>
wolffd@0	1009 <!-- Child--><td>
wolffd@0	1010 <!-- Parents--><td>
wolffd@0	1011 <!-- Comments--><td>
wolffd@0	1012 <!-- CPD_to_CPT--><td>
wolffd@0	1013 <!-- conv_to_table--><td>
wolffd@0	1014 <!-- conv_to_pot--><td>
wolffd@0	1015 <!-- sample--><td>
wolffd@0	1016 <!-- prob--><td>
wolffd@0	1017 <!-- learn--><td>
wolffd@0	1018 <!-- Bayes--><td>
wolffd@0	1019 <!-- Pearl--><td>
wolffd@0	1020
wolffd@0	1021 <tr>
wolffd@0	1022 <!-- Name--><td>boolean
wolffd@0	1023 <!-- Child--><td>B
wolffd@0	1024 <!-- Parents--><td>B
wolffd@0	1025 <!-- Comments--><td>Syntactic sugar for tabular
wolffd@0	1026 <!-- CPD_to_CPT--><td>-
wolffd@0	1027 <!-- conv_to_table--><td>-
wolffd@0	1028 <!-- conv_to_pot--><td>-
wolffd@0	1029 <!-- sample--><td>-
wolffd@0	1030 <!-- prob--><td>-
wolffd@0	1031 <!-- learn--><td>-
wolffd@0	1032 <!-- Bayes--><td>-
wolffd@0	1033 <!-- Pearl--><td>-
wolffd@0	1034
wolffd@0	1035 <tr>
wolffd@0	1036 <!-- Name--><td>deterministic
wolffd@0	1037 <!-- Child--><td>D
wolffd@0	1038 <!-- Parents--><td>D
wolffd@0	1039 <!-- Comments--><td>Syntactic sugar for tabular
wolffd@0	1040 <!-- CPD_to_CPT--><td>-
wolffd@0	1041 <!-- conv_to_table--><td>-
wolffd@0	1042 <!-- conv_to_pot--><td>-
wolffd@0	1043 <!-- sample--><td>-
wolffd@0	1044 <!-- prob--><td>-
wolffd@0	1045 <!-- learn--><td>-
wolffd@0	1046 <!-- Bayes--><td>-
wolffd@0	1047 <!-- Pearl--><td>-
wolffd@0	1048
wolffd@0	1049 <tr>
wolffd@0	1050 <!-- Name--><td>Discrete
wolffd@0	1051 <!-- Child--><td>D
wolffd@0	1052 <!-- Parents--><td>C/D
wolffd@0	1053 <!-- Comments--><td>Virtual class
wolffd@0	1054 <!-- CPD_to_CPT--><td>N
wolffd@0	1055 <!-- conv_to_table--><td>Calls CPD_to_CPT
wolffd@0	1056 <!-- conv_to_pot--><td>Calls conv_to_table
wolffd@0	1057 <!-- sample--><td>Calls conv_to_table
wolffd@0	1058 <!-- prob--><td>Calls conv_to_table
wolffd@0	1059 <!-- learn--><td>N
wolffd@0	1060 <!-- Bayes--><td>N
wolffd@0	1061 <!-- Pearl--><td>N
wolffd@0	1062
wolffd@0	1063 <tr>
wolffd@0	1064 <!-- Name--><td>Gaussian
wolffd@0	1065 <!-- Child--><td>C
wolffd@0	1066 <!-- Parents--><td>C/D
wolffd@0	1067 <!-- Comments--><td>-
wolffd@0	1068 <!-- CPD_to_CPT--><td>N
wolffd@0	1069 <!-- conv_to_table--><td>N
wolffd@0	1070 <!-- conv_to_pot--><td>Y
wolffd@0	1071 <!-- sample--><td>Y
wolffd@0	1072 <!-- prob--><td>Y
wolffd@0	1073 <!-- learn--><td>Y
wolffd@0	1074 <!-- Bayes--><td>N
wolffd@0	1075 <!-- Pearl--><td>N
wolffd@0	1076
wolffd@0	1077 <tr>
wolffd@0	1078 <!-- Name--><td>gmux
wolffd@0	1079 <!-- Child--><td>C
wolffd@0	1080 <!-- Parents--><td>C/D
wolffd@0	1081 <!-- Comments--><td>multiplexer
wolffd@0	1082 <!-- CPD_to_CPT--><td>N
wolffd@0	1083 <!-- conv_to_table--><td>N
wolffd@0	1084 <!-- conv_to_pot--><td>Y
wolffd@0	1085 <!-- sample--><td>N
wolffd@0	1086 <!-- prob--><td>N
wolffd@0	1087 <!-- learn--><td>N
wolffd@0	1088 <!-- Bayes--><td>N
wolffd@0	1089 <!-- Pearl--><td>Y
wolffd@0	1090
wolffd@0	1091
wolffd@0	1092 <tr>
wolffd@0	1093 <!-- Name--><td>MLP
wolffd@0	1094 <!-- Child--><td>D
wolffd@0	1095 <!-- Parents--><td>C/D
wolffd@0	1096 <!-- Comments--><td>multi layer perceptron
wolffd@0	1097 <!-- CPD_to_CPT--><td>N
wolffd@0	1098 <!-- conv_to_table--><td>Y
wolffd@0	1099 <!-- conv_to_pot--><td>Inherits from discrete
wolffd@0	1100 <!-- sample--><td>Inherits from discrete
wolffd@0	1101 <!-- prob--><td>Inherits from discrete
wolffd@0	1102 <!-- learn--><td>Y
wolffd@0	1103 <!-- Bayes--><td>N
wolffd@0	1104 <!-- Pearl--><td>N
wolffd@0	1105
wolffd@0	1106
wolffd@0	1107 <tr>
wolffd@0	1108 <!-- Name--><td>noisy-or
wolffd@0	1109 <!-- Child--><td>B
wolffd@0	1110 <!-- Parents--><td>B
wolffd@0	1111 <!-- Comments--><td>-
wolffd@0	1112 <!-- CPD_to_CPT--><td>Y
wolffd@0	1113 <!-- conv_to_table--><td>Inherits from discrete
wolffd@0	1114 <!-- conv_to_pot--><td>Inherits from discrete
wolffd@0	1115 <!-- sample--><td>Inherits from discrete
wolffd@0	1116 <!-- prob--><td>Inherits from discrete
wolffd@0	1117 <!-- learn--><td>N
wolffd@0	1118 <!-- Bayes--><td>N
wolffd@0	1119 <!-- Pearl--><td>Y
wolffd@0	1120
wolffd@0	1121
wolffd@0	1122 <tr>
wolffd@0	1123 <!-- Name--><td>root
wolffd@0	1124 <!-- Child--><td>C/D
wolffd@0	1125 <!-- Parents--><td>none
wolffd@0	1126 <!-- Comments--><td>no params
wolffd@0	1127 <!-- CPD_to_CPT--><td>N
wolffd@0	1128 <!-- conv_to_table--><td>N
wolffd@0	1129 <!-- conv_to_pot--><td>Y
wolffd@0	1130 <!-- sample--><td>Y
wolffd@0	1131 <!-- prob--><td>Y
wolffd@0	1132 <!-- learn--><td>N
wolffd@0	1133 <!-- Bayes--><td>N
wolffd@0	1134 <!-- Pearl--><td>N
wolffd@0	1135
wolffd@0	1136
wolffd@0	1137 <tr>
wolffd@0	1138 <!-- Name--><td>softmax
wolffd@0	1139 <!-- Child--><td>D
wolffd@0	1140 <!-- Parents--><td>C/D
wolffd@0	1141 <!-- Comments--><td>-
wolffd@0	1142 <!-- CPD_to_CPT--><td>N
wolffd@0	1143 <!-- conv_to_table--><td>Y
wolffd@0	1144 <!-- conv_to_pot--><td>Inherits from discrete
wolffd@0	1145 <!-- sample--><td>Inherits from discrete
wolffd@0	1146 <!-- prob--><td>Inherits from discrete
wolffd@0	1147 <!-- learn--><td>Y
wolffd@0	1148 <!-- Bayes--><td>N
wolffd@0	1149 <!-- Pearl--><td>N
wolffd@0	1150
wolffd@0	1151
wolffd@0	1152 <tr>
wolffd@0	1153 <!-- Name--><td>generic
wolffd@0	1154 <!-- Child--><td>C/D
wolffd@0	1155 <!-- Parents--><td>C/D
wolffd@0	1156 <!-- Comments--><td>Virtual class
wolffd@0	1157 <!-- CPD_to_CPT--><td>N
wolffd@0	1158 <!-- conv_to_table--><td>N
wolffd@0	1159 <!-- conv_to_pot--><td>N
wolffd@0	1160 <!-- sample--><td>N
wolffd@0	1161 <!-- prob--><td>N
wolffd@0	1162 <!-- learn--><td>N
wolffd@0	1163 <!-- Bayes--><td>N
wolffd@0	1164 <!-- Pearl--><td>N
wolffd@0	1165
wolffd@0	1166
wolffd@0	1167 <tr>
wolffd@0	1168 <!-- Name--><td>Tabular
wolffd@0	1169 <!-- Child--><td>D
wolffd@0	1170 <!-- Parents--><td>D
wolffd@0	1171 <!-- Comments--><td>-
wolffd@0	1172 <!-- CPD_to_CPT--><td>Y
wolffd@0	1173 <!-- conv_to_table--><td>Inherits from discrete
wolffd@0	1174 <!-- conv_to_pot--><td>Inherits from discrete
wolffd@0	1175 <!-- sample--><td>Inherits from discrete
wolffd@0	1176 <!-- prob--><td>Inherits from discrete
wolffd@0	1177 <!-- learn--><td>Y
wolffd@0	1178 <!-- Bayes--><td>Y
wolffd@0	1179 <!-- Pearl--><td>Y
wolffd@0	1180
wolffd@0	1181 </table>
wolffd@0	1182
wolffd@0	1183
wolffd@0	1184
wolffd@0	1185 <h1><a name="examples">Example models</h1>
wolffd@0	1186
wolffd@0	1187
wolffd@0	1188 <h2>Gaussian mixture models</h2>
wolffd@0	1189
wolffd@0	1190 Richard W. DeVaul has made a detailed tutorial on how to fit mixtures
wolffd@0	1191 of Gaussians using BNT. Available
wolffd@0	1192 <a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>.
wolffd@0	1193
wolffd@0	1194
wolffd@0	1195 <h2><a name="pca">PCA, ICA, and all that </h2>
wolffd@0	1196
wolffd@0	1197 In Figure (a) below, we show how Factor Analysis can be thought of as a
wolffd@0	1198 graphical model. Here, X has an N(0,I) prior, and
wolffd@0	1199 Y\|X=x ~ N(mu + Wx, Psi),
wolffd@0	1200 where Psi is diagonal and W is called the "factor loading matrix".
wolffd@0	1201 Since the noise on both X and Y is diagonal, the components of these
wolffd@0	1202 vectors are uncorrelated, and hence can be represented as individual
wolffd@0	1203 scalar nodes, as we show in (b).
wolffd@0	1204 (This is useful if parts of the observations on the Y vector are occasionally missing.)
wolffd@0	1205 We usually take k=\|X\| << \|Y\|=D, so the model tries to explain
wolffd@0	1206 many observations using a low-dimensional subspace.
wolffd@0	1207
wolffd@0	1208
wolffd@0	1209 <center>
wolffd@0	1210 <table>
wolffd@0	1211 <tr>
wolffd@0	1212 <td><img src="Figures/fa.gif">
wolffd@0	1213 <td><img src="Figures/fa_scalar.gif">
wolffd@0	1214 <td><img src="Figures/mfa.gif">
wolffd@0	1215 <td><img src="Figures/ifa.gif">
wolffd@0	1216 <tr>
wolffd@0	1217 <td align=center> (a)
wolffd@0	1218 <td align=center> (b)
wolffd@0	1219 <td align=center> (c)
wolffd@0	1220 <td align=center> (d)
wolffd@0	1221 </table>
wolffd@0	1222 </center>
wolffd@0	1223
wolffd@0	1224 <p>
wolffd@0	1225 We can create this model in BNT as follows.
wolffd@0	1226 <pre>
wolffd@0	1227 ns = [k D];
wolffd@0	1228 dag = zeros(2,2);
wolffd@0	1229 dag(1,2) = 1;
wolffd@0	1230 bnet = mk_bnet(dag, ns, 'discrete', []);
wolffd@0	1231 bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ...
wolffd@0	1232 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1);
wolffd@0	1233 bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ...
wolffd@0	1234 'cov_type', 'diag', 'clamp_mean', 1);
wolffd@0	1235 </pre>
wolffd@0	1236
wolffd@0	1237 The root node is clamped to the N(0,I) distribution, so that we will
wolffd@0	1238 not update these parameters during learning.
wolffd@0	1239 The mean of the leaf node is clamped to 0,
wolffd@0	1240 since we assume the data has been centered (had its mean subtracted
wolffd@0	1241 off); this is just for simplicity.
wolffd@0	1242 Finally, the covariance of the leaf node is constrained to be
wolffd@0	1243 diagonal. W0 and Psi0 are the initial parameter guesses.
wolffd@0	1244
wolffd@0	1245 <p>
wolffd@0	1246 We can fit this model (i.e., estimate its parameters in a maximum
wolffd@0	1247 likelihood (ML) sense) using EM, as we
wolffd@0	1248 explain <a href="#em">below</a>.
wolffd@0	1249 Not surprisingly, the ML estimates for mu and Psi turn out to be
wolffd@0	1250 identical to the
wolffd@0	1251 sample mean and variance, which can be computed directly as
wolffd@0	1252 <pre>
wolffd@0	1253 mu_ML = mean(data);
wolffd@0	1254 Psi_ML = diag(cov(data));
wolffd@0	1255 </pre>
wolffd@0	1256 Note that W can only be identified up to a rotation matrix, because of
wolffd@0	1257 the spherical symmetry of the source.
wolffd@0	1258
wolffd@0	1259 <p>
wolffd@0	1260 If we restrict Psi to be spherical, i.e., Psi = sigma*I,
wolffd@0	1261 there is a closed-form solution for W as well,
wolffd@0	1262 i.e., we do not need to use EM.
wolffd@0	1263 In particular, W contains the first \|X\| eigenvectors of the sample covariance
wolffd@0	1264 matrix, with scalings determined by the eigenvalues and sigma.
wolffd@0	1265 Classical PCA can be obtained by taking the sigma->0 limit.
wolffd@0	1266 For details, see
wolffd@0	1267
wolffd@0	1268 <ul>
wolffd@0	1269 <li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps">
wolffd@0	1270 "EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97.
wolffd@0	1271 (<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz">
wolffd@0	1272 Matlab software</a>)
wolffd@0	1273
wolffd@0	1274 <p>
wolffd@0	1275 <li>
wolffd@0	1276 <a
wolffd@0	1277 href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003>
wolffd@0	1278 "Mixtures of probabilistic principal component analyzers"</a>,
wolffd@0	1279 Tipping and Bishop, Neural Computation 11(2):443--482, 1999.
wolffd@0	1280 </ul>
wolffd@0	1281
wolffd@0	1282 <p>
wolffd@0	1283 By adding a hidden discrete variable, we can create mixtures of FA
wolffd@0	1284 models, as shown in (c).
wolffd@0	1285 Now we can explain the data using a set of subspaces.
wolffd@0	1286 We can create this model in BNT as follows.
wolffd@0	1287 <pre>
wolffd@0	1288 ns = [M k D];
wolffd@0	1289 dag = zeros(3);
wolffd@0	1290 dag(1,3) = 1;
wolffd@0	1291 dag(2,3) = 1;
wolffd@0	1292 bnet = mk_bnet(dag, ns, 'discrete', 1);
wolffd@0	1293 bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0);
wolffd@0	1294 bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ...
wolffd@0	1295 'clamp_mean', 1, 'clamp_cov', 1);
wolffd@0	1296 bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ...
wolffd@0	1297 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1);
wolffd@0	1298 </pre>
wolffd@0	1299 Notice how the covariance matrix for Y is the same for all values of
wolffd@0	1300 Q; that is, the noise level in each sub-space is assumed the same.
wolffd@0	1301 However, we allow the offset, mu, to vary.
wolffd@0	1302 For details, see
wolffd@0	1303 <ul>
wolffd@0	1304
wolffd@0	1305 <LI>
wolffd@0	1306 <a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM
wolffd@0	1307 Algorithm for Mixtures of Factor Analyzers </A>,
wolffd@0	1308 Ghahramani, Z. and Hinton, G.E. (1996),
wolffd@0	1309 University of Toronto
wolffd@0	1310 Technical Report CRG-TR-96-1.
wolffd@0	1311 (<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>)
wolffd@0	1312
wolffd@0	1313 <p>
wolffd@0	1314 <li>
wolffd@0	1315 <a
wolffd@0	1316 href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003>
wolffd@0	1317 "Mixtures of probabilistic principal component analyzers"</a>,
wolffd@0	1318 Tipping and Bishop, Neural Computation 11(2):443--482, 1999.
wolffd@0	1319 </ul>
wolffd@0	1320
wolffd@0	1321 <p>
wolffd@0	1322 I have included Zoubin's specialized MFA code (with his permission)
wolffd@0	1323 with the toolbox, so you can check that BNT gives the same results:
wolffd@0	1324 see 'BNT/examples/static/mfa1.m'.
wolffd@0	1325
wolffd@0	1326 <p>
wolffd@0	1327 Independent Factor Analysis (IFA) generalizes FA by allowing a
wolffd@0	1328 non-Gaussian prior on each component of X.
wolffd@0	1329 (Note that we can approximate a non-Gaussian prior using a mixture of
wolffd@0	1330 Gaussians.)
wolffd@0	1331 This means that the likelihood function is no longer rotationally
wolffd@0	1332 invariant, so we can uniquely identify W and the hidden
wolffd@0	1333 sources X.
wolffd@0	1334 IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y).
wolffd@0	1335 We recover classical Independent Components Analysis (ICA)
wolffd@0	1336 in the Psi -> 0 limit, and by assuming that \|X\|=\|Y\|, so that the
wolffd@0	1337 weight matrix W is square and invertible.
wolffd@0	1338 For details, see
wolffd@0	1339 <ul>
wolffd@0	1340 <li>
wolffd@0	1341 <a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor
wolffd@0	1342 Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998.
wolffd@0	1343 </ul>
wolffd@0	1344
wolffd@0	1345
wolffd@0	1346
wolffd@0	1347 <h2><a name="mixexp">Mixtures of experts</h2>
wolffd@0	1348
wolffd@0	1349 As an example of the use of the softmax function,
wolffd@0	1350 we introduce the Mixture of Experts model.
wolffd@0	1351 <!--
wolffd@0	1352 We also show
wolffd@0	1353 the Hierarchical Mixture of Experts model, where the hierarchy has two
wolffd@0	1354 levels.
wolffd@0	1355 (This is essentially a probabilistic decision tree of height two.)
wolffd@0	1356 -->
wolffd@0	1357 As before,
wolffd@0	1358 circles denote continuous-valued nodes,
wolffd@0	1359 squares denote discrete nodes, clear
wolffd@0	1360 means hidden, and shaded means observed.
wolffd@0	1361 <p>
wolffd@0	1362 <center>
wolffd@0	1363 <table>
wolffd@0	1364 <tr>
wolffd@0	1365 <td><img src="Figures/mixexp.gif">
wolffd@0	1366 <!--
wolffd@0	1367 <td><img src="Figures/hme.gif">
wolffd@0	1368 -->
wolffd@0	1369 </table>
wolffd@0	1370 </center>
wolffd@0	1371 <p>
wolffd@0	1372 X is the observed
wolffd@0	1373 input, Y is the output, and
wolffd@0	1374 the Q nodes are hidden "gating" nodes, which select the appropriate
wolffd@0	1375 set of parameters for Y. During training, Y is assumed observed,
wolffd@0	1376 but for testing, the goal is to predict Y given X.
wolffd@0	1377 Note that this is a <em>conditional</em> density model, so we don't
wolffd@0	1378 associate any parameters with X.
wolffd@0	1379 Hence X's CPD will be a root CPD, which is a way of modelling
wolffd@0	1380 exogenous nodes.
wolffd@0	1381 If the output is a continuous-valued quantity,
wolffd@0	1382 we assume the "experts" are linear-regression units,
wolffd@0	1383 and set Y's CPD to linear-Gaussian.
wolffd@0	1384 If the output is discrete, we set Y's CPD to a softmax function.
wolffd@0	1385 The Q CPDs will always be softmax functions.
wolffd@0	1386
wolffd@0	1387 <p>
wolffd@0	1388 As a concrete example, consider the mixture of experts model where X and Y are
wolffd@0	1389 scalars, and Q is binary.
wolffd@0	1390 This is just piecewise linear regression, where
wolffd@0	1391 we have two line segments, i.e.,
wolffd@0	1392 <P>
wolffd@0	1393 <IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif">
wolffd@0	1394 <P>
wolffd@0	1395 We can create this model with random parameters as follows.
wolffd@0	1396 (This code is bundled in BNT/examples/static/mixexp2.m.)
wolffd@0	1397 <PRE>
wolffd@0	1398 X = 1;
wolffd@0	1399 Q = 2;
wolffd@0	1400 Y = 3;
wolffd@0	1401 dag = zeros(3,3);
wolffd@0	1402 dag(X,[Q Y]) = 1
wolffd@0	1403 dag(Q,Y) = 1;
wolffd@0	1404 ns = [1 2 1]; % make X and Y scalars, and have 2 experts
wolffd@0	1405 onodes = [1 3];
wolffd@0	1406 bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes);
wolffd@0	1407
wolffd@0	1408 rand('state', 0);
wolffd@0	1409 randn('state', 0);
wolffd@0	1410 bnet.CPD{1} = root_CPD(bnet, 1);
wolffd@0	1411 bnet.CPD{2} = softmax_CPD(bnet, 2);
wolffd@0	1412 bnet.CPD{3} = gaussian_CPD(bnet, 3);
wolffd@0	1413 </PRE>
wolffd@0	1414 Now let us fit this model using <a href="#em">EM</a>.
wolffd@0	1415 First we <a href="#load_data">load the data</a> (1000 training cases) and plot them.
wolffd@0	1416 <P>
wolffd@0	1417 <PRE>
wolffd@0	1418 data = load('/examples/static/Misc/mixexp_data.txt', '-ascii');
wolffd@0	1419 plot(data(:,1), data(:,2), '.');
wolffd@0	1420 </PRE>
wolffd@0	1421 <p>
wolffd@0	1422 <center>
wolffd@0	1423 <IMG SRC="Figures/mixexp_data.gif">
wolffd@0	1424 </center>
wolffd@0	1425 <p>
wolffd@0	1426 This is what the model looks like before training.
wolffd@0	1427 (Thanks to Thomas Hofman for writing this plotting routine.)
wolffd@0	1428 <p>
wolffd@0	1429 <center>
wolffd@0	1430 <IMG SRC="Figures/mixexp_before.gif">
wolffd@0	1431 </center>
wolffd@0	1432 <p>
wolffd@0	1433 Now let's train the model, and plot the final performance.
wolffd@0	1434 (We will discuss how to train models in more detail <a href="#param_learning">below</a>.)
wolffd@0	1435 <P>
wolffd@0	1436 <PRE>
wolffd@0	1437 ncases = size(data, 1); % each row of data is a training case
wolffd@0	1438 cases = cell(3, ncases);
wolffd@0	1439 cases([1 3], :) = num2cell(data'); % each column of cases is a training case
wolffd@0	1440 engine = jtree_inf_engine(bnet);
wolffd@0	1441 max_iter = 20;
wolffd@0	1442 [bnet2, LLtrace] = learn_params_em(engine, cases, max_iter);
wolffd@0	1443 </PRE>
wolffd@0	1444 (We specify which nodes will be observed when we create the engine.
wolffd@0	1445 Hence BNT knows that the hidden nodes are all discrete.
wolffd@0	1446 For complex models, this can lead to a significant speedup.)
wolffd@0	1447 Below we show what the model looks like after 16 iterations of EM
wolffd@0	1448 (with 100 IRLS iterations per M step), when it converged
wolffd@0	1449 using the default convergence tolerance (that the
wolffd@0	1450 fractional change in the log-likelihood be less than 1e-3).
wolffd@0	1451 Before learning, the log-likelihood was
wolffd@0	1452 -322.927442; afterwards, it was -13.728778.
wolffd@0	1453 <p>
wolffd@0	1454 <center>
wolffd@0	1455 <IMG SRC="Figures/mixexp_after.gif">
wolffd@0	1456 </center>
wolffd@0	1457 (See BNT/examples/static/mixexp2.m for details of the code.)
wolffd@0	1458
wolffd@0	1459
wolffd@0	1460
wolffd@0	1461 <h2><a name="hme">Hierarchical mixtures of experts</h2>
wolffd@0	1462
wolffd@0	1463 A hierarchical mixture of experts (HME) extends the mixture of experts
wolffd@0	1464 model by having more than one hidden node. A two-level example is shown below, along
wolffd@0	1465 with its more traditional representation as a neural network.
wolffd@0	1466 This is like a (balanced) probabilistic decision tree of height 2.
wolffd@0	1467 <p>
wolffd@0	1468 <center>
wolffd@0	1469 <IMG SRC="Figures/HMEforMatlab.jpg">
wolffd@0	1470 </center>
wolffd@0	1471 <p>
wolffd@0	1472 <a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a>
wolffd@0	1473 has written an extensive set of routines for HMEs,
wolffd@0	1474 which are bundled with BNT: see the examples/static/HME directory.
wolffd@0	1475 These routines allow you to choose the number of hidden (gating)
wolffd@0	1476 layers, and the form of the experts (softmax or MLP).
wolffd@0	1477 See the file hmemenu, which provides a demo.
wolffd@0	1478 For example, the figure below shows the decision boundaries learned
wolffd@0	1479 for a ternary classification problem, using a 2 level HME with softmax
wolffd@0	1480 gates and softmax experts; the training set is on the left, the
wolffd@0	1481 testing set on the right.
wolffd@0	1482 <p>
wolffd@0	1483 <center>
wolffd@0	1484 <!--<IMG SRC="Figures/hme_dec_boundary.gif">-->
wolffd@0	1485 <IMG SRC="Figures/hme_dec_boundary.png">
wolffd@0	1486 </center>
wolffd@0	1487 <p>
wolffd@0	1488
wolffd@0	1489
wolffd@0	1490 <p>
wolffd@0	1491 For more details, see the following:
wolffd@0	1492 <ul>
wolffd@0	1493
wolffd@0	1494 <li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z">
wolffd@0	1495 Hierarchical mixtures of experts and the EM algorithm</a>
wolffd@0	1496 M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994.
wolffd@0	1497
wolffd@0	1498 <li> <a href =
wolffd@0	1499 "http://www.cs.berkeley.edu/~dmartin/software">David Martin's
wolffd@0	1500 matlab code for HME</a>
wolffd@0	1501
wolffd@0	1502 <li> <a
wolffd@0	1503 href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the
wolffd@0	1504 logistic function? A tutorial discussion on
wolffd@0	1505 probabilities and neural networks.</a> M. I. Jordan. MIT Computational
wolffd@0	1506 Cognitive Science Report 9503, August 1995.
wolffd@0	1507
wolffd@0	1508 <li> "Generalized Linear Models", McCullagh and Nelder, Chapman and
wolffd@0	1509 Halll, 1983.
wolffd@0	1510
wolffd@0	1511 <li>
wolffd@0	1512 "Improved learning algorithms for mixtures of experts in multiclass
wolffd@0	1513 classification".
wolffd@0	1514 K. Chen, L. Xu, H. Chi.
wolffd@0	1515 Neural Networks (1999) 12: 1229-1252.
wolffd@0	1516
wolffd@0	1517 <li> <a href="http://www.oigeeza.com/steve/">
wolffd@0	1518 Classification Using Hierarchical Mixtures of Experts</a>
wolffd@0	1519 S.R. Waterhouse and A.J. Robinson.
wolffd@0	1520 In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186
wolffd@0	1521
wolffd@0	1522 <li> <a href="http://www.idiap.ch/~perry/">
wolffd@0	1523 Localized mixtures of experts</a>,
wolffd@0	1524 P. Moerland, 1998.
wolffd@0	1525
wolffd@0	1526 <li> "Nonlinear gated experts for time series",
wolffd@0	1527 A.S. Weigend and M. Mangeas, 1995.
wolffd@0	1528
wolffd@0	1529 </ul>
wolffd@0	1530
wolffd@0	1531
wolffd@0	1532 <h2><a name="qmr">QMR</h2>
wolffd@0	1533
wolffd@0	1534 Bayes nets originally arose out of an attempt to add probabilities to
wolffd@0	1535 expert systems, and this is still the most common use for BNs.
wolffd@0	1536 A famous example is
wolffd@0	1537 QMR-DT, a decision-theoretic reformulation of the Quick Medical
wolffd@0	1538 Reference (QMR) model.
wolffd@0	1539 <p>
wolffd@0	1540 <center>
wolffd@0	1541 <IMG ALIGN=BOTTOM SRC="Figures/qmr.gif">
wolffd@0	1542 </center>
wolffd@0	1543 Here, the top layer represents hidden disease nodes, and the bottom
wolffd@0	1544 layer represents observed symptom nodes.
wolffd@0	1545 The goal is to infer the posterior probability of each disease given
wolffd@0	1546 all the symptoms (which can be present, absent or unknown).
wolffd@0	1547 Each node in the top layer has a Bernoulli prior (with a low prior
wolffd@0	1548 probability that the disease is present).
wolffd@0	1549 Since each node in the bottom layer has a high fan-in, we use a
wolffd@0	1550 noisy-OR parameterization; each disease has an independent chance of
wolffd@0	1551 causing each symptom.
wolffd@0	1552 The real QMR-DT model is copyright, but
wolffd@0	1553 we can create a random QMR-like model as follows.
wolffd@0	1554 <pre>
wolffd@0	1555 function bnet = mk_qmr_bnet(G, inhibit, leak, prior)
wolffd@0	1556 % MK_QMR_BNET Make a QMR model
wolffd@0	1557 % bnet = mk_qmr_bnet(G, inhibit, leak, prior)
wolffd@0	1558 %
wolffd@0	1559 % G(i,j) = 1 iff there is an arc from disease i to finding j
wolffd@0	1560 % inhibit(i,j) = inhibition probability on i->j arc
wolffd@0	1561 % leak(j) = inhibition prob. on leak->j arc
wolffd@0	1562 % prior(i) = prob. disease i is on
wolffd@0	1563
wolffd@0	1564 [Ndiseases Nfindings] = size(inhibit);
wolffd@0	1565 N = Ndiseases + Nfindings;
wolffd@0	1566 finding_node = Ndiseases+1:N;
wolffd@0	1567 ns = 2*ones(1,N);
wolffd@0	1568 dag = zeros(N,N);
wolffd@0	1569 dag(1:Ndiseases, finding_node) = G;
wolffd@0	1570 bnet = mk_bnet(dag, ns, 'observed', finding_node);
wolffd@0	1571
wolffd@0	1572 for d=1:Ndiseases
wolffd@0	1573 CPT = [1-prior(d) prior(d)];
wolffd@0	1574 bnet.CPD{d} = tabular_CPD(bnet, d, CPT');
wolffd@0	1575 end
wolffd@0	1576
wolffd@0	1577 for i=1:Nfindings
wolffd@0	1578 fnode = finding_node(i);
wolffd@0	1579 ps = parents(G, i);
wolffd@0	1580 bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i));
wolffd@0	1581 end
wolffd@0	1582 </pre>
wolffd@0	1583 In the file BNT/examples/static/qmr1, we create a random bipartite
wolffd@0	1584 graph G, with 5 diseases and 10 findings, and random parameters.
wolffd@0	1585 (In general, to create a random dag, use 'mk_random_dag'.)
wolffd@0	1586 We can visualize the resulting graph structure using
wolffd@0	1587 the methods discussed <a href="#graphdraw">below</a>, with the
wolffd@0	1588 following results:
wolffd@0	1589 <p>
wolffd@0	1590 <img src="Figures/qmr.rnd.jpg">
wolffd@0	1591
wolffd@0	1592 <p>
wolffd@0	1593 Now let us put some random evidence on all the leaves except the very
wolffd@0	1594 first and very last, and compute the disease posteriors.
wolffd@0	1595 <pre>
wolffd@0	1596 pos = 2:floor(Nfindings/2);
wolffd@0	1597 neg = (pos(end)+1):(Nfindings-1);
wolffd@0	1598 onodes = myunion(pos, neg);
wolffd@0	1599 evidence = cell(1, N);
wolffd@0	1600 evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos)));
wolffd@0	1601 evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg)));
wolffd@0	1602
wolffd@0	1603 engine = jtree_inf_engine(bnet);
wolffd@0	1604 [engine, ll] = enter_evidence(engine, evidence);
wolffd@0	1605 post = zeros(1, Ndiseases);
wolffd@0	1606 for i=diseases(:)'
wolffd@0	1607 m = marginal_nodes(engine, i);
wolffd@0	1608 post(i) = m.T(2);
wolffd@0	1609 end
wolffd@0	1610 </pre>
wolffd@0	1611 Junction tree can be quite slow on large QMR models.
wolffd@0	1612 Fortunately, it is possible to exploit properties of the noisy-OR
wolffd@0	1613 function to speed up exact inference using an algorithm called
wolffd@0	1614 <a href="#quickscore">quickscore</a>, discussed below.
wolffd@0	1615
wolffd@0	1616
wolffd@0	1617
wolffd@0	1618
wolffd@0	1619
wolffd@0	1620 <h2><a name="cg_model">Conditional Gaussian models</h2>
wolffd@0	1621
wolffd@0	1622 A conditional Gaussian model is one in which, conditioned on all the discrete
wolffd@0	1623 nodes, the distribution over the remaining (continuous) nodes is
wolffd@0	1624 multivariate Gaussian. This means we can have arcs from discrete (D)
wolffd@0	1625 to continuous (C) nodes, but not vice versa.
wolffd@0	1626 (We <em>are</em> allowed C->D arcs if the continuous nodes are observed,
wolffd@0	1627 as in the <a href="#mixexp">mixture of experts</a> model,
wolffd@0	1628 since this distribution can be represented with a discrete potential.)
wolffd@0	1629 <p>
wolffd@0	1630 We now give an example of a CG model, from
wolffd@0	1631 the paper "Propagation of Probabilities, Means amd
wolffd@0	1632 Variances in Mixed Graphical Association Models", Steffen Lauritzen,
wolffd@0	1633 JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert
wolffd@0	1634 Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and
wolffd@0	1635 D. J. Spiegelhalter, Springer, 1999.)
wolffd@0	1636
wolffd@0	1637 <h3>Specifying the graph</h3>
wolffd@0	1638
wolffd@0	1639 Consider the model of waste emissions from an incinerator plant shown below.
wolffd@0	1640 We follow the standard convention that shaded nodes are observed,
wolffd@0	1641 clear nodes are hidden.
wolffd@0	1642 We also use the non-standard convention that
wolffd@0	1643 square nodes are discrete (tabular) and round nodes are
wolffd@0	1644 Gaussian.
wolffd@0	1645
wolffd@0	1646 <p>
wolffd@0	1647 <center>
wolffd@0	1648 <IMG SRC="Figures/cg1.gif">
wolffd@0	1649 </center>
wolffd@0	1650 <p>
wolffd@0	1651
wolffd@0	1652 We can create this model as follows.
wolffd@0	1653 <pre>
wolffd@0	1654 F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9;
wolffd@0	1655 n = 9;
wolffd@0	1656
wolffd@0	1657 dag = zeros(n);
wolffd@0	1658 dag(F,E)=1;
wolffd@0	1659 dag(W,[E Min D]) = 1;
wolffd@0	1660 dag(E,D)=1;
wolffd@0	1661 dag(B,[C D])=1;
wolffd@0	1662 dag(D,[L Mout])=1;
wolffd@0	1663 dag(Min,Mout)=1;
wolffd@0	1664
wolffd@0	1665 % node sizes - all cts nodes are scalar, all discrete nodes are binary
wolffd@0	1666 ns = ones(1, n);
wolffd@0	1667 dnodes = [F W B];
wolffd@0	1668 cnodes = mysetdiff(1:n, dnodes);
wolffd@0	1669 ns(dnodes) = 2;
wolffd@0	1670
wolffd@0	1671 bnet = mk_bnet(dag, ns, 'discrete', dnodes);
wolffd@0	1672 </pre>
wolffd@0	1673 'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous
wolffd@0	1674 nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'.
wolffd@0	1675 <p>
wolffd@0	1676
wolffd@0	1677
wolffd@0	1678 <h3>Specifying the parameters</h3>
wolffd@0	1679
wolffd@0	1680 The parameters of the discrete nodes can be specified as follows.
wolffd@0	1681 <pre>
wolffd@0	1682 bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable
wolffd@0	1683 bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect
wolffd@0	1684 bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household
wolffd@0	1685 </pre>
wolffd@0	1686
wolffd@0	1687 <p>
wolffd@0	1688 The parameters of the continuous nodes can be specified as follows.
wolffd@0	1689 <pre>
wolffd@0	1690 bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ...
wolffd@0	1691 'cov', [0.00002 0.0001 0.00002 0.0001]);
wolffd@0	1692 bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ...
wolffd@0	1693 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]);
wolffd@0	1694 bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]);
wolffd@0	1695 bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5);
wolffd@0	1696 bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]);
wolffd@0	1697 bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]);
wolffd@0	1698 </pre>
wolffd@0	1699
wolffd@0	1700
wolffd@0	1701 <h3><a name="cg_infer">Inference</h3>
wolffd@0	1702
wolffd@0	1703 <!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.-->
wolffd@0	1704 First we compute the unconditional marginals.
wolffd@0	1705 <pre>
wolffd@0	1706 engine = jtree_inf_engine(bnet);
wolffd@0	1707 evidence = cell(1,n);
wolffd@0	1708 [engine, ll] = enter_evidence(engine, evidence);
wolffd@0	1709 marg = marginal_nodes(engine, E);
wolffd@0	1710 </pre>
wolffd@0	1711 <!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)-->
wolffd@0	1712 'marg' is a structure that contains the fields 'mu' and 'Sigma', which
wolffd@0	1713 contain the mean and (co)variance of the marginal on E.
wolffd@0	1714 In this case, they are both scalars.
wolffd@0	1715 Let us check they match the published figures (to 2 decimal places).
wolffd@0	1716 <!--(We can't expect
wolffd@0	1717 more precision than this in general because I have implemented the algorithm of
wolffd@0	1718 Lauritzen (1992), which can be numerically unstable.)-->
wolffd@0	1719 <pre>
wolffd@0	1720 tol = 1e-2;
wolffd@0	1721 assert(approxeq(marg.mu, -3.25, tol));
wolffd@0	1722 assert(approxeq(sqrt(marg.Sigma), 0.709, tol));
wolffd@0	1723 </pre>
wolffd@0	1724 We can compute the other posteriors similarly.
wolffd@0	1725 Now let us add some evidence.
wolffd@0	1726 <pre>
wolffd@0	1727 evidence = cell(1,n);
wolffd@0	1728 evidence{W} = 1; % industrial
wolffd@0	1729 evidence{L} = 1.1;
wolffd@0	1730 evidence{C} = -0.9;
wolffd@0	1731 [engine, ll] = enter_evidence(engine, evidence);
wolffd@0	1732 </pre>
wolffd@0	1733 Now we find
wolffd@0	1734 <pre>
wolffd@0	1735 marg = marginal_nodes(engine, E);
wolffd@0	1736 assert(approxeq(marg.mu, -3.8983, tol));
wolffd@0	1737 assert(approxeq(sqrt(marg.Sigma), 0.0763, tol));
wolffd@0	1738 </pre>
wolffd@0	1739
wolffd@0	1740
wolffd@0	1741 We can also compute the joint probability on a set of nodes.
wolffd@0	1742 For example, P(D, Mout \| evidence) is a 2D Gaussian:
wolffd@0	1743 <pre>
wolffd@0	1744 marg = marginal_nodes(engine, [D Mout])
wolffd@0	1745 marg =
wolffd@0	1746 domain: [6 8]
wolffd@0	1747 mu: [2x1 double]
wolffd@0	1748 Sigma: [2x2 double]
wolffd@0	1749 T: 1.0000
wolffd@0	1750 </pre>
wolffd@0	1751 The mean is
wolffd@0	1752 <pre>
wolffd@0	1753 marg.mu
wolffd@0	1754 ans =
wolffd@0	1755 3.6077
wolffd@0	1756 4.1077
wolffd@0	1757 </pre>
wolffd@0	1758 and the covariance matrix is
wolffd@0	1759 <pre>
wolffd@0	1760 marg.Sigma
wolffd@0	1761 ans =
wolffd@0	1762 0.1062 0.1062
wolffd@0	1763 0.1062 0.1182
wolffd@0	1764 </pre>
wolffd@0	1765 It is easy to visualize this posterior using standard Matlab plotting
wolffd@0	1766 functions, e.g.,
wolffd@0	1767 <pre>
wolffd@0	1768 gaussplot2d(marg.mu, marg.Sigma);
wolffd@0	1769 </pre>
wolffd@0	1770 produces the following picture.
wolffd@0	1771
wolffd@0	1772 <p>
wolffd@0	1773 <center>
wolffd@0	1774 <IMG SRC="Figures/gaussplot.png">
wolffd@0	1775 </center>
wolffd@0	1776 <p>
wolffd@0	1777
wolffd@0	1778
wolffd@0	1779 The T field indicates that the mixing weight of this Gaussian
wolffd@0	1780 component is 1.0.
wolffd@0	1781 If the joint contains discrete and continuous variables, the result
wolffd@0	1782 will be a mixture of Gaussians, e.g.,
wolffd@0	1783 <pre>
wolffd@0	1784 marg = marginal_nodes(engine, [F E])
wolffd@0	1785 domain: [1 3]
wolffd@0	1786 mu: [-3.9000 -0.4003]
wolffd@0	1787 Sigma: [1x1x2 double]
wolffd@0	1788 T: [0.9995 4.7373e-04]
wolffd@0	1789 </pre>
wolffd@0	1790 The interpretation is
wolffd@0	1791 Sigma(i,j,k) = Cov[ E(i) E(j) \| F=k ].
wolffd@0	1792 In this case, E is a scalar, so i=j=1; k specifies the mixture component.
wolffd@0	1793 <p>
wolffd@0	1794 We saw in the sprinkler network that BNT sets the effective size of
wolffd@0	1795 observed discrete nodes to 1, since they only have one legal value.
wolffd@0	1796 For continuous nodes, BNT sets their length to 0,
wolffd@0	1797 since they have been reduced to a point.
wolffd@0	1798 For example,
wolffd@0	1799 <pre>
wolffd@0	1800 marg = marginal_nodes(engine, [B C])
wolffd@0	1801 domain: [4 5]
wolffd@0	1802 mu: []
wolffd@0	1803 Sigma: []
wolffd@0	1804 T: [0.0123 0.9877]
wolffd@0	1805 </pre>
wolffd@0	1806 It is simple to post-process the output of marginal_nodes.
wolffd@0	1807 For example, the file BNT/examples/static/cg1 sets the mu term of
wolffd@0	1808 observed nodes to their observed value, and the Sigma term to 0 (since
wolffd@0	1809 observed nodes have no variance).
wolffd@0	1810
wolffd@0	1811 <p>
wolffd@0	1812 Note that the implemented version of the junction tree is numerically
wolffd@0	1813 unstable when using CG potentials
wolffd@0	1814 (which is why, in the example above, we only required our answers to agree with
wolffd@0	1815 the published ones to 2dp.)
wolffd@0	1816 This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>,
wolffd@0	1817 implemented by Shan Huang. This is described in
wolffd@0	1818
wolffd@0	1819 <ul>
wolffd@0	1820 <li> "Stable Local Computation with Conditional Gaussian Distributions",
wolffd@0	1821 S. Lauritzen and F. Jensen, Tech Report R-99-2014,
wolffd@0	1822 Dept. Math. Sciences, Allborg Univ., 1999.
wolffd@0	1823 </ul>
wolffd@0	1824
wolffd@0	1825 However, even the numerically stable version
wolffd@0	1826 can be computationally intractable if there are many hidden discrete
wolffd@0	1827 nodes, because the number of mixture components grows exponentially e.g., in a
wolffd@0	1828 <a href="usage_dbn.html#lds">switching linear dynamical system</a>.
wolffd@0	1829 In general, one must resort to approximate inference techniques: see
wolffd@0	1830 the discussion on <a href="#engines">inference engines</a> below.
wolffd@0	1831
wolffd@0	1832
wolffd@0	1833 <h2><a name="hybrid">Other hybrid models</h2>
wolffd@0	1834
wolffd@0	1835 When we have C->D arcs, where C is hidden, we need to use
wolffd@0	1836 approximate inference.
wolffd@0	1837 One approach (not implemented in BNT) is described in
wolffd@0	1838 <ul>
wolffd@0	1839 <li> <a
wolffd@0	1840 href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A
wolffd@0	1841 Variational Approximation for Bayesian Networks with
wolffd@0	1842 Discrete and Continuous Latent Variables</a>,
wolffd@0	1843 K. Murphy, UAI 99.
wolffd@0	1844 </ul>
wolffd@0	1845 Of course, one can always use <a href="#sampling">sampling</a> methods
wolffd@0	1846 for approximate inference in such models.
wolffd@0	1847
wolffd@0	1848
wolffd@0	1849
wolffd@0	1850 <h1><a name="param_learning">Parameter Learning</h1>
wolffd@0	1851
wolffd@0	1852 The parameter estimation routines in BNT can be classified into 4
wolffd@0	1853 types, depending on whether the goal is to compute
wolffd@0	1854 a full (Bayesian) posterior over the parameters or just a point
wolffd@0	1855 estimate (e.g., Maximum Likelihood or Maximum A Posteriori),
wolffd@0	1856 and whether all the variables are fully observed or there is missing
wolffd@0	1857 data/ hidden variables (partial observability).
wolffd@0	1858 <p>
wolffd@0	1859
wolffd@0	1860 <TABLE BORDER>
wolffd@0	1861 <tr>
wolffd@0	1862 <TH></TH>
wolffd@0	1863 <th>Full obs</th>
wolffd@0	1864 <th>Partial obs</th>
wolffd@0	1865 </tr>
wolffd@0	1866 <tr>
wolffd@0	1867 <th>Point</th>
wolffd@0	1868 <td><tt>learn_params</tt></td>
wolffd@0	1869 <td><tt>learn_params_em</tt></td>
wolffd@0	1870 </tr>
wolffd@0	1871 <tr>
wolffd@0	1872 <th>Bayes</th>
wolffd@0	1873 <td><tt>bayes_update_params</tt></td>
wolffd@0	1874 <td>not yet supported</td>
wolffd@0	1875 </tr>
wolffd@0	1876 </table>
wolffd@0	1877
wolffd@0	1878
wolffd@0	1879 <h2><a name="load_data">Loading data from a file</h2>
wolffd@0	1880
wolffd@0	1881 To load numeric data from an ASCII text file called 'dat.txt', where each row is a
wolffd@0	1882 case and columns are separated by white-space, such as
wolffd@0	1883 <pre>
wolffd@0	1884 011979 1626.5 0.0
wolffd@0	1885 021979 1367.0 0.0
wolffd@0	1886 ...
wolffd@0	1887 </pre>
wolffd@0	1888 you can use
wolffd@0	1889 <pre>
wolffd@0	1890 data = load('dat.txt');
wolffd@0	1891 </pre>
wolffd@0	1892 or
wolffd@0	1893 <pre>
wolffd@0	1894 load dat.txt -ascii
wolffd@0	1895 </pre>
wolffd@0	1896 In the latter case, the data is stored in a variable called 'dat' (the
wolffd@0	1897 filename minus the extension).
wolffd@0	1898 Alternatively, suppose the data is stored in a .csv file (has commas
wolffd@0	1899 separating the columns, and contains a header line), such as
wolffd@0	1900 <pre>
wolffd@0	1901 header info goes here
wolffd@0	1902 ORD,011979,1626.5,0.0
wolffd@0	1903 DSM,021979,1367.0,0.0
wolffd@0	1904 ...
wolffd@0	1905 </pre>
wolffd@0	1906 You can load this using
wolffd@0	1907 <pre>
wolffd@0	1908 [a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1);
wolffd@0	1909 </pre>
wolffd@0	1910 If your file is not in either of these formats, you can either use Perl to convert
wolffd@0	1911 it to this format, or use the Matlab scanf command.
wolffd@0	1912 Type
wolffd@0	1913 <tt>
wolffd@0	1914 help iofun
wolffd@0	1915 </tt>
wolffd@0	1916 for more information on Matlab's file functions.
wolffd@0	1917 <!--
wolffd@0	1918 <p>
wolffd@0	1919 To load data directly from Excel,
wolffd@0	1920 you should buy the
wolffd@0	1921 <a href="http://www.mathworks.com/products/excellink/">Excel Link</a>.
wolffd@0	1922 To load data directly from a relational database,
wolffd@0	1923 you should buy the
wolffd@0	1924 <a href="http://www.mathworks.com/products/database">Database
wolffd@0	1925 toolbox</a>.
wolffd@0	1926 -->
wolffd@0	1927 <p>
wolffd@0	1928 BNT learning routines require data to be stored in a cell array.
wolffd@0	1929 data{i,m} is the value of node i in case (example) m, i.e., each
wolffd@0	1930 <em>column</em> is a case.
wolffd@0	1931 If node i is not observed in case m (missing value), set
wolffd@0	1932 data{i,m} = [].
wolffd@0	1933 (Not all the learning routines can cope with such missing values, however.)
wolffd@0	1934 In the special case that all the nodes are observed and are
wolffd@0	1935 scalar-valued (as opposed to vector-valued), the data can be
wolffd@0	1936 stored in a matrix (as opposed to a cell-array).
wolffd@0	1937 <p>
wolffd@0	1938 Suppose, as in the <a href="#mixexp">mixture of experts example</a>,
wolffd@0	1939 that we have 3 nodes in the graph: X(1) is the observed input, X(3) is
wolffd@0	1940 the observed output, and X(2) is a hidden (gating) node. We can
wolffd@0	1941 create the dataset as follows.
wolffd@0	1942 <pre>
wolffd@0	1943 data = load('dat.txt');
wolffd@0	1944 ncases = size(data, 1);
wolffd@0	1945 cases = cell(3, ncases);
wolffd@0	1946 cases([1 3], :) = num2cell(data');
wolffd@0	1947 </pre>
wolffd@0	1948 Notice how we transposed the data, to convert rows into columns.
wolffd@0	1949 Also, cases{2,m} = [] for all m, since X(2) is always hidden.
wolffd@0	1950
wolffd@0	1951
wolffd@0	1952 <h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2>
wolffd@0	1953
wolffd@0	1954 As an example, let's generate some data from the sprinkler network, randomize the parameters,
wolffd@0	1955 and then try to recover the original model.
wolffd@0	1956 First we create some training data using forwards sampling.
wolffd@0	1957 <pre>
wolffd@0	1958 samples = cell(N, nsamples);
wolffd@0	1959 for i=1:nsamples
wolffd@0	1960 samples(:,i) = sample_bnet(bnet);
wolffd@0	1961 end
wolffd@0	1962 </pre>
wolffd@0	1963 samples{j,i} contains the value of the j'th node in case i.
wolffd@0	1964 sample_bnet returns a cell array because, in general, each node might
wolffd@0	1965 be a vector of different length.
wolffd@0	1966 In this case, all nodes are discrete (and hence scalars), so we
wolffd@0	1967 could have used a regular array instead (which can be quicker):
wolffd@0	1968 <pre>
wolffd@0	1969 data = cell2num(samples);
wolffd@0	1970 </pre
wolffd@0	1971 So now data(j,i) = samples{j,i}.
wolffd@0	1972 <p>
wolffd@0	1973 Now we create a network with random parameters.
wolffd@0	1974 (The initial values of bnet2 don't matter in this case, since we can find the
wolffd@0	1975 globally optimal MLE independent of where we start.)
wolffd@0	1976 <pre>
wolffd@0	1977 % Make a tabula rasa
wolffd@0	1978 bnet2 = mk_bnet(dag, node_sizes);
wolffd@0	1979 seed = 0;
wolffd@0	1980 rand('state', seed);
wolffd@0	1981 bnet2.CPD{C} = tabular_CPD(bnet2, C);
wolffd@0	1982 bnet2.CPD{R} = tabular_CPD(bnet2, R);
wolffd@0	1983 bnet2.CPD{S} = tabular_CPD(bnet2, S);
wolffd@0	1984 bnet2.CPD{W} = tabular_CPD(bnet2, W);
wolffd@0	1985 </pre>
wolffd@0	1986 Finally, we find the maximum likelihood estimates of the parameters.
wolffd@0	1987 <pre>
wolffd@0	1988 bnet3 = learn_params(bnet2, samples);
wolffd@0	1989 </pre>
wolffd@0	1990 To view the learned parameters, we use a little Matlab hackery.
wolffd@0	1991 <pre>
wolffd@0	1992 CPT3 = cell(1,N);
wolffd@0	1993 for i=1:N
wolffd@0	1994 s=struct(bnet3.CPD{i}); % violate object privacy
wolffd@0	1995 CPT3{i}=s.CPT;
wolffd@0	1996 end
wolffd@0	1997 </pre>
wolffd@0	1998 Here are the parameters learned for node 4.
wolffd@0	1999 <pre>
wolffd@0	2000 dispcpt(CPT3{4})
wolffd@0	2001 1 1 : 1.0000 0.0000
wolffd@0	2002 2 1 : 0.2000 0.8000
wolffd@0	2003 1 2 : 0.2273 0.7727
wolffd@0	2004 2 2 : 0.0000 1.0000
wolffd@0	2005 </pre>
wolffd@0	2006 So we see that the learned parameters are fairly close to the "true"
wolffd@0	2007 ones, which we display below.
wolffd@0	2008 <pre>
wolffd@0	2009 dispcpt(CPT{4})
wolffd@0	2010 1 1 : 1.0000 0.0000
wolffd@0	2011 2 1 : 0.1000 0.9000
wolffd@0	2012 1 2 : 0.1000 0.9000
wolffd@0	2013 2 2 : 0.0100 0.9900
wolffd@0	2014 </pre>
wolffd@0	2015 We can get better results by using a larger training set, or using
wolffd@0	2016 informative priors (see <a href="#prior">below</a>).
wolffd@0	2017
wolffd@0	2018
wolffd@0	2019
wolffd@0	2020 <h2><a name="prior">Parameter priors</h2>
wolffd@0	2021
wolffd@0	2022 Currently, only tabular CPDs can have priors on their parameters.
wolffd@0	2023 The conjugate prior for a multinomial is the Dirichlet.
wolffd@0	2024 (For binary random variables, the multinomial is the same as the
wolffd@0	2025 Bernoulli, and the Dirichlet is the same as the Beta.)
wolffd@0	2026 <p>
wolffd@0	2027 The Dirichlet has a simple interpretation in terms of pseudo counts.
wolffd@0	2028 If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the
wolffd@0	2029 training set, where Pa_i are the parents of X_i,
wolffd@0	2030 then the maximum likelihood (ML) estimate is
wolffd@0	2031 T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0.
wolffd@0	2032 To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this
wolffd@0	2033 event was not seen in the training set,
wolffd@0	2034 we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk)
wolffd@0	2035 of times in the past.
wolffd@0	2036 The MAP (maximum a posterior) estimate is then
wolffd@0	2037 <pre>
wolffd@0	2038 T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij)
wolffd@0	2039 </pre>
wolffd@0	2040 and is never 0 if all alpha_ijk > 0.
wolffd@0	2041 For example, consider the network A->B, where A is binary and B has 3
wolffd@0	2042 values.
wolffd@0	2043 A uniform prior for B has the form
wolffd@0	2044 <pre>
wolffd@0	2045 B=1 B=2 B=3
wolffd@0	2046 A=1 1 1 1
wolffd@0	2047 A=2 1 1 1
wolffd@0	2048 </pre>
wolffd@0	2049 which can be created using
wolffd@0	2050 <pre>
wolffd@0	2051 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif');
wolffd@0	2052 </pre>
wolffd@0	2053 This prior does not satisfy the likelihood equivalence principle,
wolffd@0	2054 which says that <a href="#markov_equiv">Markov equivalent</a> models
wolffd@0	2055 should have the same marginal likelihood.
wolffd@0	2056 A prior that does satisfy this principle is shown below.
wolffd@0	2057 Heckerman (1995) calls this the
wolffd@0	2058 BDeu prior (likelihood equivalent uniform Bayesian Dirichlet).
wolffd@0	2059 <pre>
wolffd@0	2060 B=1 B=2 B=3
wolffd@0	2061 A=1 1/6 1/6 1/6
wolffd@0	2062 A=2 1/6 1/6 1/6
wolffd@0	2063 </pre>
wolffd@0	2064 where we put N/(q*r) in each bin; N is the equivalent sample size,
wolffd@0	2065 r=\|A\|, q = \|B\|.
wolffd@0	2066 This can be created as follows
wolffd@0	2067 <pre>
wolffd@0	2068 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu');
wolffd@0	2069 </pre>
wolffd@0	2070 Here, 1 is the equivalent sample size, and is the strength of the
wolffd@0	2071 prior.
wolffd@0	2072 You can change this using
wolffd@0	2073 <pre>
wolffd@0	2074 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ...
wolffd@0	2075 'BDeu', 'dirichlet_weight', 10);
wolffd@0	2076 </pre>
wolffd@0	2077 <!--where counts is an array of pseudo-counts of the same size as the
wolffd@0	2078 CPT.-->
wolffd@0	2079 <!--
wolffd@0	2080 <p>
wolffd@0	2081 When you specify a prior, you should set row i of the CPT to the
wolffd@0	2082 normalized version of row i of the pseudo-count matrix, i.e., to the
wolffd@0	2083 expected values of the parameters. This will ensure that computing the
wolffd@0	2084 marginal likelihood sequentially (see <a
wolffd@0	2085 href="#bayes_learn">below</a>) and in batch form gives the same
wolffd@0	2086 results.
wolffd@0	2087 To do this, proceed as follows.
wolffd@0	2088 <pre>
wolffd@0	2089 tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts));
wolffd@0	2090 </pre>
wolffd@0	2091 For a non-informative prior, you can just write
wolffd@0	2092 <pre>
wolffd@0	2093 tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif');
wolffd@0	2094 </pre>
wolffd@0	2095 -->
wolffd@0	2096
wolffd@0	2097
wolffd@0	2098 <h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2>
wolffd@0	2099
wolffd@0	2100 If we use conjugate priors and have fully observed data, we can
wolffd@0	2101 compute the posterior over the parameters in batch form as follows.
wolffd@0	2102 <pre>
wolffd@0	2103 cases = sample_bnet(bnet, nsamples);
wolffd@0	2104 bnet = bayes_update_params(bnet, cases);
wolffd@0	2105 LL = log_marg_lik_complete(bnet, cases);
wolffd@0	2106 </pre>
wolffd@0	2107 bnet.CPD{i}.prior contains the new Dirichlet pseudocounts,
wolffd@0	2108 and bnet.CPD{i}.CPT is set to the mean of the posterior (the
wolffd@0	2109 normalized counts).
wolffd@0	2110 (Hence if the initial pseudo counts are 0,
wolffd@0	2111 <tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the
wolffd@0	2112 same result.)
wolffd@0	2113
wolffd@0	2114
wolffd@0	2115
wolffd@0	2116
wolffd@0	2117 <p>
wolffd@0	2118 We can compute the same result sequentially (on-line) as follows.
wolffd@0	2119 <pre>
wolffd@0	2120 LL = 0;
wolffd@0	2121 for m=1:nsamples
wolffd@0	2122 LL = LL + log_marg_lik_complete(bnet, cases(:,m));
wolffd@0	2123 bnet = bayes_update_params(bnet, cases(:,m));
wolffd@0	2124 end
wolffd@0	2125 </pre>
wolffd@0	2126
wolffd@0	2127 The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of
wolffd@0	2128 sequential model selection which uses the same idea.
wolffd@0	2129 We generate data from the model A->B
wolffd@0	2130 and compute the posterior prob of all 3 dags on 2 nodes:
wolffd@0	2131 (1) A B, (2) A <- B , (3) A -> B
wolffd@0	2132 Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from
wolffd@0	2133 observational data alone, so we expect their posteriors to be the same
wolffd@0	2134 (assuming a prior which satisfies likelihood equivalence).
wolffd@0	2135 If we use random parameters, the "true" model only gets a higher posterior after 2000 trials!
wolffd@0	2136 However, if we make B a noisy NOT gate, the true model "wins" after 12
wolffd@0	2137 trials, as shown below (red = model 1, blue/green (superimposed)
wolffd@0	2138 represents models 2/3).
wolffd@0	2139 <p>
wolffd@0	2140 <img src="Figures/model_select.png">
wolffd@0	2141 <p>
wolffd@0	2142 The use of marginal likelihood for model selection is discussed in
wolffd@0	2143 greater detail in the
wolffd@0	2144 section on <a href="structure_learning">structure learning</a>.
wolffd@0	2145
wolffd@0	2146
wolffd@0	2147
wolffd@0	2148
wolffd@0	2149 <h2><a name="em">Maximum likelihood parameter estimation with missing values</h2>
wolffd@0	2150
wolffd@0	2151 Now we consider learning when some values are not observed.
wolffd@0	2152 Let us randomly hide half the values generated from the water
wolffd@0	2153 sprinkler example.
wolffd@0	2154 <pre>
wolffd@0	2155 samples2 = samples;
wolffd@0	2156 hide = rand(N, nsamples) > 0.5;
wolffd@0	2157 [I,J]=find(hide);
wolffd@0	2158 for k=1:length(I)
wolffd@0	2159 samples2{I(k), J(k)} = [];
wolffd@0	2160 end
wolffd@0	2161 </pre>
wolffd@0	2162 samples2{i,l} is the value of node i in training case l, or [] if unobserved.
wolffd@0	2163 <p>
wolffd@0	2164 Now we will compute the MLEs using the EM algorithm.
wolffd@0	2165 We need to use an inference algorithm to compute the expected
wolffd@0	2166 sufficient statistics in the E step; the M (maximization) step is as
wolffd@0	2167 above.
wolffd@0	2168 <pre>
wolffd@0	2169 engine2 = jtree_inf_engine(bnet2);
wolffd@0	2170 max_iter = 10;
wolffd@0	2171 [bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter);
wolffd@0	2172 </pre>
wolffd@0	2173 LLtrace(i) is the log-likelihood at iteration i. We can plot this as
wolffd@0	2174 follows:
wolffd@0	2175 <pre>
wolffd@0	2176 plot(LLtrace, 'x-')
wolffd@0	2177 </pre>
wolffd@0	2178 Let's display the results after 10 iterations of EM.
wolffd@0	2179 <pre>
wolffd@0	2180 celldisp(CPT4)
wolffd@0	2181 CPT4{1} =
wolffd@0	2182 0.6616
wolffd@0	2183 0.3384
wolffd@0	2184 CPT4{2} =
wolffd@0	2185 0.6510 0.3490
wolffd@0	2186 0.8751 0.1249
wolffd@0	2187 CPT4{3} =
wolffd@0	2188 0.8366 0.1634
wolffd@0	2189 0.0197 0.9803
wolffd@0	2190 CPT4{4} =
wolffd@0	2191 (:,:,1) =
wolffd@0	2192 0.8276 0.0546
wolffd@0	2193 0.5452 0.1658
wolffd@0	2194 (:,:,2) =
wolffd@0	2195 0.1724 0.9454
wolffd@0	2196 0.4548 0.8342
wolffd@0	2197 </pre>
wolffd@0	2198 We can get improved performance by using one or more of the following
wolffd@0	2199 methods:
wolffd@0	2200 <ul>
wolffd@0	2201 <li> Increasing the size of the training set.
wolffd@0	2202 <li> Decreasing the amount of hidden data.
wolffd@0	2203 <li> Running EM for longer.
wolffd@0	2204 <li> Using informative priors.
wolffd@0	2205 <li> Initialising EM from multiple starting points.
wolffd@0	2206 </ul>
wolffd@0	2207
wolffd@0	2208 <h2><a name="gaussianNumerical">Numerical problems when fitting
wolffd@0	2209 Gaussians using EM</h2>
wolffd@0	2210
wolffd@0	2211 When fitting a <a href="#gaussian">Gaussian CPD</a> using EM, you
wolffd@0	2212 might encounter some numerical
wolffd@0	2213 problems. See <a
wolffd@0	2214 href="http://www.ai.mit.edu/~murphyk/Software/HMM/hmm.html#loglikpos">here</a>
wolffd@0	2215 for a discussion of this in the context of HMMs.
wolffd@0	2216 <!--
wolffd@0	2217 (Note: BNT does not currently support spherical covariances, although it
wolffd@0	2218 would be easy to add, since KPMstats/clg_Mstep supports this option;
wolffd@0	2219 you would just need to modify gaussian_CPD/update_ess to accumulate
wolffd@0	2220 weighted inner products.)
wolffd@0	2221 -->
wolffd@0	2222
wolffd@0	2223
wolffd@0	2224 <h2><a name="tying">Parameter tying</h2>
wolffd@0	2225
wolffd@0	2226 In networks with repeated structure (e.g., chains and grids), it is
wolffd@0	2227 common to assume that the parameters are the same at every node. This
wolffd@0	2228 is called parameter tying, and reduces the amount of data needed for
wolffd@0	2229 learning.
wolffd@0	2230 <p>
wolffd@0	2231 When we have tied parameters, there is no longer a one-to-one
wolffd@0	2232 correspondence between nodes and CPDs.
wolffd@0	2233 Rather, each CPD species the parameters for a whole equivalence class
wolffd@0	2234 of nodes.
wolffd@0	2235 It is easiest to see this by example.
wolffd@0	2236 Consider the following <a href="usage_dbn.html#hmm">hidden Markov
wolffd@0	2237 model (HMM)</a>
wolffd@0	2238 <p>
wolffd@0	2239 <img src="Figures/hmm3.gif">
wolffd@0	2240 <p>
wolffd@0	2241 <!--
wolffd@0	2242 We can create this graph structure, assuming we have T time-slices,
wolffd@0	2243 as follows.
wolffd@0	2244 (We number the nodes as shown in the figure, but we could equally well
wolffd@0	2245 number the hidden nodes 1:T, and the observed nodes T+1:2T.)
wolffd@0	2246 <pre>
wolffd@0	2247 N = 2*T;
wolffd@0	2248 dag = zeros(N);
wolffd@0	2249 hnodes = 1:2:2*T;
wolffd@0	2250 for i=1:T-1
wolffd@0	2251 dag(hnodes(i), hnodes(i+1))=1;
wolffd@0	2252 end
wolffd@0	2253 onodes = 2:2:2*T;
wolffd@0	2254 for i=1:T
wolffd@0	2255 dag(hnodes(i), onodes(i)) = 1;
wolffd@0	2256 end
wolffd@0	2257 </pre>
wolffd@0	2258 <p>
wolffd@0	2259 The hidden nodes are always discrete, and have Q possible values each,
wolffd@0	2260 but the observed nodes can be discrete or continuous, and have O possible values/length.
wolffd@0	2261 <pre>
wolffd@0	2262 if cts_obs
wolffd@0	2263 dnodes = hnodes;
wolffd@0	2264 else
wolffd@0	2265 dnodes = 1:N;
wolffd@0	2266 end
wolffd@0	2267 ns = ones(1,N);
wolffd@0	2268 ns(hnodes) = Q;
wolffd@0	2269 ns(onodes) = O;
wolffd@0	2270 </pre>
wolffd@0	2271 -->
wolffd@0	2272 When HMMs are used for semi-infinite processes like speech recognition,
wolffd@0	2273 we assume the transition matrix
wolffd@0	2274 P(H(t+1)\|H(t)) is the same for all t; this is called a time-invariant
wolffd@0	2275 or homogenous Markov chain.
wolffd@0	2276 Hence hidden nodes 2, 3, ..., T
wolffd@0	2277 are all in the same equivalence class, say class Hclass.
wolffd@0	2278 Similarly, the observation matrix P(O(t)\|H(t)) is assumed to be the
wolffd@0	2279 same for all t, so the observed nodes are all in the same equivalence
wolffd@0	2280 class, say class Oclass.
wolffd@0	2281 Finally, the prior term P(H(1)) is in a class all by itself, say class
wolffd@0	2282 H1class.
wolffd@0	2283 This is illustrated below, where we explicitly represent the
wolffd@0	2284 parameters as random variables (dotted nodes).
wolffd@0	2285 <p>
wolffd@0	2286 <img src="Figures/hmm4_params.gif">
wolffd@0	2287 <p>
wolffd@0	2288 In BNT, we cannot represent parameters as random variables (nodes).
wolffd@0	2289 Instead, we "hide" the
wolffd@0	2290 parameters inside one CPD for each equivalence class,
wolffd@0	2291 and then specify that the other CPDs should share these parameters, as
wolffd@0	2292 follows.
wolffd@0	2293 <pre>
wolffd@0	2294 hnodes = 1:2:2*T;
wolffd@0	2295 onodes = 2:2:2*T;
wolffd@0	2296 H1class = 1; Hclass = 2; Oclass = 3;
wolffd@0	2297 eclass = ones(1,N);
wolffd@0	2298 eclass(hnodes(2:end)) = Hclass;
wolffd@0	2299 eclass(hnodes(1)) = H1class;
wolffd@0	2300 eclass(onodes) = Oclass;
wolffd@0	2301 % create dag and ns in the usual way
wolffd@0	2302 bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass);
wolffd@0	2303 </pre>
wolffd@0	2304 Finally, we define the parameters for each equivalence class:
wolffd@0	2305 <pre>
wolffd@0	2306 bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior
wolffd@0	2307 bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix
wolffd@0	2308 if cts_obs
wolffd@0	2309 bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1));
wolffd@0	2310 else
wolffd@0	2311 bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1));
wolffd@0	2312 end
wolffd@0	2313 </pre>
wolffd@0	2314 In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a
wolffd@0	2315 member of e's equivalence class; that is, it is not always the case
wolffd@0	2316 that e == j. You can use bnet.rep_of_eclass(e) to return the
wolffd@0	2317 representative of equivalence class e.
wolffd@0	2318 BNT will look up the parents of j to determine the size
wolffd@0	2319 of the CPT to use. It assumes that this is the same for all members of
wolffd@0	2320 the equivalence class.
wolffd@0	2321 Click <a href="http://www.cs.berkeley.edu/~murphyk/Bayes/param_tieing.html">here</a> for
wolffd@0	2322 a more complex example of parameter tying.
wolffd@0	2323 <p>
wolffd@0	2324 Note:
wolffd@0	2325 Normally one would define an HMM as a
wolffd@0	2326 <a href = "usage_dbn.html">Dynamic Bayes Net</a>
wolffd@0	2327 (see the function BNT/examples/dynamic/mk_chmm.m).
wolffd@0	2328 However, one can define an HMM as a static BN using the function
wolffd@0	2329 BNT/examples/static/Models/mk_hmm_bnet.m.
wolffd@0	2330
wolffd@0	2331
wolffd@0	2332
wolffd@0	2333 <h1><a name="structure_learning">Structure learning</h1>
wolffd@0	2334
wolffd@0	2335 Update (9/29/03):
wolffd@0	2336 Phillipe LeRay is developing some additional structure learning code
wolffd@0	2337 on top of BNT. Click <a
wolffd@0	2338 href="http://bnt.insa-rouen.fr/ajouts.html">here</a>
wolffd@0	2339 for details.
wolffd@0	2340
wolffd@0	2341 <p>
wolffd@0	2342
wolffd@0	2343 There are two very different approaches to structure learning:
wolffd@0	2344 constraint-based and search-and-score.
wolffd@0	2345 In the <a href="#constraint">constraint-based approach</a>,
wolffd@0	2346 we start with a fully connected graph, and remove edges if certain
wolffd@0	2347 conditional independencies are measured in the data.
wolffd@0	2348 This has the disadvantage that repeated independence tests lose
wolffd@0	2349 statistical power.
wolffd@0	2350 <p>
wolffd@0	2351 In the more popular search-and-score approach,
wolffd@0	2352 we perform a search through the space of possible DAGs, and either
wolffd@0	2353 return the best one found (a point estimate), or return a sample of the
wolffd@0	2354 models found (an approximation to the Bayesian posterior).
wolffd@0	2355 <p>
wolffd@0	2356 Unfortunately, the number of DAGs as a function of the number of
wolffd@0	2357 nodes, G(n), is super-exponential in n.
wolffd@0	2358 A closed form formula for G(n) is not known, but the first few values
wolffd@0	2359 are shown below (from Cooper, 1999).
wolffd@0	2360
wolffd@0	2361 <table>
wolffd@0	2362 <tr> <th>n</th> <th align=left>G(n)</th> </tr>
wolffd@0	2363 <tr> <td>1</td> <td>1</td> </tr>
wolffd@0	2364 <tr> <td>2</td> <td>3</td> </tr>
wolffd@0	2365 <tr> <td>3</td> <td>25</td> </tr>
wolffd@0	2366 <tr> <td>4</td> <td>543</td> </tr>
wolffd@0	2367 <tr> <td>5</td> <td>29,281</td> </tr>
wolffd@0	2368 <tr> <td>6</td> <td>3,781,503</td> </tr>
wolffd@0	2369 <tr> <td>7</td> <td>1.1 x 10^9</td> </tr>
wolffd@0	2370 <tr> <td>8</td> <td>7.8 x 10^11</td> </tr>
wolffd@0	2371 <tr> <td>9</td> <td>1.2 x 10^15</td> </tr>
wolffd@0	2372 <tr> <td>10</td> <td>4.2 x 10^18</td> </tr>
wolffd@0	2373 </table>
wolffd@0	2374
wolffd@0	2375 Since the number of DAGs is super-exponential in the number of nodes,
wolffd@0	2376 we cannot exhaustively search the space, so we either use a local
wolffd@0	2377 search algorithm (e.g., greedy hill climbining, perhaps with multiple
wolffd@0	2378 restarts) or a global search algorithm (e.g., Markov Chain Monte
wolffd@0	2379 Carlo).
wolffd@0	2380 <p>
wolffd@0	2381 If we know a total ordering on the nodes,
wolffd@0	2382 finding the best structure amounts to picking the best set of parents
wolffd@0	2383 for each node independently.
wolffd@0	2384 This is what the K2 algorithm does.
wolffd@0	2385 If the ordering is unknown, we can search over orderings,
wolffd@0	2386 which is more efficient than searching over DAGs (Koller and Friedman, 2000).
wolffd@0	2387 <p>
wolffd@0	2388 In addition to the search procedure, we must specify the scoring
wolffd@0	2389 function. There are two popular choices. The Bayesian score integrates
wolffd@0	2390 out the parameters, i.e., it is the marginal likelihood of the model.
wolffd@0	2391 The BIC (Bayesian Information Criterion) is defined as
wolffd@0	2392 log P(D\|theta_hat) - 0.5dlog(N), where D is the data, theta_hat is
wolffd@0	2393 the ML estimate of the parameters, d is the number of parameters, and
wolffd@0	2394 N is the number of data cases.
wolffd@0	2395 The BIC method has the advantage of not requiring a prior.
wolffd@0	2396 <p>
wolffd@0	2397 BIC can be derived as a large sample
wolffd@0	2398 approximation to the marginal likelihood.
wolffd@0	2399 (It is also equal to the Minimum Description Length of a model.)
wolffd@0	2400 However, in practice, the sample size does not need to be very large
wolffd@0	2401 for the approximation to be good.
wolffd@0	2402 For example, in the figure below, we plot the ratio between the log marginal likelihood
wolffd@0	2403 and the BIC score against data-set size; we see that the ratio rapidly
wolffd@0	2404 approaches 1, especially for non-informative priors.
wolffd@0	2405 (This plot was generated by the file BNT/examples/static/bic1.m. It
wolffd@0	2406 uses the water sprinkler BN with BDeu Dirichlet priors with different
wolffd@0	2407 equivalent sample sizes.)
wolffd@0	2408
wolffd@0	2409 <p>
wolffd@0	2410 <center>
wolffd@0	2411 <IMG SRC="Figures/bic.png">
wolffd@0	2412 </center>
wolffd@0	2413 <p>
wolffd@0	2414
wolffd@0	2415 <p>
wolffd@0	2416 As with parameter learning, handling missing data/ hidden variables is
wolffd@0	2417 much harder than the fully observed case.
wolffd@0	2418 The structure learning routines in BNT can therefore be classified into 4
wolffd@0	2419 types, analogously to the parameter learning case.
wolffd@0	2420 <p>
wolffd@0	2421
wolffd@0	2422 <TABLE BORDER>
wolffd@0	2423 <tr>
wolffd@0	2424 <TH></TH>
wolffd@0	2425 <th>Full obs</th>
wolffd@0	2426 <th>Partial obs</th>
wolffd@0	2427 </tr>
wolffd@0	2428 <tr>
wolffd@0	2429 <th>Point</th>
wolffd@0	2430 <td><tt>learn_struct_K2</tt> <br>
wolffd@0	2431 <!-- <tt>learn_struct_hill_climb</tt></td> -->
wolffd@0	2432 <td><tt>not yet supported</tt></td>
wolffd@0	2433 </tr>
wolffd@0	2434 <tr>
wolffd@0	2435 <th>Bayes</th>
wolffd@0	2436 <td><tt>learn_struct_mcmc</tt></td>
wolffd@0	2437 <td>not yet supported</td>
wolffd@0	2438 </tr>
wolffd@0	2439 </table>
wolffd@0	2440
wolffd@0	2441
wolffd@0	2442 <h2><a name="markov_equiv">Markov equivalence</h2>
wolffd@0	2443
wolffd@0	2444 If two DAGs encode the same conditional independencies, they are
wolffd@0	2445 called Markov equivalent. The set of all DAGs can be paritioned into
wolffd@0	2446 Markov equivalence classes. Graphs within the same class can
wolffd@0	2447 have
wolffd@0	2448 the direction of some of their arcs reversed without changing any of
wolffd@0	2449 the CI relationships.
wolffd@0	2450 Each class can be represented by a PDAG
wolffd@0	2451 (partially directed acyclic graph) called an essential graph or
wolffd@0	2452 pattern. This specifies which edges must be oriented in a certain
wolffd@0	2453 direction, and which may be reversed.
wolffd@0	2454
wolffd@0	2455 <p>
wolffd@0	2456 When learning graph structure from observational data,
wolffd@0	2457 the best one can hope to do is to identify the model up to Markov
wolffd@0	2458 equivalence. To distinguish amongst graphs within the same equivalence
wolffd@0	2459 class, one needs interventional data: see the discussion on <a
wolffd@0	2460 href="#active">active learning</a> below.
wolffd@0	2461
wolffd@0	2462
wolffd@0	2463
wolffd@0	2464 <h2><a name="enumerate">Exhaustive search</h2>
wolffd@0	2465
wolffd@0	2466 The brute-force approach to structure learning is to enumerate all
wolffd@0	2467 possible DAGs, and score each one. This provides a "gold standard"
wolffd@0	2468 with which to compare other algorithms. We can do this as follows.
wolffd@0	2469 <pre>
wolffd@0	2470 dags = mk_all_dags(N);
wolffd@0	2471 score = score_dags(data, ns, dags);
wolffd@0	2472 </pre>
wolffd@0	2473 where data(i,m) is the value of node i in case m,
wolffd@0	2474 and ns(i) is the size of node i.
wolffd@0	2475 If the DAGs have a lot of families in common, we can cache the sufficient statistics,
wolffd@0	2476 making this potentially more efficient than scoring the DAGs one at a time.
wolffd@0	2477 (Caching is not currently implemented, however.)
wolffd@0	2478 <p>
wolffd@0	2479 By default, we use the Bayesian scoring metric, and assume CPDs are
wolffd@0	2480 represented by tables with BDeu(1) priors.
wolffd@0	2481 We can override these defaults as follows.
wolffd@0	2482 If we want to use uniform priors, we can say
wolffd@0	2483 <pre>
wolffd@0	2484 params = cell(1,N);
wolffd@0	2485 for i=1:N
wolffd@0	2486 params{i} = {'prior', 'unif'};
wolffd@0	2487 end
wolffd@0	2488 score = score_dags(data, ns, dags, 'params', params);
wolffd@0	2489 </pre>
wolffd@0	2490 params{i} is a cell-array, containing optional arguments that are
wolffd@0	2491 passed to the constructor for CPD i.
wolffd@0	2492 <p>
wolffd@0	2493 Now suppose we want to use different node types, e.g.,
wolffd@0	2494 Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both
wolffd@0	2495 these CPDs can support discrete and continuous parents, which is
wolffd@0	2496 necessary since all other nodes will be considered as parents).
wolffd@0	2497 The Bayesian scoring metric currently only works for tabular CPDs, so
wolffd@0	2498 we will use BIC:
wolffd@0	2499 <pre>
wolffd@0	2500 score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [],
wolffd@0	2501 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic')
wolffd@0	2502 </pre>
wolffd@0	2503 In practice, one can't enumerate all possible DAGs for N > 5,
wolffd@0	2504 but one can evaluate any reasonably-sized set of hypotheses in this
wolffd@0	2505 way (e.g., nearest neighbors of your current best guess).
wolffd@0	2506 Think of this as "computer assisted model refinement" as opposed to de
wolffd@0	2507 novo learning.
wolffd@0	2508
wolffd@0	2509
wolffd@0	2510 <h2><a name="K2">K2</h2>
wolffd@0	2511
wolffd@0	2512 The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows.
wolffd@0	2513 Initially each node has no parents. It then adds incrementally that parent whose addition most
wolffd@0	2514 increases the score of the resulting structure. When the addition of no single
wolffd@0	2515 parent can increase the score, it stops adding parents to the node.
wolffd@0	2516 Since we are using a fixed ordering, we do not need to check for
wolffd@0	2517 cycles, and can choose the parents for each node independently.
wolffd@0	2518 <p>
wolffd@0	2519 The original paper used the Bayesian scoring
wolffd@0	2520 metric with tabular CPDs and Dirichlet priors.
wolffd@0	2521 BNT generalizes this to allow any kind of CPD, and either the Bayesian
wolffd@0	2522 scoring metric or BIC, as in the example <a href="#enumerate">above</a>.
wolffd@0	2523 In addition, you can specify
wolffd@0	2524 an optional upper bound on the number of parents for each node.
wolffd@0	2525 The file BNT/examples/static/k2demo1.m gives an example of how to use K2.
wolffd@0	2526 We use the water sprinkler network and sample 100 cases from it as before.
wolffd@0	2527 Then we see how much data it takes to recover the generating structure:
wolffd@0	2528 <pre>
wolffd@0	2529 order = [C S R W];
wolffd@0	2530 max_fan_in = 2;
wolffd@0	2531 sz = 5:5:100;
wolffd@0	2532 for i=1:length(sz)
wolffd@0	2533 dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in);
wolffd@0	2534 correct(i) = isequal(dag, dag2);
wolffd@0	2535 end
wolffd@0	2536 </pre>
wolffd@0	2537 Here are the results.
wolffd@0	2538 <pre>
wolffd@0	2539 correct =
wolffd@0	2540 Columns 1 through 12
wolffd@0	2541 0 0 0 0 0 0 0 1 0 1 1 1
wolffd@0	2542 Columns 13 through 20
wolffd@0	2543 1 1 1 1 1 1 1 1
wolffd@0	2544 </pre>
wolffd@0	2545 So we see it takes about sz(10)=50 cases. (BIC behaves similarly,
wolffd@0	2546 showing that the prior doesn't matter too much.)
wolffd@0	2547 In general, we cannot hope to recover the "true" generating structure,
wolffd@0	2548 only one that is in its <a href="#markov_equiv">Markov equivalence
wolffd@0	2549 class</a>.
wolffd@0	2550
wolffd@0	2551
wolffd@0	2552 <h2><a name="hill_climb">Hill-climbing</h2>
wolffd@0	2553
wolffd@0	2554 Hill-climbing starts at a specific point in space,
wolffd@0	2555 considers all nearest neighbors, and moves to the neighbor
wolffd@0	2556 that has the highest score; if no neighbors have higher
wolffd@0	2557 score than the current point (i.e., we have reached a local maximum),
wolffd@0	2558 the algorithm stops. One can then restart in another part of the space.
wolffd@0	2559 <p>
wolffd@0	2560 A common definition of "neighbor" is all graphs that can be
wolffd@0	2561 generated from the current graph by adding, deleting or reversing a
wolffd@0	2562 single arc, subject to the acyclicity constraint.
wolffd@0	2563 Other neighborhoods are possible: see
wolffd@0	2564 <a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf">
wolffd@0	2565 Optimal Structure Identification with Greedy Search</a>, Max
wolffd@0	2566 Chickering, JMLR 2002.
wolffd@0	2567
wolffd@0	2568 <!--
wolffd@0	2569 Note: This algorithm is currently (Feb '02) being implemented by Qian
wolffd@0	2570 Diao.
wolffd@0	2571 -->
wolffd@0	2572
wolffd@0	2573
wolffd@0	2574 <h2><a name="mcmc">MCMC</h2>
wolffd@0	2575
wolffd@0	2576 We can use a Markov Chain Monte Carlo (MCMC) algorithm called
wolffd@0	2577 Metropolis-Hastings (MH) to search the space of all
wolffd@0	2578 DAGs.
wolffd@0	2579 The standard proposal distribution is to consider moving to all
wolffd@0	2580 nearest neighbors in the sense defined <a href="#hill_climb">above</a>.
wolffd@0	2581 <p>
wolffd@0	2582 The function can be called
wolffd@0	2583 as in the following example.
wolffd@0	2584 <pre>
wolffd@0	2585 [sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10);
wolffd@0	2586 </pre>
wolffd@0	2587 We can convert our set of sampled graphs to a histogram
wolffd@0	2588 (empirical posterior over all the DAGs) thus
wolffd@0	2589 <pre>
wolffd@0	2590 all_dags = mk_all_dags(N);
wolffd@0	2591 mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags);
wolffd@0	2592 </pre>
wolffd@0	2593 To see how well this performs, let us compute the exact posterior exhaustively.
wolffd@0	2594 <p>
wolffd@0	2595 <pre>
wolffd@0	2596 score = score_dags(data, ns, all_dags);
wolffd@0	2597 post = normalise(exp(score)); % assuming uniform structural prior
wolffd@0	2598 </pre>
wolffd@0	2599 We plot the results below.
wolffd@0	2600 (The data set was 100 samples drawn from a random 4 node bnet; see the
wolffd@0	2601 file BNT/examples/static/mcmc1.)
wolffd@0	2602 <pre>
wolffd@0	2603 subplot(2,1,1)
wolffd@0	2604 bar(post)
wolffd@0	2605 subplot(2,1,2)
wolffd@0	2606 bar(mcmc_post)
wolffd@0	2607 </pre>
wolffd@0	2608 <img src="Figures/mcmc_post.jpg" width="800" height="500">
wolffd@0	2609 <p>
wolffd@0	2610 We can also plot the acceptance ratio versus number of MCMC steps,
wolffd@0	2611 as a crude convergence diagnostic.
wolffd@0	2612 <pre>
wolffd@0	2613 clf
wolffd@0	2614 plot(accept_ratio)
wolffd@0	2615 </pre>
wolffd@0	2616 <img src="Figures/mcmc_accept.jpg" width="800" height="300">
wolffd@0	2617 <p>
wolffd@0	2618 Better convergence diagnostics can be found
wolffd@0	2619 <a href="<a href="http://www.lce.hut.fi/~ave/code/mcmcdiag/">here</a>
wolffd@0	2620 <p>
wolffd@0	2621 Even though the number of samples needed by MCMC is theoretically
wolffd@0	2622 polynomial (not exponential) in the dimensionality of the search space, in practice it has been
wolffd@0	2623 found that MCMC does not converge in reasonable time for graphs with
wolffd@0	2624 more than about 10 nodes.
wolffd@0	2625 <p>
wolffd@0	2626 <a href="http://www.bioss.sari.ac.uk/~dirk/software/DBmcmc/">
wolffd@0	2627 Dirk Husmeier has extended MCMC model selection to DBNs</a>.
wolffd@0	2628
wolffd@0	2629
wolffd@0	2630 <h2><a name="active">Active structure learning</h2>
wolffd@0	2631
wolffd@0	2632 As was mentioned <a href="#markov_equiv">above</a>,
wolffd@0	2633 one can only learn a DAG up to Markov equivalence, even given infinite data.
wolffd@0	2634 If one is interested in learning the structure of a causal network,
wolffd@0	2635 one needs interventional data.
wolffd@0	2636 (By "intervention" we mean forcing a node to take on a specific value,
wolffd@0	2637 thereby effectively severing its incoming arcs.)
wolffd@0	2638 <p>
wolffd@0	2639 Most of the scoring functions accept an optional argument
wolffd@0	2640 that specifies whether a node was observed to have a certain value, or
wolffd@0	2641 was forced to have that value: we set clamped(i,m)=1 if node i was
wolffd@0	2642 forced in training case m. e.g., see the file
wolffd@0	2643 BNT/examples/static/cooper_yoo.
wolffd@0	2644 <p>
wolffd@0	2645 An interesting question is to decide which interventions to perform
wolffd@0	2646 (c.f., design of experiments). For details, see the following tech
wolffd@0	2647 report
wolffd@0	2648 <ul>
wolffd@0	2649 <li> <a href = "../../Papers/alearn.ps.gz">
wolffd@0	2650 Active learning of causal Bayes net structure</a>, Kevin Murphy, March
wolffd@0	2651 2001.
wolffd@0	2652 </ul>
wolffd@0	2653
wolffd@0	2654
wolffd@0	2655 <h2><a name="struct_em">Structural EM</h2>
wolffd@0	2656
wolffd@0	2657 Computing the Bayesian score when there is partial observability is
wolffd@0	2658 computationally challenging, because the parameter posterior becomes
wolffd@0	2659 multimodal (the hidden nodes induce a mixture distribution).
wolffd@0	2660 One therefore needs to use approximations such as BIC.
wolffd@0	2661 Unfortunately, search algorithms are still expensive, because we need
wolffd@0	2662 to run EM at each step to compute the MLE, which is needed to compute
wolffd@0	2663 the score of each model. An alternative approach is
wolffd@0	2664 to do the local search steps inside of the M step of EM, which is more
wolffd@0	2665 efficient since the data has been "filled in" - this is
wolffd@0	2666 called the structural EM algorithm (Friedman 1997), and provably
wolffd@0	2667 converges to a local maximum of the BIC score.
wolffd@0	2668 <p>
wolffd@0	2669 Wei Hu has implemented SEM for discrete nodes.
wolffd@0	2670 You can download his package from
wolffd@0	2671 <a href="../SEM.zip">here</a>.
wolffd@0	2672 Please address all questions about this code to
wolffd@0	2673 wei.hu@intel.com.
wolffd@0	2674 See also <a href="#phl">Phl's implementation of SEM</a>.
wolffd@0	2675
wolffd@0	2676 <!--
wolffd@0	2677 <h2><a name="reveal">REVEAL algorithm</h2>
wolffd@0	2678
wolffd@0	2679 A simple way to learn the structure of a fully observed, discrete,
wolffd@0	2680 factored DBN from a time series is described <a
wolffd@0	2681 href="usage_dbn.html#struct_learn">here</a>.
wolffd@0	2682 -->
wolffd@0	2683
wolffd@0	2684
wolffd@0	2685 <h2><a name="graphdraw">Visualizing the graph</h2>
wolffd@0	2686
wolffd@0	2687 You can visualize an arbitrary graph (such as one learned using the
wolffd@0	2688 structure learning routines) with Matlab code contributed by
wolffd@0	2689 <a href="http://www.mbfys.kun.nl/~cemgil/matlab/layout.html">Ali
wolffd@0	2690 Taylan Cemgil</a>
wolffd@0	2691 from the University of Nijmegen.
wolffd@0	2692 For static BNs, call it as follows:
wolffd@0	2693 <pre>
wolffd@0	2694 draw_graph(bnet.dag);
wolffd@0	2695 </pre>
wolffd@0	2696 For example, this is the output produced on a
wolffd@0	2697 <a href="#qmr">random QMR-like model</a>:
wolffd@0	2698 <p>
wolffd@0	2699 <img src="Figures/qmr.rnd.jpg">
wolffd@0	2700 <p>
wolffd@0	2701 If you install the excellent <a
wolffd@0	2702 href="http://www.research.att.com/sw/tools/graphviz">graphhviz</a>, an
wolffd@0	2703 open-source graph visualization package from AT&T,
wolffd@0	2704 you can create a much better visualization as follows
wolffd@0	2705 <pre>
wolffd@0	2706 graph_to_dot(bnet.dag)
wolffd@0	2707 </pre>
wolffd@0	2708 This works by converting the adjacency matrix to a file suitable
wolffd@0	2709 for input to graphviz (using the dot format),
wolffd@0	2710 then converting the output of graphviz to postscript, and displaying the results using
wolffd@0	2711 ghostview.
wolffd@0	2712 You can do each of these steps separately for more control, as shown
wolffd@0	2713 below.
wolffd@0	2714 <pre>
wolffd@0	2715 graph_to_dot(bnet.dag, 'filename', 'foo.dot');
wolffd@0	2716 dot -Tps foo.dot -o foo.ps
wolffd@0	2717 ghostview foo.ps &
wolffd@0	2718 </pre>
wolffd@0	2719
wolffd@0	2720 <h2><a name = "constraint">Constraint-based methods</h2>
wolffd@0	2721
wolffd@0	2722 The IC algorithm (Pearl and Verma, 1991),
wolffd@0	2723 and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993),
wolffd@0	2724 computes many conditional independence tests,
wolffd@0	2725 and combines these constraints into a
wolffd@0	2726 PDAG to represent the whole
wolffd@0	2727 <a href="#markov_equiv">Markov equivalence class</a>.
wolffd@0	2728 <p>
wolffd@0	2729 IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>.
wolffd@0	2730 (IC stands for inductive causation; PC stands for Peter and Clark,
wolffd@0	2731 the first names of Spirtes and Glymour; FCI stands for fast causal
wolffd@0	2732 inference.
wolffd@0	2733 What we, following Pearl (2000), call IC* was called
wolffd@0	2734 IC in the original Pearl and Verma paper.)
wolffd@0	2735 For details, see
wolffd@0	2736 <ul>
wolffd@0	2737 <li>
wolffd@0	2738 <a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation,
wolffd@0	2739 Prediction, and Search</a>, Spirtes, Glymour and
wolffd@0	2740 Scheines (SGS), 2001 (2nd edition), MIT Press.
wolffd@0	2741 <li>
wolffd@0	2742 <a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl,
wolffd@0	2743 2000, Cambridge University Press.
wolffd@0	2744 </ul>
wolffd@0	2745
wolffd@0	2746 <p>
wolffd@0	2747
wolffd@0	2748 The PC algorithm takes as arguments a function f, the number of nodes N,
wolffd@0	2749 the maximum fan in K, and additional arguments A which are passed to f.
wolffd@0	2750 The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0
wolffd@0	2751 otherwise.
wolffd@0	2752 For example, suppose we cheat by
wolffd@0	2753 passing in a CI "oracle" which has access to the true DAG; the oracle
wolffd@0	2754 tests for d-separation in this DAG, i.e.,
wolffd@0	2755 f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows.
wolffd@0	2756 <pre>
wolffd@0	2757 pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag);
wolffd@0	2758 </pre>
wolffd@0	2759 pdag(i,j) = -1 if there is definitely an i->j arc,
wolffd@0	2760 and pdag(i,j) = 1 if there is either an i->j or and i<-j arc.
wolffd@0	2761 <p>
wolffd@0	2762 Applied to the sprinkler network, this returns
wolffd@0	2763 <pre>
wolffd@0	2764 pdag =
wolffd@0	2765 0 1 1 0
wolffd@0	2766 1 0 0 -1
wolffd@0	2767 1 0 0 -1
wolffd@0	2768 0 0 0 0
wolffd@0	2769 </pre>
wolffd@0	2770 So as expected, we see that the V-structure at the W node is uniquely identified,
wolffd@0	2771 but the other arcs have ambiguous orientation.
wolffd@0	2772 <p>
wolffd@0	2773 We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS
wolffd@0	2774 book.
wolffd@0	2775 This example concerns the female orgasm.
wolffd@0	2776 We are given a correlation matrix C between 7 measured factors (such
wolffd@0	2777 as subjective experiences of coital and masturbatory experiences),
wolffd@0	2778 derived from 281 samples, and want to learn a causal model of the
wolffd@0	2779 data. We will not discuss the merits of this type of work here, but
wolffd@0	2780 merely show how to reproduce the results in the SGS book.
wolffd@0	2781 Their program,
wolffd@0	2782 <a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>,
wolffd@0	2783 makes use of the Fisher Z-test for conditional
wolffd@0	2784 independence, so we do the same:
wolffd@0	2785 <pre>
wolffd@0	2786 max_fan_in = 4;
wolffd@0	2787 nsamples = 281;
wolffd@0	2788 alpha = 0.05;
wolffd@0	2789 pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha);
wolffd@0	2790 </pre>
wolffd@0	2791 In this case, the CI test is
wolffd@0	2792 <pre>
wolffd@0	2793 f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha)
wolffd@0	2794 </pre>
wolffd@0	2795 The results match those of Fig 12a of SGS apart from two edge
wolffd@0	2796 differences; presumably this is due to rounding error (although it
wolffd@0	2797 could be a bug, either in BNT or in Tetrad).
wolffd@0	2798 This example can be found in the file BNT/examples/static/pc2.m.
wolffd@0	2799
wolffd@0	2800 <p>
wolffd@0	2801
wolffd@0	2802 The IC* algorithm (Pearl and Verma, 1991),
wolffd@0	2803 and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993),
wolffd@0	2804 are like the IC/PC algorithm, except that they can detect the presence
wolffd@0	2805 of latent variables.
wolffd@0	2806 See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar
wolffd@0	2807 Kushnir. The output is a matrix P, defined as follows
wolffd@0	2808 (see Pearl (2000), p52 for details):
wolffd@0	2809 <pre>
wolffd@0	2810 % P(i,j) = -1 if there is either a latent variable L such that i <-L->j OR there is a directed edge from i->j.
wolffd@0	2811 % P(i,j) = -2 if there is a marked directed i-*>j edge.
wolffd@0	2812 % P(i,j) = P(j,i) = 1 if there is and undirected edge i--j
wolffd@0	2813 % P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j.
wolffd@0	2814 </pre>
wolffd@0	2815
wolffd@0	2816
wolffd@0	2817 <h2><a name="phl">Philippe Leray's structure learning package</h2>
wolffd@0	2818
wolffd@0	2819 Philippe Leray has written a
wolffd@0	2820 <a href="http://bnt.insa-rouen.fr/ajouts.html">
wolffd@0	2821 structure learning package</a> that uses BNT.
wolffd@0	2822
wolffd@0	2823 It currently (Juen 2003) has the following features:
wolffd@0	2824 <ul>
wolffd@0	2825 <li>PC with Chi2 statistical test
wolffd@0	2826 <li> MWST : Maximum weighted Spanning Tree
wolffd@0	2827 <li> Hill Climbing
wolffd@0	2828 <li> Greedy Search
wolffd@0	2829 <li> Structural EM
wolffd@0	2830 <li> hist_ic : optimal Histogram based on IC information criterion
wolffd@0	2831 <li> cpdag_to_dag
wolffd@0	2832 <li> dag_to_cpdag
wolffd@0	2833 <li> ...
wolffd@0	2834 </ul>
wolffd@0	2835
wolffd@0	2836
wolffd@0	2837 </a>
wolffd@0	2838
wolffd@0	2839
wolffd@0	2840 <!--
wolffd@0	2841 <h2><a name="read_learning">Further reading on learning</h2>
wolffd@0	2842
wolffd@0	2843 I recommend the following tutorials for more details on learning.
wolffd@0	2844 <ul>
wolffd@0	2845 <li> <a
wolffd@0	2846 href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short
wolffd@0	2847 tutorial</a> on graphical models, which contains an overview of learning.
wolffd@0	2848
wolffd@0	2849 <li>
wolffd@0	2850 <A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS">
wolffd@0	2851 A tutorial on learning with Bayesian networks</a>, D. Heckerman,
wolffd@0	2852 Microsoft Research Tech Report, 1995.
wolffd@0	2853
wolffd@0	2854 <li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja">
wolffd@0	2855 Operations for Learning with Graphical Models</a>,
wolffd@0	2856 W. L. Buntine, JAIR'94, 159--225.
wolffd@0	2857 </ul>
wolffd@0	2858 <p>
wolffd@0	2859 -->
wolffd@0	2860
wolffd@0	2861
wolffd@0	2862
wolffd@0	2863
wolffd@0	2864
wolffd@0	2865 <h1><a name="engines">Inference engines</h1>
wolffd@0	2866
wolffd@0	2867 Up until now, we have used the junction tree algorithm for inference.
wolffd@0	2868 However, sometimes this is too slow, or not even applicable.
wolffd@0	2869 In general, there are many inference algorithms each of which make
wolffd@0	2870 different tradeoffs between speed, accuracy, complexity and
wolffd@0	2871 generality. Furthermore, there might be many implementations of the
wolffd@0	2872 same algorithm; for instance, a general purpose, readable version,
wolffd@0	2873 and a highly-optimized, specialized one.
wolffd@0	2874 To cope with this variety, we treat each inference algorithm as an
wolffd@0	2875 object, which we call an inference engine.
wolffd@0	2876
wolffd@0	2877 <p>
wolffd@0	2878 An inference engine is an object that contains a bnet and supports the
wolffd@0	2879 'enter_evidence' and 'marginal_nodes' methods. The engine constructor
wolffd@0	2880 takes the bnet as argument and may do some model-specific processing.
wolffd@0	2881 When 'enter_evidence' is called, the engine may do some
wolffd@0	2882 evidence-specific processing. Finally, when 'marginal_nodes' is
wolffd@0	2883 called, the engine may do some query-specific processing.
wolffd@0	2884
wolffd@0	2885 <p>
wolffd@0	2886 The amount of work done when each stage is specified -- structure,
wolffd@0	2887 parameters, evidence, and query -- depends on the engine. The cost of
wolffd@0	2888 work done early in this sequence can be amortized. On the other hand,
wolffd@0	2889 one can make better optimizations if one waits until later in the
wolffd@0	2890 sequence.
wolffd@0	2891 For example, the parameters might imply
wolffd@0	2892 conditional indpendencies that are not evident in the graph structure,
wolffd@0	2893 but can nevertheless be exploited; the evidence indicates which nodes
wolffd@0	2894 are observed and hence can effectively be disconnected from the
wolffd@0	2895 graph; and the query might indicate that large parts of the network
wolffd@0	2896 are d-separated from the query nodes. (Since it is not the actual
wolffd@0	2897 <em>values</em> of the evidence that matters, just which nodes are observed,
wolffd@0	2898 many engines allow you to specify which nodes will be observed when they are constructed,
wolffd@0	2899 i.e., before calling 'enter_evidence'. Some engines can still cope if
wolffd@0	2900 the actual pattern of evidence is different, e.g., if there is missing
wolffd@0	2901 data.)
wolffd@0	2902 <p>
wolffd@0	2903
wolffd@0	2904 Although being maximally lazy (i.e., only doing work when a query is
wolffd@0	2905 issued) may seem desirable,
wolffd@0	2906 this is not always the most efficient.
wolffd@0	2907 For example,
wolffd@0	2908 when learning using EM, we need to call marginal_nodes N times, where N is the
wolffd@0	2909 number of nodes. <a href="varelim">Variable elimination</a> would end
wolffd@0	2910 up repeating a lot of work
wolffd@0	2911 each time marginal_nodes is called, making it inefficient for
wolffd@0	2912 learning. The junction tree algorithm, by contrast, uses dynamic
wolffd@0	2913 programming to avoid this redundant computation --- it calculates all
wolffd@0	2914 marginals in two passes during 'enter_evidence', so calling
wolffd@0	2915 'marginal_nodes' takes constant time.
wolffd@0	2916 <p>
wolffd@0	2917 We will discuss some of the inference algorithms implemented in BNT
wolffd@0	2918 below, and finish with a <a href="#engine_summary">summary</a> of all
wolffd@0	2919 of them.
wolffd@0	2920
wolffd@0	2921
wolffd@0	2922
wolffd@0	2923
wolffd@0	2924
wolffd@0	2925
wolffd@0	2926
wolffd@0	2927 <h2><a name="varelim">Variable elimination</h2>
wolffd@0	2928
wolffd@0	2929 The variable elimination algorithm, also known as bucket elimination
wolffd@0	2930 or peeling, is one of the simplest inference algorithms.
wolffd@0	2931 The basic idea is to "push sums inside of products"; this is explained
wolffd@0	2932 in more detail
wolffd@0	2933 <a
wolffd@0	2934 href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>.
wolffd@0	2935 <p>
wolffd@0	2936 The principle of distributing sums over products can be generalized
wolffd@0	2937 greatly to apply to any commutative semiring.
wolffd@0	2938 This forms the basis of many common algorithms, such as Viterbi
wolffd@0	2939 decoding and the Fast Fourier Transform. For details, see
wolffd@0	2940
wolffd@0	2941 <ul>
wolffd@0	2942 <li> R. McEliece and S. M. Aji, 2000.
wolffd@0	2943 <!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">-->
wolffd@0	2944 <a href="GDL.pdf">
wolffd@0	2945 The Generalized Distributive Law</a>,
wolffd@0	2946 IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000),
wolffd@0	2947 pp. 325--343.
wolffd@0	2948
wolffd@0	2949
wolffd@0	2950 <li>
wolffd@0	2951 F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001.
wolffd@0	2952 <a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html">
wolffd@0	2953 Factor graphs and the sum-product algorithm</a>
wolffd@0	2954 IEEE Transactions on Information Theory, February, 2001.
wolffd@0	2955
wolffd@0	2956 </ul>
wolffd@0	2957
wolffd@0	2958 <p>
wolffd@0	2959 Choosing an order in which to sum out the variables so as to minimize
wolffd@0	2960 computational cost is known to be NP-hard.
wolffd@0	2961 The implementation of this algorithm in
wolffd@0	2962 <tt>var_elim_inf_engine</tt> makes no attempt to optimize this
wolffd@0	2963 ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a
wolffd@0	2964 greedy search procedure to find a good ordering).
wolffd@0	2965 <p>
wolffd@0	2966 Note: unlike most algorithms, var_elim does all its computational work
wolffd@0	2967 inside of <tt>marginal_nodes</tt>, not inside of
wolffd@0	2968 <tt>enter_evidence</tt>.
wolffd@0	2969
wolffd@0	2970
wolffd@0	2971
wolffd@0	2972
wolffd@0	2973 <h2><a name="global">Global inference methods</h2>
wolffd@0	2974
wolffd@0	2975 The simplest inference algorithm of all is to explicitely construct
wolffd@0	2976 the joint distribution over all the nodes, and then to marginalize it.
wolffd@0	2977 This is implemented in <tt>global_joint_inf_engine</tt>.
wolffd@0	2978 Since the size of the joint is exponential in the
wolffd@0	2979 number of discrete (hidden) nodes, this is not a very practical algorithm.
wolffd@0	2980 It is included merely for pedagogical and debugging purposes.
wolffd@0	2981 <p>
wolffd@0	2982 Three specialized versions of this algorithm have also been implemented,
wolffd@0	2983 corresponding to the cases where all the nodes are discrete (D), all
wolffd@0	2984 are Gaussian (G), and some are discrete and some Gaussian (CG).
wolffd@0	2985 They are called <tt>enumerative_inf_engine</tt>,
wolffd@0	2986 <tt>gaussian_inf_engine</tt>,
wolffd@0	2987 and <tt>cond_gauss_inf_engine</tt> respectively.
wolffd@0	2988 <p>
wolffd@0	2989 Note: unlike most algorithms, these global inference algorithms do all their computational work
wolffd@0	2990 inside of <tt>marginal_nodes</tt>, not inside of
wolffd@0	2991 <tt>enter_evidence</tt>.
wolffd@0	2992
wolffd@0	2993
wolffd@0	2994 <h2><a name="quickscore">Quickscore</h2>
wolffd@0	2995
wolffd@0	2996 The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network,
wolffd@0	2997 since the cliques are so big.
wolffd@0	2998 One simple trick we can use is to notice that hidden leaves do not
wolffd@0	2999 affect the posteriors on the roots, and hence do not need to be
wolffd@0	3000 included in the network.
wolffd@0	3001 A second trick is to notice that the negative findings can be
wolffd@0	3002 "absorbed" into the prior:
wolffd@0	3003 see the file
wolffd@0	3004 BNT/examples/static/mk_minimal_qmr_bnet for details.
wolffd@0	3005 <p>
wolffd@0	3006
wolffd@0	3007 A much more significant speedup is obtained by exploiting special
wolffd@0	3008 properties of the noisy-or node, as done by the quickscore
wolffd@0	3009 algorithm. For details, see
wolffd@0	3010 <ul>
wolffd@0	3011 <li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89.
wolffd@0	3012 <li> Rish and Dechter, "On the impact of causal independence", UCI
wolffd@0	3013 tech report, 1998.
wolffd@0	3014 </ul>
wolffd@0	3015
wolffd@0	3016 This has been implemented in BNT as a special-purpose inference
wolffd@0	3017 engine, which can be created and used as follows:
wolffd@0	3018 <pre>
wolffd@0	3019 engine = quickscore_inf_engine(inhibit, leak, prior);
wolffd@0	3020 engine = enter_evidence(engine, pos, neg);
wolffd@0	3021 m = marginal_nodes(engine, i);
wolffd@0	3022 </pre>
wolffd@0	3023
wolffd@0	3024
wolffd@0	3025 <h2><a name="belprop">Belief propagation</h2>
wolffd@0	3026
wolffd@0	3027 Even using quickscore, exact inference takes time that is exponential
wolffd@0	3028 in the number of positive findings.
wolffd@0	3029 Hence for large networks we need to resort to approximate inference techniques.
wolffd@0	3030 See for example
wolffd@0	3031 <ul>
wolffd@0	3032 <li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the
wolffd@0	3033 QMR-DT network", JAIR 10, 1999.
wolffd@0	3034
wolffd@0	3035 <li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study",
wolffd@0	3036 UAI 99.
wolffd@0	3037 </ul>
wolffd@0	3038 The latter approximation
wolffd@0	3039 entails applying Pearl's belief propagation algorithm to a model even
wolffd@0	3040 if it has loops (hence the name loopy belief propagation).
wolffd@0	3041 Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives
wolffd@0	3042 exact results when applied to singly-connected graphs
wolffd@0	3043 (a.k.a. polytrees, since
wolffd@0	3044 the underlying undirected topology is a tree, but a node may have
wolffd@0	3045 multiple parents).
wolffd@0	3046 To apply this algorithm to a graph with loops,
wolffd@0	3047 use <tt>pearl_inf_engine</tt>.
wolffd@0	3048 This can use a centralized or distributed message passing protocol.
wolffd@0	3049 You can use it as in the following example.
wolffd@0	3050 <pre>
wolffd@0	3051 engine = pearl_inf_engine(bnet, 'max_iter', 30);
wolffd@0	3052 engine = enter_evidence(engine, evidence);
wolffd@0	3053 m = marginal_nodes(engine, i);
wolffd@0	3054 </pre>
wolffd@0	3055 We found that this algorithm often converges, and when it does, often
wolffd@0	3056 is very accurate, but it depends on the precise setting of the
wolffd@0	3057 parameter values of the network.
wolffd@0	3058 (See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.)
wolffd@0	3059 Understanding when and why belief propagation converges/ works
wolffd@0	3060 is a topic of ongoing research.
wolffd@0	3061 <p>
wolffd@0	3062 <tt>pearl_inf_engine</tt> can exploit special structure in noisy-or
wolffd@0	3063 and gmux nodes to compute messages efficiently.
wolffd@0	3064 <p>
wolffd@0	3065 <tt>belprop_inf_engine</tt> is like pearl, but uses potentials to
wolffd@0	3066 represent messages. Hence this is slower.
wolffd@0	3067 <p>
wolffd@0	3068 <tt>belprop_fg_inf_engine</tt> is like belprop,
wolffd@0	3069 but is designed for factor graphs.
wolffd@0	3070
wolffd@0	3071
wolffd@0	3072
wolffd@0	3073 <h2><a name="sampling">Sampling</h2>
wolffd@0	3074
wolffd@0	3075 BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms:
wolffd@0	3076 <ul>
wolffd@0	3077 <li> <tt>likelihood_weighting_inf_engine</tt> which does importance
wolffd@0	3078 sampling and can handle any node type.
wolffd@0	3079 <li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi.
wolffd@0	3080 Currently this can only handle tabular CPDs.
wolffd@0	3081 For a much faster and more powerful Gibbs sampling program, see
wolffd@0	3082 <a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>.
wolffd@0	3083 </ul>
wolffd@0	3084 Note: To generate samples from a network (which is not the same as inference!),
wolffd@0	3085 use <tt>sample_bnet</tt>.
wolffd@0	3086
wolffd@0	3087
wolffd@0	3088
wolffd@0	3089 <h2><a name="engine_summary">Summary of inference engines</h2>
wolffd@0	3090
wolffd@0	3091
wolffd@0	3092 The inference engines differ in many ways. Here are
wolffd@0	3093 some of the major "axes":
wolffd@0	3094 <ul>
wolffd@0	3095 <li> Works for all topologies or makes restrictions?
wolffd@0	3096 <li> Works for all node types or makes restrictions?
wolffd@0	3097 <li> Exact or approximate inference?
wolffd@0	3098 </ul>
wolffd@0	3099
wolffd@0	3100 <p>
wolffd@0	3101 In terms of topology, most engines handle any kind of DAG.
wolffd@0	3102 <tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which
wolffd@0	3103 can be used to represent directed, undirected, and mixed (chain)
wolffd@0	3104 graphs.
wolffd@0	3105 (In the future, we plan to support exact inference on chain graphs.)
wolffd@0	3106 <tt>quickscore</tt> only works on QMR-like models.
wolffd@0	3107 <p>
wolffd@0	3108 In terms of node types: algorithms that use potentials can handle
wolffd@0	3109 discrete (D), Gaussian (G) or conditional Gaussian (CG) models.
wolffd@0	3110 Sampling algorithms can essentially handle any kind of node (distribution).
wolffd@0	3111 Other algorithms make more restrictive assumptions in exchange for
wolffd@0	3112 speed.
wolffd@0	3113 <p>
wolffd@0	3114 Finally, most algorithms are designed to give the exact answer.
wolffd@0	3115 The belief propagation algorithms are exact if applied to trees, and
wolffd@0	3116 in some other cases.
wolffd@0	3117 Sampling is considered approximate, even though, in the limit of an
wolffd@0	3118 infinite number of samples, it gives the exact answer.
wolffd@0	3119
wolffd@0	3120 <p>
wolffd@0	3121
wolffd@0	3122 Here is a summary of the properties
wolffd@0	3123 of all the engines in BNT which work on static networks.
wolffd@0	3124 <p>
wolffd@0	3125 <table>
wolffd@0	3126 <table border units = pixels><tr>
wolffd@0	3127 <td align=left width=0>Name
wolffd@0	3128 <td align=left width=0>Exact?
wolffd@0	3129 <td align=left width=0>Node type?
wolffd@0	3130 <td align=left width=0>topology
wolffd@0	3131 <tr>
wolffd@0	3132 <tr>
wolffd@0	3133 <td align=left> belprop
wolffd@0	3134 <td align=left> approx
wolffd@0	3135 <td align=left> D
wolffd@0	3136 <td align=left> DAG
wolffd@0	3137 <tr>
wolffd@0	3138 <td align=left> belprop_fg
wolffd@0	3139 <td align=left> approx
wolffd@0	3140 <td align=left> D
wolffd@0	3141 <td align=left> factor graph
wolffd@0	3142 <tr>
wolffd@0	3143 <td align=left> cond_gauss
wolffd@0	3144 <td align=left> exact
wolffd@0	3145 <td align=left> CG
wolffd@0	3146 <td align=left> DAG
wolffd@0	3147 <tr>
wolffd@0	3148 <td align=left> enumerative
wolffd@0	3149 <td align=left> exact
wolffd@0	3150 <td align=left> D
wolffd@0	3151 <td align=left> DAG
wolffd@0	3152 <tr>
wolffd@0	3153 <td align=left> gaussian
wolffd@0	3154 <td align=left> exact
wolffd@0	3155 <td align=left> G
wolffd@0	3156 <td align=left> DAG
wolffd@0	3157 <tr>
wolffd@0	3158 <td align=left> gibbs
wolffd@0	3159 <td align=left> approx
wolffd@0	3160 <td align=left> D
wolffd@0	3161 <td align=left> DAG
wolffd@0	3162 <tr>
wolffd@0	3163 <td align=left> global_joint
wolffd@0	3164 <td align=left> exact
wolffd@0	3165 <td align=left> D,G,CG
wolffd@0	3166 <td align=left> DAG
wolffd@0	3167 <tr>
wolffd@0	3168 <td align=left> jtree
wolffd@0	3169 <td align=left> exact
wolffd@0	3170 <td align=left> D,G,CG
wolffd@0	3171 <td align=left> DAG
wolffd@0	3172 b<tr>
wolffd@0	3173 <td align=left> likelihood_weighting
wolffd@0	3174 <td align=left> approx
wolffd@0	3175 <td align=left> any
wolffd@0	3176 <td align=left> DAG
wolffd@0	3177 <tr>
wolffd@0	3178 <td align=left> pearl
wolffd@0	3179 <td align=left> approx
wolffd@0	3180 <td align=left> D,G
wolffd@0	3181 <td align=left> DAG
wolffd@0	3182 <tr>
wolffd@0	3183 <td align=left> pearl
wolffd@0	3184 <td align=left> exact
wolffd@0	3185 <td align=left> D,G
wolffd@0	3186 <td align=left> polytree
wolffd@0	3187 <tr>
wolffd@0	3188 <td align=left> quickscore
wolffd@0	3189 <td align=left> exact
wolffd@0	3190 <td align=left> noisy-or
wolffd@0	3191 <td align=left> QMR
wolffd@0	3192 <tr>
wolffd@0	3193 <td align=left> stab_cond_gauss
wolffd@0	3194 <td align=left> exact
wolffd@0	3195 <td align=left> CG
wolffd@0	3196 <td align=left> DAG
wolffd@0	3197 <tr>
wolffd@0	3198 <td align=left> var_elim
wolffd@0	3199 <td align=left> exact
wolffd@0	3200 <td align=left> D,G,CG
wolffd@0	3201 <td align=left> DAG
wolffd@0	3202 </table>
wolffd@0	3203
wolffd@0	3204
wolffd@0	3205
wolffd@0	3206 <h1><a name="influence">Influence diagrams/ decision making</h1>
wolffd@0	3207
wolffd@0	3208 BNT implements an exact algorithm for solving LIMIDs (limited memory
wolffd@0	3209 influence diagrams), described in
wolffd@0	3210 <ul>
wolffd@0	3211 <li> S. L. Lauritzen and D. Nilsson.
wolffd@0	3212 <a href="http://www.math.auc.dk/~steffen/papers/limids.pdf">
wolffd@0	3213 Representing and solving decision problems with limited
wolffd@0	3214 information</a>
wolffd@0	3215 Management Science, 47, 1238 - 1251. September 2001.
wolffd@0	3216 </ul>
wolffd@0	3217 LIMIDs explicitely show all information arcs, rather than implicitely
wolffd@0	3218 assuming no forgetting. This allows them to model forgetful
wolffd@0	3219 controllers.
wolffd@0	3220 <p>
wolffd@0	3221 See the examples in <tt>BNT/examples/limids</tt> for details.
wolffd@0	3222
wolffd@0	3223
wolffd@0	3224
wolffd@0	3225
wolffd@0	3226 <h1>DBNs, HMMs, Kalman filters and all that</h1>
wolffd@0	3227
wolffd@0	3228 Click <a href="usage_dbn.html">here</a> for documentation about how to
wolffd@0	3229 use BNT for dynamical systems and sequence data.
wolffd@0	3230
wolffd@0	3231
wolffd@0	3232 </BODY>

Mercurial > hg > camir-aes2014

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