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