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Daniel@0 1 <HEAD>
Daniel@0 2 <TITLE>How to use the Bayes Net Toolbox</TITLE>
Daniel@0 3 </HEAD>
Daniel@0 4
Daniel@0 5 <BODY BGCOLOR="#FFFFFF">
Daniel@0 6 <!-- white background is better for the pictures and equations -->
Daniel@0 7
Daniel@0 8 <h1>How to use the Bayes Net Toolbox</h1>
Daniel@0 9
Daniel@0 10 This documentation was last updated on 29 October 2007.
Daniel@0 11 <br>
Daniel@0 12 Click
Daniel@0 13 <a href="http://bnt.insa-rouen.fr/">here</a>
Daniel@0 14 for a French version of this documentation (last updated in 2005).
Daniel@0 15 <p>
Daniel@0 16
Daniel@0 17 <ul>
Daniel@0 18 <li> <a href="install.html">Installation</a>
Daniel@0 19
Daniel@0 20 <li> <a href="#basics">Creating your first Bayes net</a>
Daniel@0 21 <ul>
Daniel@0 22 <li> <a href="#basics">Creating a model by hand</a>
Daniel@0 23 <li> <a href="#file">Loading a model from a file</a>
Daniel@0 24 <li> <a href="#GUI">Creating a model using a GUI</a>
Daniel@0 25 <li> <a href="graphviz.html">Graph visualization</a>
Daniel@0 26 </ul>
Daniel@0 27
Daniel@0 28 <li> <a href="#inference">Inference</a>
Daniel@0 29 <ul>
Daniel@0 30 <li> <a href="#marginal">Computing marginal distributions</a>
Daniel@0 31 <li> <a href="#joint">Computing joint distributions</a>
Daniel@0 32 <li> <a href="#soft">Soft/virtual evidence</a>
Daniel@0 33 <li> <a href="#mpe">Most probable explanation</a>
Daniel@0 34 </ul>
Daniel@0 35
Daniel@0 36 <li> <a href="#cpd">Conditional Probability Distributions</a>
Daniel@0 37 <ul>
Daniel@0 38 <li> <a href="#tabular">Tabular (multinomial) nodes</a>
Daniel@0 39 <li> <a href="#noisyor">Noisy-or nodes</a>
Daniel@0 40 <li> <a href="#deterministic">Other (noisy) deterministic nodes</a>
Daniel@0 41 <li> <a href="#softmax">Softmax (multinomial logit) nodes</a>
Daniel@0 42 <li> <a href="#mlp">Neural network nodes</a>
Daniel@0 43 <li> <a href="#root">Root nodes</a>
Daniel@0 44 <li> <a href="#gaussian">Gaussian nodes</a>
Daniel@0 45 <li> <a href="#glm">Generalized linear model nodes</a>
Daniel@0 46 <li> <a href="#dtree">Classification/regression tree nodes</a>
Daniel@0 47 <li> <a href="#nongauss">Other continuous distributions</a>
Daniel@0 48 <li> <a href="#cpd_summary">Summary of CPD types</a>
Daniel@0 49 </ul>
Daniel@0 50
Daniel@0 51 <li> <a href="#examples">Example models</a>
Daniel@0 52 <ul>
Daniel@0 53 <li> <a
Daniel@0 54 href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">
Daniel@0 55 Gaussian mixture models</a>
Daniel@0 56 <li> <a href="#pca">PCA, ICA, and all that</a>
Daniel@0 57 <li> <a href="#mixep">Mixtures of experts</a>
Daniel@0 58 <li> <a href="#hme">Hierarchical mixtures of experts</a>
Daniel@0 59 <li> <a href="#qmr">QMR</a>
Daniel@0 60 <li> <a href="#cg_model">Conditional Gaussian models</a>
Daniel@0 61 <li> <a href="#hybrid">Other hybrid models</a>
Daniel@0 62 </ul>
Daniel@0 63
Daniel@0 64 <li> <a href="#param_learning">Parameter learning</a>
Daniel@0 65 <ul>
Daniel@0 66 <li> <a href="#load_data">Loading data from a file</a>
Daniel@0 67 <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a>
Daniel@0 68 <li> <a href="#prior">Parameter priors</a>
Daniel@0 69 <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a>
Daniel@0 70 <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a>
Daniel@0 71 <li> <a href="#tying">Parameter tying</a>
Daniel@0 72 </ul>
Daniel@0 73
Daniel@0 74 <li> <a href="#structure_learning">Structure learning</a>
Daniel@0 75 <ul>
Daniel@0 76 <li> <a href="#enumerate">Exhaustive search</a>
Daniel@0 77 <li> <a href="#K2">K2</a>
Daniel@0 78 <li> <a href="#hill_climb">Hill-climbing</a>
Daniel@0 79 <li> <a href="#mcmc">MCMC</a>
Daniel@0 80 <li> <a href="#active">Active learning</a>
Daniel@0 81 <li> <a href="#struct_em">Structural EM</a>
Daniel@0 82 <li> <a href="#graphdraw">Visualizing the learned graph structure</a>
Daniel@0 83 <li> <a href="#constraint">Constraint-based methods</a>
Daniel@0 84 </ul>
Daniel@0 85
Daniel@0 86
Daniel@0 87 <li> <a href="#engines">Inference engines</a>
Daniel@0 88 <ul>
Daniel@0 89 <li> <a href="#jtree">Junction tree</a>
Daniel@0 90 <li> <a href="#varelim">Variable elimination</a>
Daniel@0 91 <li> <a href="#global">Global inference methods</a>
Daniel@0 92 <li> <a href="#quickscore">Quickscore</a>
Daniel@0 93 <li> <a href="#belprop">Belief propagation</a>
Daniel@0 94 <li> <a href="#sampling">Sampling (Monte Carlo)</a>
Daniel@0 95 <li> <a href="#engine_summary">Summary of inference engines</a>
Daniel@0 96 </ul>
Daniel@0 97
Daniel@0 98
Daniel@0 99 <li> <a href="#influence">Influence diagrams/ decision making</a>
Daniel@0 100
Daniel@0 101
Daniel@0 102 <li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a>
Daniel@0 103 </ul>
Daniel@0 104
Daniel@0 105 </ul>
Daniel@0 106
Daniel@0 107
Daniel@0 108
Daniel@0 109 <h1><a name="basics">Creating your first Bayes net</h1>
Daniel@0 110
Daniel@0 111 To define a Bayes net, you must specify the graph structure and then
Daniel@0 112 the parameters. We look at each in turn, using a simple example
Daniel@0 113 (adapted from Russell and
Daniel@0 114 Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall,
Daniel@0 115 1995, p454).
Daniel@0 116
Daniel@0 117
Daniel@0 118 <h2>Graph structure</h2>
Daniel@0 119
Daniel@0 120
Daniel@0 121 Consider the following network.
Daniel@0 122
Daniel@0 123 <p>
Daniel@0 124 <center>
Daniel@0 125 <IMG SRC="Figures/sprinkler.gif">
Daniel@0 126 </center>
Daniel@0 127 <p>
Daniel@0 128
Daniel@0 129 <P>
Daniel@0 130 To specify this directed acyclic graph (dag), we create an adjacency matrix:
Daniel@0 131 <PRE>
Daniel@0 132 N = 4;
Daniel@0 133 dag = zeros(N,N);
Daniel@0 134 C = 1; S = 2; R = 3; W = 4;
Daniel@0 135 dag(C,[R S]) = 1;
Daniel@0 136 dag(R,W) = 1;
Daniel@0 137 dag(S,W)=1;
Daniel@0 138 </PRE>
Daniel@0 139 <P>
Daniel@0 140 We have numbered the nodes as follows:
Daniel@0 141 Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4.
Daniel@0 142 <b>The nodes must always be numbered in topological order, i.e.,
Daniel@0 143 ancestors before descendants.</b>
Daniel@0 144 For a more complicated graph, this is a little inconvenient: we will
Daniel@0 145 see how to get around this <a href="usage_dbn.html#bat">below</a>.
Daniel@0 146 <p>
Daniel@0 147 In Matlab 6, you can use logical arrays instead of double arrays,
Daniel@0 148 which are 4 times smaller:
Daniel@0 149 <pre>
Daniel@0 150 dag = false(N,N);
Daniel@0 151 dag(C,[R S]) = true;
Daniel@0 152 ...
Daniel@0 153 </pre>
Daniel@0 154 However, <b>some graph functions (eg acyclic) do not work on
Daniel@0 155 logical arrays</b>!
Daniel@0 156 <p>
Daniel@0 157 You can visualize the resulting graph structure using
Daniel@0 158 the methods discussed <a href="#graphdraw">below</a>.
Daniel@0 159 For details on GUIs,
Daniel@0 160 click <a href="#GUI">here</a>.
Daniel@0 161
Daniel@0 162 <h2>Creating the Bayes net shell</h2>
Daniel@0 163
Daniel@0 164 In addition to specifying the graph structure,
Daniel@0 165 we must specify the size and type of each node.
Daniel@0 166 If a node is discrete, its size is the
Daniel@0 167 number of possible values
Daniel@0 168 each node can take on; if a node is continuous,
Daniel@0 169 it can be a vector, and its size is the length of this vector.
Daniel@0 170 In this case, we will assume all nodes are discrete and binary.
Daniel@0 171 <PRE>
Daniel@0 172 discrete_nodes = 1:N;
Daniel@0 173 node_sizes = 2*ones(1,N);
Daniel@0 174 </pre>
Daniel@0 175 If the nodes were not binary, you could type e.g.,
Daniel@0 176 <pre>
Daniel@0 177 node_sizes = [4 2 3 5];
Daniel@0 178 </pre>
Daniel@0 179 meaning that Cloudy has 4 possible values,
Daniel@0 180 Sprinkler has 2 possible values, etc.
Daniel@0 181 Note that these are cardinal values, not ordinal, i.e.,
Daniel@0 182 they are not ordered in any way, like 'low', 'medium', 'high'.
Daniel@0 183 <p>
Daniel@0 184 We are now ready to make the Bayes net:
Daniel@0 185 <pre>
Daniel@0 186 bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes);
Daniel@0 187 </PRE>
Daniel@0 188 By default, all nodes are assumed to be discrete, so we can also just
Daniel@0 189 write
Daniel@0 190 <pre>
Daniel@0 191 bnet = mk_bnet(dag, node_sizes);
Daniel@0 192 </PRE>
Daniel@0 193 You may also specify which nodes will be observed.
Daniel@0 194 If you don't know, or if this not fixed in advance,
Daniel@0 195 just use the empty list (the default).
Daniel@0 196 <pre>
Daniel@0 197 onodes = [];
Daniel@0 198 bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes);
Daniel@0 199 </PRE>
Daniel@0 200 Note that optional arguments are specified using a name/value syntax.
Daniel@0 201 This is common for many BNT functions.
Daniel@0 202 In general, to find out more about a function (e.g., which optional
Daniel@0 203 arguments it takes), please see its
Daniel@0 204 documentation string by typing
Daniel@0 205 <pre>
Daniel@0 206 help mk_bnet
Daniel@0 207 </pre>
Daniel@0 208 See also other <a href="matlab_tips.html">useful Matlab tips</a>.
Daniel@0 209 <p>
Daniel@0 210 It is possible to associate names with nodes, as follows:
Daniel@0 211 <pre>
Daniel@0 212 bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4);
Daniel@0 213 </pre>
Daniel@0 214 You can then refer to a node by its name:
Daniel@0 215 <pre>
Daniel@0 216 C = bnet.names('cloudy'); % bnet.names is an associative array
Daniel@0 217 bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]);
Daniel@0 218 </pre>
Daniel@0 219 This feature uses my own associative array class.
Daniel@0 220
Daniel@0 221
Daniel@0 222 <h2><a name="cpt">Parameters</h2>
Daniel@0 223
Daniel@0 224 A model consists of the graph structure and the parameters.
Daniel@0 225 The parameters are represented by CPD objects (CPD = Conditional
Daniel@0 226 Probability Distribution), which define the probability distribution
Daniel@0 227 of a node given its parents.
Daniel@0 228 (We will use the terms "node" and "random variable" interchangeably.)
Daniel@0 229 The simplest kind of CPD is a table (multi-dimensional array), which
Daniel@0 230 is suitable when all the nodes are discrete-valued. Note that the discrete
Daniel@0 231 values are not assumed to be ordered in any way; that is, they
Daniel@0 232 represent categorical quantities, like male and female, rather than
Daniel@0 233 ordinal quantities, like low, medium and high.
Daniel@0 234 (We will discuss CPDs in more detail <a href="#cpd">below</a>.)
Daniel@0 235 <p>
Daniel@0 236 Tabular CPDs, also called CPTs (conditional probability tables),
Daniel@0 237 are stored as multidimensional arrays, where the dimensions
Daniel@0 238 are arranged in the same order as the nodes, e.g., the CPT for node 4
Daniel@0 239 (WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself.
Daniel@0 240 Hence the child is always the last dimension.
Daniel@0 241 If a node has no parents, its CPT is a column vector representing its
Daniel@0 242 prior.
Daniel@0 243 Note that in Matlab (unlike C), arrays are indexed
Daniel@0 244 from 1, and are layed out in memory such that the first index toggles
Daniel@0 245 fastest, e.g., the CPT for node 4 (WetGrass) is as follows
Daniel@0 246 <P>
Daniel@0 247 <P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P>
Daniel@0 248 <P>
Daniel@0 249 where we have used the convention that false==1, true==2.
Daniel@0 250 We can create this CPT in Matlab as follows
Daniel@0 251 <PRE>
Daniel@0 252 CPT = zeros(2,2,2);
Daniel@0 253 CPT(1,1,1) = 1.0;
Daniel@0 254 CPT(2,1,1) = 0.1;
Daniel@0 255 ...
Daniel@0 256 </PRE>
Daniel@0 257 Here is an easier way:
Daniel@0 258 <PRE>
Daniel@0 259 CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]);
Daniel@0 260 </PRE>
Daniel@0 261 In fact, we don't need to reshape the array, since the CPD constructor
Daniel@0 262 will do that for us. So we can just write
Daniel@0 263 <pre>
Daniel@0 264 bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]);
Daniel@0 265 </pre>
Daniel@0 266 The other nodes are created similarly (using the old syntax for
Daniel@0 267 optional parameters)
Daniel@0 268 <PRE>
Daniel@0 269 bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]);
Daniel@0 270 bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]);
Daniel@0 271 bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]);
Daniel@0 272 bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]);
Daniel@0 273 </PRE>
Daniel@0 274
Daniel@0 275
Daniel@0 276 <h2><a name="rnd_cpt">Random Parameters</h2>
Daniel@0 277
Daniel@0 278 If we do not specify the CPT, random parameters will be
Daniel@0 279 created, i.e., each "row" of the CPT will be drawn from the uniform distribution.
Daniel@0 280 To ensure repeatable results, use
Daniel@0 281 <pre>
Daniel@0 282 rand('state', seed);
Daniel@0 283 randn('state', seed);
Daniel@0 284 </pre>
Daniel@0 285 To control the degree of randomness (entropy),
Daniel@0 286 you can sample each row of the CPT from a Dirichlet(p,p,...) distribution.
Daniel@0 287 If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0).
Daniel@0 288 If p = 1, each entry is drawn from U[0,1].
Daniel@0 289 If p >> 1, the entries will all be near 1/k, where k is the arity of
Daniel@0 290 this node, i.e., each row will be nearly uniform.
Daniel@0 291 You can do this as follows, assuming this node
Daniel@0 292 is number i, and ns is the node_sizes.
Daniel@0 293 <pre>
Daniel@0 294 k = ns(i);
Daniel@0 295 ps = parents(dag, i);
Daniel@0 296 psz = prod(ns(ps));
Daniel@0 297 CPT = sample_dirichlet(p*ones(1,k), psz);
Daniel@0 298 bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT);
Daniel@0 299 </pre>
Daniel@0 300
Daniel@0 301
Daniel@0 302 <h2><a name="file">Loading a network from a file</h2>
Daniel@0 303
Daniel@0 304 If you already have a Bayes net represented in the XML-based
Daniel@0 305 <a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/">
Daniel@0 306 Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the
Daniel@0 307 <a
Daniel@0 308 href="http://www.cs.huji.ac.il/labs/compbio/Repository">
Daniel@0 309 Bayes Net repository</a>),
Daniel@0 310 you can convert it to BNT format using
Daniel@0 311 the
Daniel@0 312 <a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java
Daniel@0 313 program</a> written by Ken Shan.
Daniel@0 314 (This is not necessarily up-to-date.)
Daniel@0 315 <p>
Daniel@0 316 <b>It is currently not possible to save/load a BNT matlab object to
Daniel@0 317 file</b>, but this is easily fixed if you modify all the constructors
Daniel@0 318 for all the classes (see matlab documentation).
Daniel@0 319
Daniel@0 320 <h2><a name="GUI">Creating a model using a GUI</h2>
Daniel@0 321
Daniel@0 322 <ul>
Daniel@0 323 <li>Senthil Nachimuthu
Daniel@0 324 has started (Oct 07) an open source
Daniel@0 325 GUI for BNT called
Daniel@0 326 <a href="http://projeny.sourceforge.net">projeny</a>
Daniel@0 327 using Java. This is a successor to BNJ.
Daniel@0 328
Daniel@0 329 <li>
Daniel@0 330 Philippe LeRay has written (Sep 05)
Daniel@0 331 a
Daniel@0 332 <a href="http://banquiseasi.insa-rouen.fr/projects/bnt-editor/">
Daniel@0 333 BNT GUI</a> in matlab.
Daniel@0 334
Daniel@0 335 <li>
Daniel@0 336 <a
Daniel@0 337 href="http://www.dataonstage.com/BNT/PACKAGES/LinkStrength/index.html">
Daniel@0 338 LinkStrength</a>,
Daniel@0 339 a package by Imme Ebert-Uphoff for visualizing the strength of
Daniel@0 340 dependencies between nodes.
Daniel@0 341
Daniel@0 342 </ul>
Daniel@0 343
Daniel@0 344 <h2>Graph visualization</h2>
Daniel@0 345
Daniel@0 346 Click <a href="graphviz.html">here</a> for more information
Daniel@0 347 on graph visualization.
Daniel@0 348
Daniel@0 349
Daniel@0 350 <h1><a name="inference">Inference</h1>
Daniel@0 351
Daniel@0 352 Having created the BN, we can now use it for inference.
Daniel@0 353 There are many different algorithms for doing inference in Bayes nets,
Daniel@0 354 that make different tradeoffs between speed,
Daniel@0 355 complexity, generality, and accuracy.
Daniel@0 356 BNT therefore offers a variety of different inference
Daniel@0 357 "engines". We will discuss these
Daniel@0 358 in more detail <a href="#engines">below</a>.
Daniel@0 359 For now, we will use the junction tree
Daniel@0 360 engine, which is the mother of all exact inference algorithms.
Daniel@0 361 This can be created as follows.
Daniel@0 362 <pre>
Daniel@0 363 engine = jtree_inf_engine(bnet);
Daniel@0 364 </pre>
Daniel@0 365 The other engines have similar constructors, but might take
Daniel@0 366 additional, algorithm-specific parameters.
Daniel@0 367 All engines are used in the same way, once they have been created.
Daniel@0 368 We illustrate this in the following sections.
Daniel@0 369
Daniel@0 370
Daniel@0 371 <h2><a name="marginal">Computing marginal distributions</h2>
Daniel@0 372
Daniel@0 373 Suppose we want to compute the probability that the sprinker was on
Daniel@0 374 given that the grass is wet.
Daniel@0 375 The evidence consists of the fact that W=2. All the other nodes
Daniel@0 376 are hidden (unobserved). We can specify this as follows.
Daniel@0 377 <pre>
Daniel@0 378 evidence = cell(1,N);
Daniel@0 379 evidence{W} = 2;
Daniel@0 380 </pre>
Daniel@0 381 We use a 1D cell array instead of a vector to
Daniel@0 382 cope with the fact that nodes can be vectors of different lengths.
Daniel@0 383 In addition, the value [] can be used
Daniel@0 384 to denote 'no evidence', instead of having to specify the observation
Daniel@0 385 pattern as a separate argument.
Daniel@0 386 (Click <a href="cellarray.html">here</a> for a quick tutorial on cell
Daniel@0 387 arrays in matlab.)
Daniel@0 388 <p>
Daniel@0 389 We are now ready to add the evidence to the engine.
Daniel@0 390 <pre>
Daniel@0 391 [engine, loglik] = enter_evidence(engine, evidence);
Daniel@0 392 </pre>
Daniel@0 393 The behavior of this function is algorithm-specific, and is discussed
Daniel@0 394 in more detail <a href="#engines">below</a>.
Daniel@0 395 In the case of the jtree engine,
Daniel@0 396 enter_evidence implements a two-pass message-passing scheme.
Daniel@0 397 The first return argument contains the modified engine, which
Daniel@0 398 incorporates the evidence. The second return argument contains the
Daniel@0 399 log-likelihood of the evidence. (Not all engines are capable of
Daniel@0 400 computing the log-likelihood.)
Daniel@0 401 <p>
Daniel@0 402 Finally, we can compute p=P(S=2|W=2) as follows.
Daniel@0 403 <PRE>
Daniel@0 404 marg = marginal_nodes(engine, S);
Daniel@0 405 marg.T
Daniel@0 406 ans =
Daniel@0 407 0.57024
Daniel@0 408 0.42976
Daniel@0 409 p = marg.T(2);
Daniel@0 410 </PRE>
Daniel@0 411 We see that p = 0.4298.
Daniel@0 412 <p>
Daniel@0 413 Now let us add the evidence that it was raining, and see what
Daniel@0 414 difference it makes.
Daniel@0 415 <PRE>
Daniel@0 416 evidence{R} = 2;
Daniel@0 417 [engine, loglik] = enter_evidence(engine, evidence);
Daniel@0 418 marg = marginal_nodes(engine, S);
Daniel@0 419 p = marg.T(2);
Daniel@0 420 </PRE>
Daniel@0 421 We find that p = P(S=2|W=2,R=2) = 0.1945,
Daniel@0 422 which is lower than
Daniel@0 423 before, because the rain can ``explain away'' the
Daniel@0 424 fact that the grass is wet.
Daniel@0 425 <p>
Daniel@0 426 You can plot a marginal distribution over a discrete variable
Daniel@0 427 as a barchart using the built 'bar' function:
Daniel@0 428 <pre>
Daniel@0 429 bar(marg.T)
Daniel@0 430 </pre>
Daniel@0 431 This is what it looks like
Daniel@0 432
Daniel@0 433 <p>
Daniel@0 434 <center>
Daniel@0 435 <IMG SRC="Figures/sprinkler_bar.gif">
Daniel@0 436 </center>
Daniel@0 437 <p>
Daniel@0 438
Daniel@0 439 <h2><a name="observed">Observed nodes</h2>
Daniel@0 440
Daniel@0 441 What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)?
Daniel@0 442 An observed discrete node effectively only has 1 value (the observed
Daniel@0 443 one) --- all other values would result in 0 probability.
Daniel@0 444 For efficiency, BNT treats observed (discrete) nodes as if they were
Daniel@0 445 set to 1, as we see below:
Daniel@0 446 <pre>
Daniel@0 447 evidence = cell(1,N);
Daniel@0 448 evidence{W} = 2;
Daniel@0 449 engine = enter_evidence(engine, evidence);
Daniel@0 450 m = marginal_nodes(engine, W);
Daniel@0 451 m.T
Daniel@0 452 ans =
Daniel@0 453 1
Daniel@0 454 </pre>
Daniel@0 455 This can get a little confusing, since we assigned W=2.
Daniel@0 456 So we can ask BNT to add the evidence back in by passing in an optional argument:
Daniel@0 457 <pre>
Daniel@0 458 m = marginal_nodes(engine, W, 1);
Daniel@0 459 m.T
Daniel@0 460 ans =
Daniel@0 461 0
Daniel@0 462 1
Daniel@0 463 </pre>
Daniel@0 464 This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1.
Daniel@0 465
Daniel@0 466
Daniel@0 467
Daniel@0 468 <h2><a name="joint">Computing joint distributions</h2>
Daniel@0 469
Daniel@0 470 We can compute the joint probability on a set of nodes as in the
Daniel@0 471 following example.
Daniel@0 472 <pre>
Daniel@0 473 evidence = cell(1,N);
Daniel@0 474 [engine, ll] = enter_evidence(engine, evidence);
Daniel@0 475 m = marginal_nodes(engine, [S R W]);
Daniel@0 476 </pre>
Daniel@0 477 m is a structure. The 'T' field is a multi-dimensional array (in
Daniel@0 478 this case, 3-dimensional) that contains the joint probability
Daniel@0 479 distribution on the specified nodes.
Daniel@0 480 <pre>
Daniel@0 481 >> m.T
Daniel@0 482 ans(:,:,1) =
Daniel@0 483 0.2900 0.0410
Daniel@0 484 0.0210 0.0009
Daniel@0 485 ans(:,:,2) =
Daniel@0 486 0 0.3690
Daniel@0 487 0.1890 0.0891
Daniel@0 488 </pre>
Daniel@0 489 We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass
Daniel@0 490 to be wet if both the rain and sprinkler are off.
Daniel@0 491 <p>
Daniel@0 492 Let us now add some evidence to R.
Daniel@0 493 <pre>
Daniel@0 494 evidence{R} = 2;
Daniel@0 495 [engine, ll] = enter_evidence(engine, evidence);
Daniel@0 496 m = marginal_nodes(engine, [S R W])
Daniel@0 497 m =
Daniel@0 498 domain: [2 3 4]
Daniel@0 499 T: [2x1x2 double]
Daniel@0 500 >> m.T
Daniel@0 501 m.T
Daniel@0 502 ans(:,:,1) =
Daniel@0 503 0.0820
Daniel@0 504 0.0018
Daniel@0 505 ans(:,:,2) =
Daniel@0 506 0.7380
Daniel@0 507 0.1782
Daniel@0 508 </pre>
Daniel@0 509 The joint T(i,j,k) = P(S=i,R=j,W=k|evidence)
Daniel@0 510 should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible
Daniel@0 511 with the evidence that R=2.
Daniel@0 512 Instead of creating large tables with many 0s, BNT sets the effective
Daniel@0 513 size of observed (discrete) nodes to 1, as explained above.
Daniel@0 514 This is why m.T has size 2x1x2.
Daniel@0 515 To get a 2x2x2 table, type
Daniel@0 516 <pre>
Daniel@0 517 m = marginal_nodes(engine, [S R W], 1)
Daniel@0 518 m =
Daniel@0 519 domain: [2 3 4]
Daniel@0 520 T: [2x2x2 double]
Daniel@0 521 >> m.T
Daniel@0 522 m.T
Daniel@0 523 ans(:,:,1) =
Daniel@0 524 0 0.082
Daniel@0 525 0 0.0018
Daniel@0 526 ans(:,:,2) =
Daniel@0 527 0 0.738
Daniel@0 528 0 0.1782
Daniel@0 529 </pre>
Daniel@0 530
Daniel@0 531 <p>
Daniel@0 532 Note: It is not always possible to compute the joint on arbitrary
Daniel@0 533 sets of nodes: it depends on which inference engine you use, as discussed
Daniel@0 534 in more detail <a href="#engines">below</a>.
Daniel@0 535
Daniel@0 536
Daniel@0 537 <h2><a name="soft">Soft/virtual evidence</h2>
Daniel@0 538
Daniel@0 539 Sometimes a node is not observed, but we have some distribution over
Daniel@0 540 its possible values; this is often called "soft" or "virtual"
Daniel@0 541 evidence.
Daniel@0 542 One can use this as follows
Daniel@0 543 <pre>
Daniel@0 544 [engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence);
Daniel@0 545 </pre>
Daniel@0 546 where soft_evidence{i} is either [] (if node i has no soft evidence)
Daniel@0 547 or is a vector representing the probability distribution over i's
Daniel@0 548 possible values.
Daniel@0 549 For example, if we don't know i's exact value, but we know its
Daniel@0 550 likelihood ratio is 60/40, we can write evidence{i} = [] and
Daniel@0 551 soft_evidence{i} = [0.6 0.4].
Daniel@0 552 <p>
Daniel@0 553 Currently only jtree_inf_engine supports this option.
Daniel@0 554 It assumes that all hidden nodes, and all nodes for
Daniel@0 555 which we have soft evidence, are discrete.
Daniel@0 556 For a longer example, see BNT/examples/static/softev1.m.
Daniel@0 557
Daniel@0 558
Daniel@0 559 <h2><a name="mpe">Most probable explanation</h2>
Daniel@0 560
Daniel@0 561 To compute the most probable explanation (MPE) of the evidence (i.e.,
Daniel@0 562 the most probable assignment, or a mode of the joint), use
Daniel@0 563 <pre>
Daniel@0 564 [mpe, ll] = calc_mpe(engine, evidence);
Daniel@0 565 </pre>
Daniel@0 566 mpe{i} is the most likely value of node i.
Daniel@0 567 This calls enter_evidence with the 'maximize' flag set to 1, which
Daniel@0 568 causes the engine to do max-product instead of sum-product.
Daniel@0 569 The resulting max-marginals are then thresholded.
Daniel@0 570 If there is more than one maximum probability assignment, we must take
Daniel@0 571 care to break ties in a consistent manner (thresholding the
Daniel@0 572 max-marginals may give the wrong result). To force this behavior,
Daniel@0 573 type
Daniel@0 574 <pre>
Daniel@0 575 [mpe, ll] = calc_mpe(engine, evidence, 1);
Daniel@0 576 </pre>
Daniel@0 577 Note that computing the MPE is someties called abductive reasoning.
Daniel@0 578
Daniel@0 579 <p>
Daniel@0 580 You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar,
Daniel@0 581 that does a forwards max-product pass, and then a backwards traceback
Daniel@0 582 pass, which is how Viterbi is traditionally implemented.
Daniel@0 583
Daniel@0 584
Daniel@0 585
Daniel@0 586 <h1><a name="cpd">Conditional Probability Distributions</h1>
Daniel@0 587
Daniel@0 588 A Conditional Probability Distributions (CPD)
Daniel@0 589 defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i))
Daniel@0 590 are the parents of node i. There are many ways to represent this
Daniel@0 591 distribution, which depend in part on whether X(i) and X(Pa(i)) are
Daniel@0 592 discrete, continuous, or a combination.
Daniel@0 593 We will discuss various representations below.
Daniel@0 594
Daniel@0 595
Daniel@0 596 <h2><a name="tabular">Tabular nodes</h2>
Daniel@0 597
Daniel@0 598 If the CPD is represented as a table (i.e., if it is a multinomial
Daniel@0 599 distribution), it has a number of parameters that is exponential in
Daniel@0 600 the number of parents. See the example <a href="#cpt">above</a>.
Daniel@0 601
Daniel@0 602
Daniel@0 603 <h2><a name="noisyor">Noisy-or nodes</h2>
Daniel@0 604
Daniel@0 605 A noisy-OR node is like a regular logical OR gate except that
Daniel@0 606 sometimes the effects of parents that are on get inhibited.
Daniel@0 607 Let the prob. that parent i gets inhibited be q(i).
Daniel@0 608 Then a node, C, with 2 parents, A and B, has the following CPD, where
Daniel@0 609 we use F and T to represent off and on (1 and 2 in BNT).
Daniel@0 610 <pre>
Daniel@0 611 A B P(C=off) P(C=on)
Daniel@0 612 ---------------------------
Daniel@0 613 F F 1.0 0.0
Daniel@0 614 T F q(A) 1-q(A)
Daniel@0 615 F T q(B) 1-q(B)
Daniel@0 616 T T q(A)q(B) 1-q(A)q(B)
Daniel@0 617 </pre>
Daniel@0 618 Thus we see that the causes get inhibited independently.
Daniel@0 619 It is common to associate a "leak" node with a noisy-or CPD, which is
Daniel@0 620 like a parent that is always on. This can account for all other unmodelled
Daniel@0 621 causes which might turn the node on.
Daniel@0 622 <p>
Daniel@0 623 The noisy-or distribution is similar to the logistic distribution.
Daniel@0 624 To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j)
Daniel@0 625 be the prob. that j inhibits i. Then
Daniel@0 626 <pre>
Daniel@0 627 Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j)
Daniel@0 628 </pre>
Daniel@0 629 Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then
Daniel@0 630 <pre>
Daniel@0 631 Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j))
Daniel@0 632 </pre>
Daniel@0 633 For a sigmoid node, we have
Daniel@0 634 <pre>
Daniel@0 635 Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j))
Daniel@0 636 </pre>
Daniel@0 637 where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of
Daniel@0 638 the activation function (although both are monotonically increasing).
Daniel@0 639 In addition, in the case of a noisy-or, the weights are constrained to be
Daniel@0 640 positive, since they derive from probabilities q(i,j).
Daniel@0 641 In both cases, the number of parameters is <em>linear</em> in the
Daniel@0 642 number of parents, unlike the case of a multinomial distribution,
Daniel@0 643 where the number of parameters is exponential in the number of parents.
Daniel@0 644 We will see an example of noisy-OR nodes <a href="#qmr">below</a>.
Daniel@0 645
Daniel@0 646
Daniel@0 647 <h2><a name="deterministic">Other (noisy) deterministic nodes</h2>
Daniel@0 648
Daniel@0 649 Deterministic CPDs for discrete random variables can be created using
Daniel@0 650 the deterministic_CPD class. It is also possible to 'flip' the output
Daniel@0 651 of the function with some probability, to simulate noise.
Daniel@0 652 The boolean_CPD class is just a special case of a
Daniel@0 653 deterministic CPD, where the parents and child are all binary.
Daniel@0 654 <p>
Daniel@0 655 Both of these classes are just "syntactic sugar" for the tabular_CPD
Daniel@0 656 class.
Daniel@0 657
Daniel@0 658
Daniel@0 659
Daniel@0 660 <h2><a name="softmax">Softmax nodes</h2>
Daniel@0 661
Daniel@0 662 If we have a discrete node with a continuous parent,
Daniel@0 663 we can define its CPD using a softmax function
Daniel@0 664 (also known as the multinomial logit function).
Daniel@0 665 This acts like a soft thresholding operator, and is defined as follows:
Daniel@0 666 <pre>
Daniel@0 667 exp(w(:,i)'*x + b(i))
Daniel@0 668 Pr(Q=i | X=x) = -----------------------------
Daniel@0 669 sum_j exp(w(:,j)'*x + b(j))
Daniel@0 670
Daniel@0 671 </pre>
Daniel@0 672 The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the
Daniel@0 673 following interpretation: w(:,i)-w(:,j) is the normal vector to the
Daniel@0 674 decision boundary between classes i and j,
Daniel@0 675 and b(i)-b(j) is its offset (bias). For example, suppose
Daniel@0 676 X is a 2-vector, and Q is binary. Then
Daniel@0 677 <pre>
Daniel@0 678 w = [1 -1;
Daniel@0 679 0 0];
Daniel@0 680
Daniel@0 681 b = [0 0];
Daniel@0 682 </pre>
Daniel@0 683 means class 1 are points in the 2D plane with positive x coordinate,
Daniel@0 684 and class 2 are points in the 2D plane with negative x coordinate.
Daniel@0 685 If w has large magnitude, the decision boundary is sharp, otherwise it
Daniel@0 686 is soft.
Daniel@0 687 In the special case that Q is binary (0/1), the softmax function reduces to the logistic
Daniel@0 688 (sigmoid) function.
Daniel@0 689 <p>
Daniel@0 690 Fitting a softmax function can be done using the iteratively reweighted
Daniel@0 691 least squares (IRLS) algorithm.
Daniel@0 692 We use the implementation from
Daniel@0 693 <a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>.
Daniel@0 694 Note that since
Daniel@0 695 the softmax distribution is not in the exponential family, it does not
Daniel@0 696 have finite sufficient statistics, and hence we must store all the
Daniel@0 697 training data in uncompressed form.
Daniel@0 698 If this takes too much space, one should use online (stochastic) gradient
Daniel@0 699 descent (not implemented in BNT).
Daniel@0 700 <p>
Daniel@0 701 If a softmax node also has discrete parents,
Daniel@0 702 we use a different set of w/b parameters for each combination of
Daniel@0 703 parent values, as in the <a href="#gaussian">conditional linear
Daniel@0 704 Gaussian CPD</a>.
Daniel@0 705 This feature was implemented by Pierpaolo Brutti.
Daniel@0 706 He is currently extending it so that discrete parents can be treated
Daniel@0 707 as if they were continuous, by adding indicator variables to the X
Daniel@0 708 vector.
Daniel@0 709 <p>
Daniel@0 710 We will see an example of softmax nodes <a href="#mixexp">below</a>.
Daniel@0 711
Daniel@0 712
Daniel@0 713 <h2><a name="mlp">Neural network nodes</h2>
Daniel@0 714
Daniel@0 715 Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron
Daniel@0 716 to implement a mapping from continuous parents to discrete children,
Daniel@0 717 similar to the softmax function.
Daniel@0 718 (If there are also discrete parents, it creates a mixture of MLPs.)
Daniel@0 719 It uses code from <a
Daniel@0 720 href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>.
Daniel@0 721 This is work in progress.
Daniel@0 722
Daniel@0 723 <h2><a name="root">Root nodes</h2>
Daniel@0 724
Daniel@0 725 A root node has no parents and no parameters; it can be used to model
Daniel@0 726 an observed, exogeneous input variable, i.e., one which is "outside"
Daniel@0 727 the model.
Daniel@0 728 This is useful for conditional density models.
Daniel@0 729 We will see an example of root nodes <a href="#mixexp">below</a>.
Daniel@0 730
Daniel@0 731
Daniel@0 732 <h2><a name="gaussian">Gaussian nodes</h2>
Daniel@0 733
Daniel@0 734 We now consider a distribution suitable for the continuous-valued nodes.
Daniel@0 735 Suppose the node is called Y, its continuous parents (if any) are
Daniel@0 736 called X, and its discrete parents (if any) are called Q.
Daniel@0 737 The distribution on Y is defined as follows:
Daniel@0 738 <pre>
Daniel@0 739 - no parents: Y ~ N(mu, Sigma)
Daniel@0 740 - cts parents : Y|X=x ~ N(mu + W x, Sigma)
Daniel@0 741 - discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i))
Daniel@0 742 - cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i))
Daniel@0 743 </pre>
Daniel@0 744 where N(mu, Sigma) denotes a Normal distribution with mean mu and
Daniel@0 745 covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q
Daniel@0 746 respectively.
Daniel@0 747 If there are no discrete parents, |Q|=1; if there is
Daniel@0 748 more than one, then |Q| = a vector of the sizes of each discrete parent.
Daniel@0 749 If there are no continuous parents, |X|=0; if there is more than one,
Daniel@0 750 then |X| = the sum of their sizes.
Daniel@0 751 Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive
Daniel@0 752 semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight)
Daniel@0 753 matrix.
Daniel@0 754 <p>
Daniel@0 755 We can create a Gaussian node with random parameters as follows.
Daniel@0 756 <pre>
Daniel@0 757 bnet.CPD{i} = gaussian_CPD(bnet, i);
Daniel@0 758 </pre>
Daniel@0 759 We can specify the value of one or more of the parameters as in the
Daniel@0 760 following example, in which |Y|=2, and |Q|=1.
Daniel@0 761 <pre>
Daniel@0 762 bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y));
Daniel@0 763 </pre>
Daniel@0 764 <p>
Daniel@0 765 We will see an example of conditional linear Gaussian nodes <a
Daniel@0 766 href="#cg_model">below</a>.
Daniel@0 767 <p>
Daniel@0 768 <b>When learning Gaussians from data</b>, it is helpful to ensure the
Daniel@0 769 data has a small magnitde
Daniel@0 770 (see e.g., KPMstats/standardize) to prevent numerical problems.
Daniel@0 771 Unless you have a lot of data, it is also a very good idea to use
Daniel@0 772 diagonal instead of full covariance matrices.
Daniel@0 773 (BNT does not currently support spherical covariances, although it
Daniel@0 774 would be easy to add, since KPMstats/clg_Mstep supports this option;
Daniel@0 775 you would just need to modify gaussian_CPD/update_ess to accumulate
Daniel@0 776 weighted inner products.)
Daniel@0 777
Daniel@0 778
Daniel@0 779
Daniel@0 780 <h2><a name="nongauss">Other continuous distributions</h2>
Daniel@0 781
Daniel@0 782 Currently BNT does not support any CPDs for continuous nodes other
Daniel@0 783 than the Gaussian.
Daniel@0 784 However, you can use a mixture of Gaussians to
Daniel@0 785 approximate other continuous distributions. We will see some an example
Daniel@0 786 of this with the IFA model <a href="#pca">below</a>.
Daniel@0 787
Daniel@0 788
Daniel@0 789 <h2><a name="glm">Generalized linear model nodes</h2>
Daniel@0 790
Daniel@0 791 In the future, we may incorporate some of the functionality of
Daniel@0 792 <a href =
Daniel@0 793 "http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a>
Daniel@0 794 into BNT.
Daniel@0 795
Daniel@0 796
Daniel@0 797 <h2><a name="dtree">Classification/regression tree nodes</h2>
Daniel@0 798
Daniel@0 799 We plan to add classification and regression trees to define CPDs for
Daniel@0 800 discrete and continuous nodes, respectively.
Daniel@0 801 Trees have many advantages: they are easy to interpret, they can do
Daniel@0 802 feature selection, they can
Daniel@0 803 handle discrete and continuous inputs, they do not make strong
Daniel@0 804 assumptions about the form of the distribution, the number of
Daniel@0 805 parameters can grow in a data-dependent way (i.e., they are
Daniel@0 806 semi-parametric), they can handle missing data, etc.
Daniel@0 807 However, they are not yet implemented.
Daniel@0 808 <!--
Daniel@0 809 Yimin Zhang is currently (Feb '02) implementing this.
Daniel@0 810 -->
Daniel@0 811
Daniel@0 812
Daniel@0 813 <h2><a name="cpd_summary">Summary of CPD types</h2>
Daniel@0 814
Daniel@0 815 We list all the different types of CPDs supported by BNT.
Daniel@0 816 For each CPD, we specify if the child and parents can be discrete (D) or
Daniel@0 817 continuous (C) (Binary (B) nodes are a special case).
Daniel@0 818 We also specify which methods each class supports.
Daniel@0 819 If a method is inherited, the name of the parent class is mentioned.
Daniel@0 820 If a parent class calls a child method, this is mentioned.
Daniel@0 821 <p>
Daniel@0 822 The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this
Daniel@0 823 requires that the child and all parents are discrete.
Daniel@0 824 The CPT might be exponentially big...
Daniel@0 825 <tt>convert_to_table</tt> evaluates a CPD with evidence, and
Daniel@0 826 represents the the resulting potential as an array.
Daniel@0 827 This requires that the child is discrete, and any continuous parents
Daniel@0 828 are observed.
Daniel@0 829 <tt>convert_to_pot</tt> evaluates a CPD with evidence, and
Daniel@0 830 represents the resulting potential as a dpot, gpot, cgpot or upot, as
Daniel@0 831 requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u =
Daniel@0 832 utility).
Daniel@0 833
Daniel@0 834 <p>
Daniel@0 835 When we sample a node, all the parents are observed.
Daniel@0 836 When we compute the (log) probability of a node, all the parents and
Daniel@0 837 the child are observed.
Daniel@0 838 <p>
Daniel@0 839 We also specify if the parameters are learnable.
Daniel@0 840 For learning with EM, we require
Daniel@0 841 the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and
Daniel@0 842 <tt>maximize_params</tt>.
Daniel@0 843 For learning from fully observed data, we require
Daniel@0 844 the method <tt>learn_params</tt>.
Daniel@0 845 By default, all classes inherit this from generic_CPD, which simply
Daniel@0 846 calls <tt>update_ess</tt> N times, once for each data case, followed
Daniel@0 847 by <tt>maximize_params</tt>, i.e., it is like EM, without the E step.
Daniel@0 848 Some classes implement a batch formula, which is quicker.
Daniel@0 849 <p>
Daniel@0 850 Bayesian learning means computing a posterior over the parameters
Daniel@0 851 given fully observed data.
Daniel@0 852 <p>
Daniel@0 853 Pearl means we implement the methods <tt>compute_pi</tt> and
Daniel@0 854 <tt>compute_lambda_msg</tt>, used by
Daniel@0 855 <tt>pearl_inf_engine</tt>, which runs on directed graphs.
Daniel@0 856 <tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H
Daniel@0 857 The pearl methods can exploit special properties of the CPDs for
Daniel@0 858 computing the messages efficiently, whereas belprop does not.
Daniel@0 859 <p>
Daniel@0 860 The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>,
Daniel@0 861 which is not shown, since it is not very interesting.
Daniel@0 862
Daniel@0 863
Daniel@0 864 <p>
Daniel@0 865
Daniel@0 866
Daniel@0 867 <table>
Daniel@0 868 <table border units = pixels><tr>
Daniel@0 869 <td align=center>Name
Daniel@0 870 <td align=center>Child
Daniel@0 871 <td align=center>Parents
Daniel@0 872 <td align=center>Comments
Daniel@0 873 <td align=center>CPD_to_CPT
Daniel@0 874 <td align=center>conv_to_table
Daniel@0 875 <td align=center>conv_to_pot
Daniel@0 876 <td align=center>sample
Daniel@0 877 <td align=center>prob
Daniel@0 878 <td align=center>learn
Daniel@0 879 <td align=center>Bayes
Daniel@0 880 <td align=center>Pearl
Daniel@0 881
Daniel@0 882
Daniel@0 883 <tr>
Daniel@0 884 <!-- Name--><td>
Daniel@0 885 <!-- Child--><td>
Daniel@0 886 <!-- Parents--><td>
Daniel@0 887 <!-- Comments--><td>
Daniel@0 888 <!-- CPD_to_CPT--><td>
Daniel@0 889 <!-- conv_to_table--><td>
Daniel@0 890 <!-- conv_to_pot--><td>
Daniel@0 891 <!-- sample--><td>
Daniel@0 892 <!-- prob--><td>
Daniel@0 893 <!-- learn--><td>
Daniel@0 894 <!-- Bayes--><td>
Daniel@0 895 <!-- Pearl--><td>
Daniel@0 896
Daniel@0 897 <tr>
Daniel@0 898 <!-- Name--><td>boolean
Daniel@0 899 <!-- Child--><td>B
Daniel@0 900 <!-- Parents--><td>B
Daniel@0 901 <!-- Comments--><td>Syntactic sugar for tabular
Daniel@0 902 <!-- CPD_to_CPT--><td>-
Daniel@0 903 <!-- conv_to_table--><td>-
Daniel@0 904 <!-- conv_to_pot--><td>-
Daniel@0 905 <!-- sample--><td>-
Daniel@0 906 <!-- prob--><td>-
Daniel@0 907 <!-- learn--><td>-
Daniel@0 908 <!-- Bayes--><td>-
Daniel@0 909 <!-- Pearl--><td>-
Daniel@0 910
Daniel@0 911 <tr>
Daniel@0 912 <!-- Name--><td>deterministic
Daniel@0 913 <!-- Child--><td>D
Daniel@0 914 <!-- Parents--><td>D
Daniel@0 915 <!-- Comments--><td>Syntactic sugar for tabular
Daniel@0 916 <!-- CPD_to_CPT--><td>-
Daniel@0 917 <!-- conv_to_table--><td>-
Daniel@0 918 <!-- conv_to_pot--><td>-
Daniel@0 919 <!-- sample--><td>-
Daniel@0 920 <!-- prob--><td>-
Daniel@0 921 <!-- learn--><td>-
Daniel@0 922 <!-- Bayes--><td>-
Daniel@0 923 <!-- Pearl--><td>-
Daniel@0 924
Daniel@0 925 <tr>
Daniel@0 926 <!-- Name--><td>Discrete
Daniel@0 927 <!-- Child--><td>D
Daniel@0 928 <!-- Parents--><td>C/D
Daniel@0 929 <!-- Comments--><td>Virtual class
Daniel@0 930 <!-- CPD_to_CPT--><td>N
Daniel@0 931 <!-- conv_to_table--><td>Calls CPD_to_CPT
Daniel@0 932 <!-- conv_to_pot--><td>Calls conv_to_table
Daniel@0 933 <!-- sample--><td>Calls conv_to_table
Daniel@0 934 <!-- prob--><td>Calls conv_to_table
Daniel@0 935 <!-- learn--><td>N
Daniel@0 936 <!-- Bayes--><td>N
Daniel@0 937 <!-- Pearl--><td>N
Daniel@0 938
Daniel@0 939 <tr>
Daniel@0 940 <!-- Name--><td>Gaussian
Daniel@0 941 <!-- Child--><td>C
Daniel@0 942 <!-- Parents--><td>C/D
Daniel@0 943 <!-- Comments--><td>-
Daniel@0 944 <!-- CPD_to_CPT--><td>N
Daniel@0 945 <!-- conv_to_table--><td>N
Daniel@0 946 <!-- conv_to_pot--><td>Y
Daniel@0 947 <!-- sample--><td>Y
Daniel@0 948 <!-- prob--><td>Y
Daniel@0 949 <!-- learn--><td>Y
Daniel@0 950 <!-- Bayes--><td>N
Daniel@0 951 <!-- Pearl--><td>N
Daniel@0 952
Daniel@0 953 <tr>
Daniel@0 954 <!-- Name--><td>gmux
Daniel@0 955 <!-- Child--><td>C
Daniel@0 956 <!-- Parents--><td>C/D
Daniel@0 957 <!-- Comments--><td>multiplexer
Daniel@0 958 <!-- CPD_to_CPT--><td>N
Daniel@0 959 <!-- conv_to_table--><td>N
Daniel@0 960 <!-- conv_to_pot--><td>Y
Daniel@0 961 <!-- sample--><td>N
Daniel@0 962 <!-- prob--><td>N
Daniel@0 963 <!-- learn--><td>N
Daniel@0 964 <!-- Bayes--><td>N
Daniel@0 965 <!-- Pearl--><td>Y
Daniel@0 966
Daniel@0 967
Daniel@0 968 <tr>
Daniel@0 969 <!-- Name--><td>MLP
Daniel@0 970 <!-- Child--><td>D
Daniel@0 971 <!-- Parents--><td>C/D
Daniel@0 972 <!-- Comments--><td>multi layer perceptron
Daniel@0 973 <!-- CPD_to_CPT--><td>N
Daniel@0 974 <!-- conv_to_table--><td>Y
Daniel@0 975 <!-- conv_to_pot--><td>Inherits from discrete
Daniel@0 976 <!-- sample--><td>Inherits from discrete
Daniel@0 977 <!-- prob--><td>Inherits from discrete
Daniel@0 978 <!-- learn--><td>Y
Daniel@0 979 <!-- Bayes--><td>N
Daniel@0 980 <!-- Pearl--><td>N
Daniel@0 981
Daniel@0 982
Daniel@0 983 <tr>
Daniel@0 984 <!-- Name--><td>noisy-or
Daniel@0 985 <!-- Child--><td>B
Daniel@0 986 <!-- Parents--><td>B
Daniel@0 987 <!-- Comments--><td>-
Daniel@0 988 <!-- CPD_to_CPT--><td>Y
Daniel@0 989 <!-- conv_to_table--><td>Inherits from discrete
Daniel@0 990 <!-- conv_to_pot--><td>Inherits from discrete
Daniel@0 991 <!-- sample--><td>Inherits from discrete
Daniel@0 992 <!-- prob--><td>Inherits from discrete
Daniel@0 993 <!-- learn--><td>N
Daniel@0 994 <!-- Bayes--><td>N
Daniel@0 995 <!-- Pearl--><td>Y
Daniel@0 996
Daniel@0 997
Daniel@0 998 <tr>
Daniel@0 999 <!-- Name--><td>root
Daniel@0 1000 <!-- Child--><td>C/D
Daniel@0 1001 <!-- Parents--><td>none
Daniel@0 1002 <!-- Comments--><td>no params
Daniel@0 1003 <!-- CPD_to_CPT--><td>N
Daniel@0 1004 <!-- conv_to_table--><td>N
Daniel@0 1005 <!-- conv_to_pot--><td>Y
Daniel@0 1006 <!-- sample--><td>Y
Daniel@0 1007 <!-- prob--><td>Y
Daniel@0 1008 <!-- learn--><td>N
Daniel@0 1009 <!-- Bayes--><td>N
Daniel@0 1010 <!-- Pearl--><td>N
Daniel@0 1011
Daniel@0 1012
Daniel@0 1013 <tr>
Daniel@0 1014 <!-- Name--><td>softmax
Daniel@0 1015 <!-- Child--><td>D
Daniel@0 1016 <!-- Parents--><td>C/D
Daniel@0 1017 <!-- Comments--><td>-
Daniel@0 1018 <!-- CPD_to_CPT--><td>N
Daniel@0 1019 <!-- conv_to_table--><td>Y
Daniel@0 1020 <!-- conv_to_pot--><td>Inherits from discrete
Daniel@0 1021 <!-- sample--><td>Inherits from discrete
Daniel@0 1022 <!-- prob--><td>Inherits from discrete
Daniel@0 1023 <!-- learn--><td>Y
Daniel@0 1024 <!-- Bayes--><td>N
Daniel@0 1025 <!-- Pearl--><td>N
Daniel@0 1026
Daniel@0 1027
Daniel@0 1028 <tr>
Daniel@0 1029 <!-- Name--><td>generic
Daniel@0 1030 <!-- Child--><td>C/D
Daniel@0 1031 <!-- Parents--><td>C/D
Daniel@0 1032 <!-- Comments--><td>Virtual class
Daniel@0 1033 <!-- CPD_to_CPT--><td>N
Daniel@0 1034 <!-- conv_to_table--><td>N
Daniel@0 1035 <!-- conv_to_pot--><td>N
Daniel@0 1036 <!-- sample--><td>N
Daniel@0 1037 <!-- prob--><td>N
Daniel@0 1038 <!-- learn--><td>N
Daniel@0 1039 <!-- Bayes--><td>N
Daniel@0 1040 <!-- Pearl--><td>N
Daniel@0 1041
Daniel@0 1042
Daniel@0 1043 <tr>
Daniel@0 1044 <!-- Name--><td>Tabular
Daniel@0 1045 <!-- Child--><td>D
Daniel@0 1046 <!-- Parents--><td>D
Daniel@0 1047 <!-- Comments--><td>-
Daniel@0 1048 <!-- CPD_to_CPT--><td>Y
Daniel@0 1049 <!-- conv_to_table--><td>Inherits from discrete
Daniel@0 1050 <!-- conv_to_pot--><td>Inherits from discrete
Daniel@0 1051 <!-- sample--><td>Inherits from discrete
Daniel@0 1052 <!-- prob--><td>Inherits from discrete
Daniel@0 1053 <!-- learn--><td>Y
Daniel@0 1054 <!-- Bayes--><td>Y
Daniel@0 1055 <!-- Pearl--><td>Y
Daniel@0 1056
Daniel@0 1057 </table>
Daniel@0 1058
Daniel@0 1059
Daniel@0 1060
Daniel@0 1061 <h1><a name="examples">Example models</h1>
Daniel@0 1062
Daniel@0 1063
Daniel@0 1064 <h2>Gaussian mixture models</h2>
Daniel@0 1065
Daniel@0 1066 Richard W. DeVaul has made a detailed tutorial on how to fit mixtures
Daniel@0 1067 of Gaussians using BNT. Available
Daniel@0 1068 <a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>.
Daniel@0 1069
Daniel@0 1070
Daniel@0 1071 <h2><a name="pca">PCA, ICA, and all that </h2>
Daniel@0 1072
Daniel@0 1073 In Figure (a) below, we show how Factor Analysis can be thought of as a
Daniel@0 1074 graphical model. Here, X has an N(0,I) prior, and
Daniel@0 1075 Y|X=x ~ N(mu + Wx, Psi),
Daniel@0 1076 where Psi is diagonal and W is called the "factor loading matrix".
Daniel@0 1077 Since the noise on both X and Y is diagonal, the components of these
Daniel@0 1078 vectors are uncorrelated, and hence can be represented as individual
Daniel@0 1079 scalar nodes, as we show in (b).
Daniel@0 1080 (This is useful if parts of the observations on the Y vector are occasionally missing.)
Daniel@0 1081 We usually take k=|X| << |Y|=D, so the model tries to explain
Daniel@0 1082 many observations using a low-dimensional subspace.
Daniel@0 1083
Daniel@0 1084
Daniel@0 1085 <center>
Daniel@0 1086 <table>
Daniel@0 1087 <tr>
Daniel@0 1088 <td><img src="Figures/fa.gif">
Daniel@0 1089 <td><img src="Figures/fa_scalar.gif">
Daniel@0 1090 <td><img src="Figures/mfa.gif">
Daniel@0 1091 <td><img src="Figures/ifa.gif">
Daniel@0 1092 <tr>
Daniel@0 1093 <td align=center> (a)
Daniel@0 1094 <td align=center> (b)
Daniel@0 1095 <td align=center> (c)
Daniel@0 1096 <td align=center> (d)
Daniel@0 1097 </table>
Daniel@0 1098 </center>
Daniel@0 1099
Daniel@0 1100 <p>
Daniel@0 1101 We can create this model in BNT as follows.
Daniel@0 1102 <pre>
Daniel@0 1103 ns = [k D];
Daniel@0 1104 dag = zeros(2,2);
Daniel@0 1105 dag(1,2) = 1;
Daniel@0 1106 bnet = mk_bnet(dag, ns, 'discrete', []);
Daniel@0 1107 bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ...
Daniel@0 1108 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1);
Daniel@0 1109 bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ...
Daniel@0 1110 'cov_type', 'diag', 'clamp_mean', 1);
Daniel@0 1111 </pre>
Daniel@0 1112
Daniel@0 1113 The root node is clamped to the N(0,I) distribution, so that we will
Daniel@0 1114 not update these parameters during learning.
Daniel@0 1115 The mean of the leaf node is clamped to 0,
Daniel@0 1116 since we assume the data has been centered (had its mean subtracted
Daniel@0 1117 off); this is just for simplicity.
Daniel@0 1118 Finally, the covariance of the leaf node is constrained to be
Daniel@0 1119 diagonal. W0 and Psi0 are the initial parameter guesses.
Daniel@0 1120
Daniel@0 1121 <p>
Daniel@0 1122 We can fit this model (i.e., estimate its parameters in a maximum
Daniel@0 1123 likelihood (ML) sense) using EM, as we
Daniel@0 1124 explain <a href="#em">below</a>.
Daniel@0 1125 Not surprisingly, the ML estimates for mu and Psi turn out to be
Daniel@0 1126 identical to the
Daniel@0 1127 sample mean and variance, which can be computed directly as
Daniel@0 1128 <pre>
Daniel@0 1129 mu_ML = mean(data);
Daniel@0 1130 Psi_ML = diag(cov(data));
Daniel@0 1131 </pre>
Daniel@0 1132 Note that W can only be identified up to a rotation matrix, because of
Daniel@0 1133 the spherical symmetry of the source.
Daniel@0 1134
Daniel@0 1135 <p>
Daniel@0 1136 If we restrict Psi to be spherical, i.e., Psi = sigma*I,
Daniel@0 1137 there is a closed-form solution for W as well,
Daniel@0 1138 i.e., we do not need to use EM.
Daniel@0 1139 In particular, W contains the first |X| eigenvectors of the sample covariance
Daniel@0 1140 matrix, with scalings determined by the eigenvalues and sigma.
Daniel@0 1141 Classical PCA can be obtained by taking the sigma->0 limit.
Daniel@0 1142 For details, see
Daniel@0 1143
Daniel@0 1144 <ul>
Daniel@0 1145 <li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps">
Daniel@0 1146 "EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97.
Daniel@0 1147 (<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz">
Daniel@0 1148 Matlab software</a>)
Daniel@0 1149
Daniel@0 1150 <p>
Daniel@0 1151 <li>
Daniel@0 1152 <a
Daniel@0 1153 href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003>
Daniel@0 1154 "Mixtures of probabilistic principal component analyzers"</a>,
Daniel@0 1155 Tipping and Bishop, Neural Computation 11(2):443--482, 1999.
Daniel@0 1156 </ul>
Daniel@0 1157
Daniel@0 1158 <p>
Daniel@0 1159 By adding a hidden discrete variable, we can create mixtures of FA
Daniel@0 1160 models, as shown in (c).
Daniel@0 1161 Now we can explain the data using a set of subspaces.
Daniel@0 1162 We can create this model in BNT as follows.
Daniel@0 1163 <pre>
Daniel@0 1164 ns = [M k D];
Daniel@0 1165 dag = zeros(3);
Daniel@0 1166 dag(1,3) = 1;
Daniel@0 1167 dag(2,3) = 1;
Daniel@0 1168 bnet = mk_bnet(dag, ns, 'discrete', 1);
Daniel@0 1169 bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0);
Daniel@0 1170 bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ...
Daniel@0 1171 'clamp_mean', 1, 'clamp_cov', 1);
Daniel@0 1172 bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ...
Daniel@0 1173 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1);
Daniel@0 1174 </pre>
Daniel@0 1175 Notice how the covariance matrix for Y is the same for all values of
Daniel@0 1176 Q; that is, the noise level in each sub-space is assumed the same.
Daniel@0 1177 However, we allow the offset, mu, to vary.
Daniel@0 1178 For details, see
Daniel@0 1179 <ul>
Daniel@0 1180
Daniel@0 1181 <LI>
Daniel@0 1182 <a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM
Daniel@0 1183 Algorithm for Mixtures of Factor Analyzers </A>,
Daniel@0 1184 Ghahramani, Z. and Hinton, G.E. (1996),
Daniel@0 1185 University of Toronto
Daniel@0 1186 Technical Report CRG-TR-96-1.
Daniel@0 1187 (<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>)
Daniel@0 1188
Daniel@0 1189 <p>
Daniel@0 1190 <li>
Daniel@0 1191 <a
Daniel@0 1192 href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003>
Daniel@0 1193 "Mixtures of probabilistic principal component analyzers"</a>,
Daniel@0 1194 Tipping and Bishop, Neural Computation 11(2):443--482, 1999.
Daniel@0 1195 </ul>
Daniel@0 1196
Daniel@0 1197 <p>
Daniel@0 1198 I have included Zoubin's specialized MFA code (with his permission)
Daniel@0 1199 with the toolbox, so you can check that BNT gives the same results:
Daniel@0 1200 see 'BNT/examples/static/mfa1.m'.
Daniel@0 1201
Daniel@0 1202 <p>
Daniel@0 1203 Independent Factor Analysis (IFA) generalizes FA by allowing a
Daniel@0 1204 non-Gaussian prior on each component of X.
Daniel@0 1205 (Note that we can approximate a non-Gaussian prior using a mixture of
Daniel@0 1206 Gaussians.)
Daniel@0 1207 This means that the likelihood function is no longer rotationally
Daniel@0 1208 invariant, so we can uniquely identify W and the hidden
Daniel@0 1209 sources X.
Daniel@0 1210 IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y).
Daniel@0 1211 We recover classical Independent Components Analysis (ICA)
Daniel@0 1212 in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the
Daniel@0 1213 weight matrix W is square and invertible.
Daniel@0 1214 For details, see
Daniel@0 1215 <ul>
Daniel@0 1216 <li>
Daniel@0 1217 <a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor
Daniel@0 1218 Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998.
Daniel@0 1219 </ul>
Daniel@0 1220
Daniel@0 1221
Daniel@0 1222
Daniel@0 1223 <h2><a name="mixexp">Mixtures of experts</h2>
Daniel@0 1224
Daniel@0 1225 As an example of the use of the softmax function,
Daniel@0 1226 we introduce the Mixture of Experts model.
Daniel@0 1227 <!--
Daniel@0 1228 We also show
Daniel@0 1229 the Hierarchical Mixture of Experts model, where the hierarchy has two
Daniel@0 1230 levels.
Daniel@0 1231 (This is essentially a probabilistic decision tree of height two.)
Daniel@0 1232 -->
Daniel@0 1233 As before,
Daniel@0 1234 circles denote continuous-valued nodes,
Daniel@0 1235 squares denote discrete nodes, clear
Daniel@0 1236 means hidden, and shaded means observed.
Daniel@0 1237 <p>
Daniel@0 1238 <center>
Daniel@0 1239 <table>
Daniel@0 1240 <tr>
Daniel@0 1241 <td><img src="Figures/mixexp.gif">
Daniel@0 1242 <!--
Daniel@0 1243 <td><img src="Figures/hme.gif">
Daniel@0 1244 -->
Daniel@0 1245 </table>
Daniel@0 1246 </center>
Daniel@0 1247 <p>
Daniel@0 1248 X is the observed
Daniel@0 1249 input, Y is the output, and
Daniel@0 1250 the Q nodes are hidden "gating" nodes, which select the appropriate
Daniel@0 1251 set of parameters for Y. During training, Y is assumed observed,
Daniel@0 1252 but for testing, the goal is to predict Y given X.
Daniel@0 1253 Note that this is a <em>conditional</em> density model, so we don't
Daniel@0 1254 associate any parameters with X.
Daniel@0 1255 Hence X's CPD will be a root CPD, which is a way of modelling
Daniel@0 1256 exogenous nodes.
Daniel@0 1257 If the output is a continuous-valued quantity,
Daniel@0 1258 we assume the "experts" are linear-regression units,
Daniel@0 1259 and set Y's CPD to linear-Gaussian.
Daniel@0 1260 If the output is discrete, we set Y's CPD to a softmax function.
Daniel@0 1261 The Q CPDs will always be softmax functions.
Daniel@0 1262
Daniel@0 1263 <p>
Daniel@0 1264 As a concrete example, consider the mixture of experts model where X and Y are
Daniel@0 1265 scalars, and Q is binary.
Daniel@0 1266 This is just piecewise linear regression, where
Daniel@0 1267 we have two line segments, i.e.,
Daniel@0 1268 <P>
Daniel@0 1269 <IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif">
Daniel@0 1270 <P>
Daniel@0 1271 We can create this model with random parameters as follows.
Daniel@0 1272 (This code is bundled in BNT/examples/static/mixexp2.m.)
Daniel@0 1273 <PRE>
Daniel@0 1274 X = 1;
Daniel@0 1275 Q = 2;
Daniel@0 1276 Y = 3;
Daniel@0 1277 dag = zeros(3,3);
Daniel@0 1278 dag(X,[Q Y]) = 1
Daniel@0 1279 dag(Q,Y) = 1;
Daniel@0 1280 ns = [1 2 1]; % make X and Y scalars, and have 2 experts
Daniel@0 1281 onodes = [1 3];
Daniel@0 1282 bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes);
Daniel@0 1283
Daniel@0 1284 rand('state', 0);
Daniel@0 1285 randn('state', 0);
Daniel@0 1286 bnet.CPD{1} = root_CPD(bnet, 1);
Daniel@0 1287 bnet.CPD{2} = softmax_CPD(bnet, 2);
Daniel@0 1288 bnet.CPD{3} = gaussian_CPD(bnet, 3);
Daniel@0 1289 </PRE>
Daniel@0 1290 Now let us fit this model using <a href="#em">EM</a>.
Daniel@0 1291 First we <a href="#load_data">load the data</a> (1000 training cases) and plot them.
Daniel@0 1292 <P>
Daniel@0 1293 <PRE>
Daniel@0 1294 data = load('/examples/static/Misc/mixexp_data.txt', '-ascii');
Daniel@0 1295 plot(data(:,1), data(:,2), '.');
Daniel@0 1296 </PRE>
Daniel@0 1297 <p>
Daniel@0 1298 <center>
Daniel@0 1299 <IMG SRC="Figures/mixexp_data.gif">
Daniel@0 1300 </center>
Daniel@0 1301 <p>
Daniel@0 1302 This is what the model looks like before training.
Daniel@0 1303 (Thanks to Thomas Hofman for writing this plotting routine.)
Daniel@0 1304 <p>
Daniel@0 1305 <center>
Daniel@0 1306 <IMG SRC="Figures/mixexp_before.gif">
Daniel@0 1307 </center>
Daniel@0 1308 <p>
Daniel@0 1309 Now let's train the model, and plot the final performance.
Daniel@0 1310 (We will discuss how to train models in more detail <a href="#param_learning">below</a>.)
Daniel@0 1311 <P>
Daniel@0 1312 <PRE>
Daniel@0 1313 ncases = size(data, 1); % each row of data is a training case
Daniel@0 1314 cases = cell(3, ncases);
Daniel@0 1315 cases([1 3], :) = num2cell(data'); % each column of cases is a training case
Daniel@0 1316 engine = jtree_inf_engine(bnet);
Daniel@0 1317 max_iter = 20;
Daniel@0 1318 [bnet2, LLtrace] = learn_params_em(engine, cases, max_iter);
Daniel@0 1319 </PRE>
Daniel@0 1320 (We specify which nodes will be observed when we create the engine.
Daniel@0 1321 Hence BNT knows that the hidden nodes are all discrete.
Daniel@0 1322 For complex models, this can lead to a significant speedup.)
Daniel@0 1323 Below we show what the model looks like after 16 iterations of EM
Daniel@0 1324 (with 100 IRLS iterations per M step), when it converged
Daniel@0 1325 using the default convergence tolerance (that the
Daniel@0 1326 fractional change in the log-likelihood be less than 1e-3).
Daniel@0 1327 Before learning, the log-likelihood was
Daniel@0 1328 -322.927442; afterwards, it was -13.728778.
Daniel@0 1329 <p>
Daniel@0 1330 <center>
Daniel@0 1331 <IMG SRC="Figures/mixexp_after.gif">
Daniel@0 1332 </center>
Daniel@0 1333 (See BNT/examples/static/mixexp2.m for details of the code.)
Daniel@0 1334
Daniel@0 1335
Daniel@0 1336
Daniel@0 1337 <h2><a name="hme">Hierarchical mixtures of experts</h2>
Daniel@0 1338
Daniel@0 1339 A hierarchical mixture of experts (HME) extends the mixture of experts
Daniel@0 1340 model by having more than one hidden node. A two-level example is shown below, along
Daniel@0 1341 with its more traditional representation as a neural network.
Daniel@0 1342 This is like a (balanced) probabilistic decision tree of height 2.
Daniel@0 1343 <p>
Daniel@0 1344 <center>
Daniel@0 1345 <IMG SRC="Figures/HMEforMatlab.jpg">
Daniel@0 1346 </center>
Daniel@0 1347 <p>
Daniel@0 1348 <a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a>
Daniel@0 1349 has written an extensive set of routines for HMEs,
Daniel@0 1350 which are bundled with BNT: see the examples/static/HME directory.
Daniel@0 1351 These routines allow you to choose the number of hidden (gating)
Daniel@0 1352 layers, and the form of the experts (softmax or MLP).
Daniel@0 1353 See the file hmemenu, which provides a demo.
Daniel@0 1354 For example, the figure below shows the decision boundaries learned
Daniel@0 1355 for a ternary classification problem, using a 2 level HME with softmax
Daniel@0 1356 gates and softmax experts; the training set is on the left, the
Daniel@0 1357 testing set on the right.
Daniel@0 1358 <p>
Daniel@0 1359 <center>
Daniel@0 1360 <!--<IMG SRC="Figures/hme_dec_boundary.gif">-->
Daniel@0 1361 <IMG SRC="Figures/hme_dec_boundary.png">
Daniel@0 1362 </center>
Daniel@0 1363 <p>
Daniel@0 1364
Daniel@0 1365
Daniel@0 1366 <p>
Daniel@0 1367 For more details, see the following:
Daniel@0 1368 <ul>
Daniel@0 1369
Daniel@0 1370 <li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z">
Daniel@0 1371 Hierarchical mixtures of experts and the EM algorithm</a>
Daniel@0 1372 M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994.
Daniel@0 1373
Daniel@0 1374 <li> <a href =
Daniel@0 1375 "http://www.cs.berkeley.edu/~dmartin/software">David Martin's
Daniel@0 1376 matlab code for HME</a>
Daniel@0 1377
Daniel@0 1378 <li> <a
Daniel@0 1379 href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the
Daniel@0 1380 logistic function? A tutorial discussion on
Daniel@0 1381 probabilities and neural networks.</a> M. I. Jordan. MIT Computational
Daniel@0 1382 Cognitive Science Report 9503, August 1995.
Daniel@0 1383
Daniel@0 1384 <li> "Generalized Linear Models", McCullagh and Nelder, Chapman and
Daniel@0 1385 Halll, 1983.
Daniel@0 1386
Daniel@0 1387 <li>
Daniel@0 1388 "Improved learning algorithms for mixtures of experts in multiclass
Daniel@0 1389 classification".
Daniel@0 1390 K. Chen, L. Xu, H. Chi.
Daniel@0 1391 Neural Networks (1999) 12: 1229-1252.
Daniel@0 1392
Daniel@0 1393 <li> <a href="http://www.oigeeza.com/steve/">
Daniel@0 1394 Classification Using Hierarchical Mixtures of Experts</a>
Daniel@0 1395 S.R. Waterhouse and A.J. Robinson.
Daniel@0 1396 In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186
Daniel@0 1397
Daniel@0 1398 <li> <a href="http://www.idiap.ch/~perry/">
Daniel@0 1399 Localized mixtures of experts</a>,
Daniel@0 1400 P. Moerland, 1998.
Daniel@0 1401
Daniel@0 1402 <li> "Nonlinear gated experts for time series",
Daniel@0 1403 A.S. Weigend and M. Mangeas, 1995.
Daniel@0 1404
Daniel@0 1405 </ul>
Daniel@0 1406
Daniel@0 1407
Daniel@0 1408 <h2><a name="qmr">QMR</h2>
Daniel@0 1409
Daniel@0 1410 Bayes nets originally arose out of an attempt to add probabilities to
Daniel@0 1411 expert systems, and this is still the most common use for BNs.
Daniel@0 1412 A famous example is
Daniel@0 1413 QMR-DT, a decision-theoretic reformulation of the Quick Medical
Daniel@0 1414 Reference (QMR) model.
Daniel@0 1415 <p>
Daniel@0 1416 <center>
Daniel@0 1417 <IMG ALIGN=BOTTOM SRC="Figures/qmr.gif">
Daniel@0 1418 </center>
Daniel@0 1419 Here, the top layer represents hidden disease nodes, and the bottom
Daniel@0 1420 layer represents observed symptom nodes.
Daniel@0 1421 The goal is to infer the posterior probability of each disease given
Daniel@0 1422 all the symptoms (which can be present, absent or unknown).
Daniel@0 1423 Each node in the top layer has a Bernoulli prior (with a low prior
Daniel@0 1424 probability that the disease is present).
Daniel@0 1425 Since each node in the bottom layer has a high fan-in, we use a
Daniel@0 1426 noisy-OR parameterization; each disease has an independent chance of
Daniel@0 1427 causing each symptom.
Daniel@0 1428 The real QMR-DT model is copyright, but
Daniel@0 1429 we can create a random QMR-like model as follows.
Daniel@0 1430 <pre>
Daniel@0 1431 function bnet = mk_qmr_bnet(G, inhibit, leak, prior)
Daniel@0 1432 % MK_QMR_BNET Make a QMR model
Daniel@0 1433 % bnet = mk_qmr_bnet(G, inhibit, leak, prior)
Daniel@0 1434 %
Daniel@0 1435 % G(i,j) = 1 iff there is an arc from disease i to finding j
Daniel@0 1436 % inhibit(i,j) = inhibition probability on i->j arc
Daniel@0 1437 % leak(j) = inhibition prob. on leak->j arc
Daniel@0 1438 % prior(i) = prob. disease i is on
Daniel@0 1439
Daniel@0 1440 [Ndiseases Nfindings] = size(inhibit);
Daniel@0 1441 N = Ndiseases + Nfindings;
Daniel@0 1442 finding_node = Ndiseases+1:N;
Daniel@0 1443 ns = 2*ones(1,N);
Daniel@0 1444 dag = zeros(N,N);
Daniel@0 1445 dag(1:Ndiseases, finding_node) = G;
Daniel@0 1446 bnet = mk_bnet(dag, ns, 'observed', finding_node);
Daniel@0 1447
Daniel@0 1448 for d=1:Ndiseases
Daniel@0 1449 CPT = [1-prior(d) prior(d)];
Daniel@0 1450 bnet.CPD{d} = tabular_CPD(bnet, d, CPT');
Daniel@0 1451 end
Daniel@0 1452
Daniel@0 1453 for i=1:Nfindings
Daniel@0 1454 fnode = finding_node(i);
Daniel@0 1455 ps = parents(G, i);
Daniel@0 1456 bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i));
Daniel@0 1457 end
Daniel@0 1458 </pre>
Daniel@0 1459 In the file BNT/examples/static/qmr1, we create a random bipartite
Daniel@0 1460 graph G, with 5 diseases and 10 findings, and random parameters.
Daniel@0 1461 (In general, to create a random dag, use 'mk_random_dag'.)
Daniel@0 1462 We can visualize the resulting graph structure using
Daniel@0 1463 the methods discussed <a href="#graphdraw">below</a>, with the
Daniel@0 1464 following results:
Daniel@0 1465 <p>
Daniel@0 1466 <img src="Figures/qmr.rnd.jpg">
Daniel@0 1467
Daniel@0 1468 <p>
Daniel@0 1469 Now let us put some random evidence on all the leaves except the very
Daniel@0 1470 first and very last, and compute the disease posteriors.
Daniel@0 1471 <pre>
Daniel@0 1472 pos = 2:floor(Nfindings/2);
Daniel@0 1473 neg = (pos(end)+1):(Nfindings-1);
Daniel@0 1474 onodes = myunion(pos, neg);
Daniel@0 1475 evidence = cell(1, N);
Daniel@0 1476 evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos)));
Daniel@0 1477 evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg)));
Daniel@0 1478
Daniel@0 1479 engine = jtree_inf_engine(bnet);
Daniel@0 1480 [engine, ll] = enter_evidence(engine, evidence);
Daniel@0 1481 post = zeros(1, Ndiseases);
Daniel@0 1482 for i=diseases(:)'
Daniel@0 1483 m = marginal_nodes(engine, i);
Daniel@0 1484 post(i) = m.T(2);
Daniel@0 1485 end
Daniel@0 1486 </pre>
Daniel@0 1487 Junction tree can be quite slow on large QMR models.
Daniel@0 1488 Fortunately, it is possible to exploit properties of the noisy-OR
Daniel@0 1489 function to speed up exact inference using an algorithm called
Daniel@0 1490 <a href="#quickscore">quickscore</a>, discussed below.
Daniel@0 1491
Daniel@0 1492
Daniel@0 1493
Daniel@0 1494
Daniel@0 1495
Daniel@0 1496 <h2><a name="cg_model">Conditional Gaussian models</h2>
Daniel@0 1497
Daniel@0 1498 A conditional Gaussian model is one in which, conditioned on all the discrete
Daniel@0 1499 nodes, the distribution over the remaining (continuous) nodes is
Daniel@0 1500 multivariate Gaussian. This means we can have arcs from discrete (D)
Daniel@0 1501 to continuous (C) nodes, but not vice versa.
Daniel@0 1502 (We <em>are</em> allowed C->D arcs if the continuous nodes are observed,
Daniel@0 1503 as in the <a href="#mixexp">mixture of experts</a> model,
Daniel@0 1504 since this distribution can be represented with a discrete potential.)
Daniel@0 1505 <p>
Daniel@0 1506 We now give an example of a CG model, from
Daniel@0 1507 the paper "Propagation of Probabilities, Means amd
Daniel@0 1508 Variances in Mixed Graphical Association Models", Steffen Lauritzen,
Daniel@0 1509 JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert
Daniel@0 1510 Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and
Daniel@0 1511 D. J. Spiegelhalter, Springer, 1999.)
Daniel@0 1512
Daniel@0 1513 <h3>Specifying the graph</h3>
Daniel@0 1514
Daniel@0 1515 Consider the model of waste emissions from an incinerator plant shown below.
Daniel@0 1516 We follow the standard convention that shaded nodes are observed,
Daniel@0 1517 clear nodes are hidden.
Daniel@0 1518 We also use the non-standard convention that
Daniel@0 1519 square nodes are discrete (tabular) and round nodes are
Daniel@0 1520 Gaussian.
Daniel@0 1521
Daniel@0 1522 <p>
Daniel@0 1523 <center>
Daniel@0 1524 <IMG SRC="Figures/cg1.gif">
Daniel@0 1525 </center>
Daniel@0 1526 <p>
Daniel@0 1527
Daniel@0 1528 We can create this model as follows.
Daniel@0 1529 <pre>
Daniel@0 1530 F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9;
Daniel@0 1531 n = 9;
Daniel@0 1532
Daniel@0 1533 dag = zeros(n);
Daniel@0 1534 dag(F,E)=1;
Daniel@0 1535 dag(W,[E Min D]) = 1;
Daniel@0 1536 dag(E,D)=1;
Daniel@0 1537 dag(B,[C D])=1;
Daniel@0 1538 dag(D,[L Mout])=1;
Daniel@0 1539 dag(Min,Mout)=1;
Daniel@0 1540
Daniel@0 1541 % node sizes - all cts nodes are scalar, all discrete nodes are binary
Daniel@0 1542 ns = ones(1, n);
Daniel@0 1543 dnodes = [F W B];
Daniel@0 1544 cnodes = mysetdiff(1:n, dnodes);
Daniel@0 1545 ns(dnodes) = 2;
Daniel@0 1546
Daniel@0 1547 bnet = mk_bnet(dag, ns, 'discrete', dnodes);
Daniel@0 1548 </pre>
Daniel@0 1549 'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous
Daniel@0 1550 nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'.
Daniel@0 1551 <p>
Daniel@0 1552
Daniel@0 1553
Daniel@0 1554 <h3>Specifying the parameters</h3>
Daniel@0 1555
Daniel@0 1556 The parameters of the discrete nodes can be specified as follows.
Daniel@0 1557 <pre>
Daniel@0 1558 bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable
Daniel@0 1559 bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect
Daniel@0 1560 bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household
Daniel@0 1561 </pre>
Daniel@0 1562
Daniel@0 1563 <p>
Daniel@0 1564 The parameters of the continuous nodes can be specified as follows.
Daniel@0 1565 <pre>
Daniel@0 1566 bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ...
Daniel@0 1567 'cov', [0.00002 0.0001 0.00002 0.0001]);
Daniel@0 1568 bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ...
Daniel@0 1569 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]);
Daniel@0 1570 bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]);
Daniel@0 1571 bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5);
Daniel@0 1572 bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]);
Daniel@0 1573 bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]);
Daniel@0 1574 </pre>
Daniel@0 1575
Daniel@0 1576
Daniel@0 1577 <h3><a name="cg_infer">Inference</h3>
Daniel@0 1578
Daniel@0 1579 <!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.-->
Daniel@0 1580 First we compute the unconditional marginals.
Daniel@0 1581 <pre>
Daniel@0 1582 engine = jtree_inf_engine(bnet);
Daniel@0 1583 evidence = cell(1,n);
Daniel@0 1584 [engine, ll] = enter_evidence(engine, evidence);
Daniel@0 1585 marg = marginal_nodes(engine, E);
Daniel@0 1586 </pre>
Daniel@0 1587 <!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)-->
Daniel@0 1588 'marg' is a structure that contains the fields 'mu' and 'Sigma', which
Daniel@0 1589 contain the mean and (co)variance of the marginal on E.
Daniel@0 1590 In this case, they are both scalars.
Daniel@0 1591 Let us check they match the published figures (to 2 decimal places).
Daniel@0 1592 <!--(We can't expect
Daniel@0 1593 more precision than this in general because I have implemented the algorithm of
Daniel@0 1594 Lauritzen (1992), which can be numerically unstable.)-->
Daniel@0 1595 <pre>
Daniel@0 1596 tol = 1e-2;
Daniel@0 1597 assert(approxeq(marg.mu, -3.25, tol));
Daniel@0 1598 assert(approxeq(sqrt(marg.Sigma), 0.709, tol));
Daniel@0 1599 </pre>
Daniel@0 1600 We can compute the other posteriors similarly.
Daniel@0 1601 Now let us add some evidence.
Daniel@0 1602 <pre>
Daniel@0 1603 evidence = cell(1,n);
Daniel@0 1604 evidence{W} = 1; % industrial
Daniel@0 1605 evidence{L} = 1.1;
Daniel@0 1606 evidence{C} = -0.9;
Daniel@0 1607 [engine, ll] = enter_evidence(engine, evidence);
Daniel@0 1608 </pre>
Daniel@0 1609 Now we find
Daniel@0 1610 <pre>
Daniel@0 1611 marg = marginal_nodes(engine, E);
Daniel@0 1612 assert(approxeq(marg.mu, -3.8983, tol));
Daniel@0 1613 assert(approxeq(sqrt(marg.Sigma), 0.0763, tol));
Daniel@0 1614 </pre>
Daniel@0 1615
Daniel@0 1616
Daniel@0 1617 We can also compute the joint probability on a set of nodes.
Daniel@0 1618 For example, P(D, Mout | evidence) is a 2D Gaussian:
Daniel@0 1619 <pre>
Daniel@0 1620 marg = marginal_nodes(engine, [D Mout])
Daniel@0 1621 marg =
Daniel@0 1622 domain: [6 8]
Daniel@0 1623 mu: [2x1 double]
Daniel@0 1624 Sigma: [2x2 double]
Daniel@0 1625 T: 1.0000
Daniel@0 1626 </pre>
Daniel@0 1627 The mean is
Daniel@0 1628 <pre>
Daniel@0 1629 marg.mu
Daniel@0 1630 ans =
Daniel@0 1631 3.6077
Daniel@0 1632 4.1077
Daniel@0 1633 </pre>
Daniel@0 1634 and the covariance matrix is
Daniel@0 1635 <pre>
Daniel@0 1636 marg.Sigma
Daniel@0 1637 ans =
Daniel@0 1638 0.1062 0.1062
Daniel@0 1639 0.1062 0.1182
Daniel@0 1640 </pre>
Daniel@0 1641 It is easy to visualize this posterior using standard Matlab plotting
Daniel@0 1642 functions, e.g.,
Daniel@0 1643 <pre>
Daniel@0 1644 gaussplot2d(marg.mu, marg.Sigma);
Daniel@0 1645 </pre>
Daniel@0 1646 produces the following picture.
Daniel@0 1647
Daniel@0 1648 <p>
Daniel@0 1649 <center>
Daniel@0 1650 <IMG SRC="Figures/gaussplot.png">
Daniel@0 1651 </center>
Daniel@0 1652 <p>
Daniel@0 1653
Daniel@0 1654
Daniel@0 1655 The T field indicates that the mixing weight of this Gaussian
Daniel@0 1656 component is 1.0.
Daniel@0 1657 If the joint contains discrete and continuous variables, the result
Daniel@0 1658 will be a mixture of Gaussians, e.g.,
Daniel@0 1659 <pre>
Daniel@0 1660 marg = marginal_nodes(engine, [F E])
Daniel@0 1661 domain: [1 3]
Daniel@0 1662 mu: [-3.9000 -0.4003]
Daniel@0 1663 Sigma: [1x1x2 double]
Daniel@0 1664 T: [0.9995 4.7373e-04]
Daniel@0 1665 </pre>
Daniel@0 1666 The interpretation is
Daniel@0 1667 Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ].
Daniel@0 1668 In this case, E is a scalar, so i=j=1; k specifies the mixture component.
Daniel@0 1669 <p>
Daniel@0 1670 We saw in the sprinkler network that BNT sets the effective size of
Daniel@0 1671 observed discrete nodes to 1, since they only have one legal value.
Daniel@0 1672 For continuous nodes, BNT sets their length to 0,
Daniel@0 1673 since they have been reduced to a point.
Daniel@0 1674 For example,
Daniel@0 1675 <pre>
Daniel@0 1676 marg = marginal_nodes(engine, [B C])
Daniel@0 1677 domain: [4 5]
Daniel@0 1678 mu: []
Daniel@0 1679 Sigma: []
Daniel@0 1680 T: [0.0123 0.9877]
Daniel@0 1681 </pre>
Daniel@0 1682 It is simple to post-process the output of marginal_nodes.
Daniel@0 1683 For example, the file BNT/examples/static/cg1 sets the mu term of
Daniel@0 1684 observed nodes to their observed value, and the Sigma term to 0 (since
Daniel@0 1685 observed nodes have no variance).
Daniel@0 1686
Daniel@0 1687 <p>
Daniel@0 1688 Note that the implemented version of the junction tree is numerically
Daniel@0 1689 unstable when using CG potentials
Daniel@0 1690 (which is why, in the example above, we only required our answers to agree with
Daniel@0 1691 the published ones to 2dp.)
Daniel@0 1692 This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>,
Daniel@0 1693 implemented by Shan Huang. This is described in
Daniel@0 1694
Daniel@0 1695 <ul>
Daniel@0 1696 <li> "Stable Local Computation with Conditional Gaussian Distributions",
Daniel@0 1697 S. Lauritzen and F. Jensen, Tech Report R-99-2014,
Daniel@0 1698 Dept. Math. Sciences, Allborg Univ., 1999.
Daniel@0 1699 </ul>
Daniel@0 1700
Daniel@0 1701 However, even the numerically stable version
Daniel@0 1702 can be computationally intractable if there are many hidden discrete
Daniel@0 1703 nodes, because the number of mixture components grows exponentially e.g., in a
Daniel@0 1704 <a href="usage_dbn.html#lds">switching linear dynamical system</a>.
Daniel@0 1705 In general, one must resort to approximate inference techniques: see
Daniel@0 1706 the discussion on <a href="#engines">inference engines</a> below.
Daniel@0 1707
Daniel@0 1708
Daniel@0 1709 <h2><a name="hybrid">Other hybrid models</h2>
Daniel@0 1710
Daniel@0 1711 When we have C->D arcs, where C is hidden, we need to use
Daniel@0 1712 approximate inference.
Daniel@0 1713 One approach (not implemented in BNT) is described in
Daniel@0 1714 <ul>
Daniel@0 1715 <li> <a
Daniel@0 1716 href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A
Daniel@0 1717 Variational Approximation for Bayesian Networks with
Daniel@0 1718 Discrete and Continuous Latent Variables</a>,
Daniel@0 1719 K. Murphy, UAI 99.
Daniel@0 1720 </ul>
Daniel@0 1721 Of course, one can always use <a href="#sampling">sampling</a> methods
Daniel@0 1722 for approximate inference in such models.
Daniel@0 1723
Daniel@0 1724
Daniel@0 1725
Daniel@0 1726 <h1><a name="param_learning">Parameter Learning</h1>
Daniel@0 1727
Daniel@0 1728 The parameter estimation routines in BNT can be classified into 4
Daniel@0 1729 types, depending on whether the goal is to compute
Daniel@0 1730 a full (Bayesian) posterior over the parameters or just a point
Daniel@0 1731 estimate (e.g., Maximum Likelihood or Maximum A Posteriori),
Daniel@0 1732 and whether all the variables are fully observed or there is missing
Daniel@0 1733 data/ hidden variables (partial observability).
Daniel@0 1734 <p>
Daniel@0 1735
Daniel@0 1736 <TABLE BORDER>
Daniel@0 1737 <tr>
Daniel@0 1738 <TH></TH>
Daniel@0 1739 <th>Full obs</th>
Daniel@0 1740 <th>Partial obs</th>
Daniel@0 1741 </tr>
Daniel@0 1742 <tr>
Daniel@0 1743 <th>Point</th>
Daniel@0 1744 <td><tt>learn_params</tt></td>
Daniel@0 1745 <td><tt>learn_params_em</tt></td>
Daniel@0 1746 </tr>
Daniel@0 1747 <tr>
Daniel@0 1748 <th>Bayes</th>
Daniel@0 1749 <td><tt>bayes_update_params</tt></td>
Daniel@0 1750 <td>not yet supported</td>
Daniel@0 1751 </tr>
Daniel@0 1752 </table>
Daniel@0 1753
Daniel@0 1754
Daniel@0 1755 <h2><a name="load_data">Loading data from a file</h2>
Daniel@0 1756
Daniel@0 1757 To load numeric data from an ASCII text file called 'dat.txt', where each row is a
Daniel@0 1758 case and columns are separated by white-space, such as
Daniel@0 1759 <pre>
Daniel@0 1760 011979 1626.5 0.0
Daniel@0 1761 021979 1367.0 0.0
Daniel@0 1762 ...
Daniel@0 1763 </pre>
Daniel@0 1764 you can use
Daniel@0 1765 <pre>
Daniel@0 1766 data = load('dat.txt');
Daniel@0 1767 </pre>
Daniel@0 1768 or
Daniel@0 1769 <pre>
Daniel@0 1770 load dat.txt -ascii
Daniel@0 1771 </pre>
Daniel@0 1772 In the latter case, the data is stored in a variable called 'dat' (the
Daniel@0 1773 filename minus the extension).
Daniel@0 1774 Alternatively, suppose the data is stored in a .csv file (has commas
Daniel@0 1775 separating the columns, and contains a header line), such as
Daniel@0 1776 <pre>
Daniel@0 1777 header info goes here
Daniel@0 1778 ORD,011979,1626.5,0.0
Daniel@0 1779 DSM,021979,1367.0,0.0
Daniel@0 1780 ...
Daniel@0 1781 </pre>
Daniel@0 1782 You can load this using
Daniel@0 1783 <pre>
Daniel@0 1784 [a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1);
Daniel@0 1785 </pre>
Daniel@0 1786 If your file is not in either of these formats, you can either use Perl to convert
Daniel@0 1787 it to this format, or use the Matlab scanf command.
Daniel@0 1788 Type
Daniel@0 1789 <tt>
Daniel@0 1790 help iofun
Daniel@0 1791 </tt>
Daniel@0 1792 for more information on Matlab's file functions.
Daniel@0 1793 <!--
Daniel@0 1794 <p>
Daniel@0 1795 To load data directly from Excel,
Daniel@0 1796 you should buy the
Daniel@0 1797 <a href="http://www.mathworks.com/products/excellink/">Excel Link</a>.
Daniel@0 1798 To load data directly from a relational database,
Daniel@0 1799 you should buy the
Daniel@0 1800 <a href="http://www.mathworks.com/products/database">Database
Daniel@0 1801 toolbox</a>.
Daniel@0 1802 -->
Daniel@0 1803 <p>
Daniel@0 1804 BNT learning routines require data to be stored in a cell array.
Daniel@0 1805 data{i,m} is the value of node i in case (example) m, i.e., each
Daniel@0 1806 <em>column</em> is a case.
Daniel@0 1807 If node i is not observed in case m (missing value), set
Daniel@0 1808 data{i,m} = [].
Daniel@0 1809 (Not all the learning routines can cope with such missing values, however.)
Daniel@0 1810 In the special case that all the nodes are observed and are
Daniel@0 1811 scalar-valued (as opposed to vector-valued), the data can be
Daniel@0 1812 stored in a matrix (as opposed to a cell-array).
Daniel@0 1813 <p>
Daniel@0 1814 Suppose, as in the <a href="#mixexp">mixture of experts example</a>,
Daniel@0 1815 that we have 3 nodes in the graph: X(1) is the observed input, X(3) is
Daniel@0 1816 the observed output, and X(2) is a hidden (gating) node. We can
Daniel@0 1817 create the dataset as follows.
Daniel@0 1818 <pre>
Daniel@0 1819 data = load('dat.txt');
Daniel@0 1820 ncases = size(data, 1);
Daniel@0 1821 cases = cell(3, ncases);
Daniel@0 1822 cases([1 3], :) = num2cell(data');
Daniel@0 1823 </pre>
Daniel@0 1824 Notice how we transposed the data, to convert rows into columns.
Daniel@0 1825 Also, cases{2,m} = [] for all m, since X(2) is always hidden.
Daniel@0 1826
Daniel@0 1827
Daniel@0 1828 <h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2>
Daniel@0 1829
Daniel@0 1830 As an example, let's generate some data from the sprinkler network, randomize the parameters,
Daniel@0 1831 and then try to recover the original model.
Daniel@0 1832 First we create some training data using forwards sampling.
Daniel@0 1833 <pre>
Daniel@0 1834 samples = cell(N, nsamples);
Daniel@0 1835 for i=1:nsamples
Daniel@0 1836 samples(:,i) = sample_bnet(bnet);
Daniel@0 1837 end
Daniel@0 1838 </pre>
Daniel@0 1839 samples{j,i} contains the value of the j'th node in case i.
Daniel@0 1840 sample_bnet returns a cell array because, in general, each node might
Daniel@0 1841 be a vector of different length.
Daniel@0 1842 In this case, all nodes are discrete (and hence scalars), so we
Daniel@0 1843 could have used a regular array instead (which can be quicker):
Daniel@0 1844 <pre>
Daniel@0 1845 data = cell2num(samples);
Daniel@0 1846 </pre
Daniel@0 1847 So now data(j,i) = samples{j,i}.
Daniel@0 1848 <p>
Daniel@0 1849 Now we create a network with random parameters.
Daniel@0 1850 (The initial values of bnet2 don't matter in this case, since we can find the
Daniel@0 1851 globally optimal MLE independent of where we start.)
Daniel@0 1852 <pre>
Daniel@0 1853 % Make a tabula rasa
Daniel@0 1854 bnet2 = mk_bnet(dag, node_sizes);
Daniel@0 1855 seed = 0;
Daniel@0 1856 rand('state', seed);
Daniel@0 1857 bnet2.CPD{C} = tabular_CPD(bnet2, C);
Daniel@0 1858 bnet2.CPD{R} = tabular_CPD(bnet2, R);
Daniel@0 1859 bnet2.CPD{S} = tabular_CPD(bnet2, S);
Daniel@0 1860 bnet2.CPD{W} = tabular_CPD(bnet2, W);
Daniel@0 1861 </pre>
Daniel@0 1862 Finally, we find the maximum likelihood estimates of the parameters.
Daniel@0 1863 <pre>
Daniel@0 1864 bnet3 = learn_params(bnet2, samples);
Daniel@0 1865 </pre>
Daniel@0 1866 To view the learned parameters, we use a little Matlab hackery.
Daniel@0 1867 <pre>
Daniel@0 1868 CPT3 = cell(1,N);
Daniel@0 1869 for i=1:N
Daniel@0 1870 s=struct(bnet3.CPD{i}); % violate object privacy
Daniel@0 1871 CPT3{i}=s.CPT;
Daniel@0 1872 end
Daniel@0 1873 </pre>
Daniel@0 1874 Here are the parameters learned for node 4.
Daniel@0 1875 <pre>
Daniel@0 1876 dispcpt(CPT3{4})
Daniel@0 1877 1 1 : 1.0000 0.0000
Daniel@0 1878 2 1 : 0.2000 0.8000
Daniel@0 1879 1 2 : 0.2273 0.7727
Daniel@0 1880 2 2 : 0.0000 1.0000
Daniel@0 1881 </pre>
Daniel@0 1882 So we see that the learned parameters are fairly close to the "true"
Daniel@0 1883 ones, which we display below.
Daniel@0 1884 <pre>
Daniel@0 1885 dispcpt(CPT{4})
Daniel@0 1886 1 1 : 1.0000 0.0000
Daniel@0 1887 2 1 : 0.1000 0.9000
Daniel@0 1888 1 2 : 0.1000 0.9000
Daniel@0 1889 2 2 : 0.0100 0.9900
Daniel@0 1890 </pre>
Daniel@0 1891 We can get better results by using a larger training set, or using
Daniel@0 1892 informative priors (see <a href="#prior">below</a>).
Daniel@0 1893
Daniel@0 1894
Daniel@0 1895
Daniel@0 1896 <h2><a name="prior">Parameter priors</h2>
Daniel@0 1897
Daniel@0 1898 Currently, only tabular CPDs can have priors on their parameters.
Daniel@0 1899 The conjugate prior for a multinomial is the Dirichlet.
Daniel@0 1900 (For binary random variables, the multinomial is the same as the
Daniel@0 1901 Bernoulli, and the Dirichlet is the same as the Beta.)
Daniel@0 1902 <p>
Daniel@0 1903 The Dirichlet has a simple interpretation in terms of pseudo counts.
Daniel@0 1904 If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the
Daniel@0 1905 training set, where Pa_i are the parents of X_i,
Daniel@0 1906 then the maximum likelihood (ML) estimate is
Daniel@0 1907 T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0.
Daniel@0 1908 To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this
Daniel@0 1909 event was not seen in the training set,
Daniel@0 1910 we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk)
Daniel@0 1911 of times in the past.
Daniel@0 1912 The MAP (maximum a posterior) estimate is then
Daniel@0 1913 <pre>
Daniel@0 1914 T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij)
Daniel@0 1915 </pre>
Daniel@0 1916 and is never 0 if all alpha_ijk > 0.
Daniel@0 1917 For example, consider the network A->B, where A is binary and B has 3
Daniel@0 1918 values.
Daniel@0 1919 A uniform prior for B has the form
Daniel@0 1920 <pre>
Daniel@0 1921 B=1 B=2 B=3
Daniel@0 1922 A=1 1 1 1
Daniel@0 1923 A=2 1 1 1
Daniel@0 1924 </pre>
Daniel@0 1925 which can be created using
Daniel@0 1926 <pre>
Daniel@0 1927 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif');
Daniel@0 1928 </pre>
Daniel@0 1929 This prior does not satisfy the likelihood equivalence principle,
Daniel@0 1930 which says that <a href="#markov_equiv">Markov equivalent</a> models
Daniel@0 1931 should have the same marginal likelihood.
Daniel@0 1932 A prior that does satisfy this principle is shown below.
Daniel@0 1933 Heckerman (1995) calls this the
Daniel@0 1934 BDeu prior (likelihood equivalent uniform Bayesian Dirichlet).
Daniel@0 1935 <pre>
Daniel@0 1936 B=1 B=2 B=3
Daniel@0 1937 A=1 1/6 1/6 1/6
Daniel@0 1938 A=2 1/6 1/6 1/6
Daniel@0 1939 </pre>
Daniel@0 1940 where we put N/(q*r) in each bin; N is the equivalent sample size,
Daniel@0 1941 r=|A|, q = |B|.
Daniel@0 1942 This can be created as follows
Daniel@0 1943 <pre>
Daniel@0 1944 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu');
Daniel@0 1945 </pre>
Daniel@0 1946 Here, 1 is the equivalent sample size, and is the strength of the
Daniel@0 1947 prior.
Daniel@0 1948 You can change this using
Daniel@0 1949 <pre>
Daniel@0 1950 tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ...
Daniel@0 1951 'BDeu', 'dirichlet_weight', 10);
Daniel@0 1952 </pre>
Daniel@0 1953 <!--where counts is an array of pseudo-counts of the same size as the
Daniel@0 1954 CPT.-->
Daniel@0 1955 <!--
Daniel@0 1956 <p>
Daniel@0 1957 When you specify a prior, you should set row i of the CPT to the
Daniel@0 1958 normalized version of row i of the pseudo-count matrix, i.e., to the
Daniel@0 1959 expected values of the parameters. This will ensure that computing the
Daniel@0 1960 marginal likelihood sequentially (see <a
Daniel@0 1961 href="#bayes_learn">below</a>) and in batch form gives the same
Daniel@0 1962 results.
Daniel@0 1963 To do this, proceed as follows.
Daniel@0 1964 <pre>
Daniel@0 1965 tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts));
Daniel@0 1966 </pre>
Daniel@0 1967 For a non-informative prior, you can just write
Daniel@0 1968 <pre>
Daniel@0 1969 tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif');
Daniel@0 1970 </pre>
Daniel@0 1971 -->
Daniel@0 1972
Daniel@0 1973
Daniel@0 1974 <h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2>
Daniel@0 1975
Daniel@0 1976 If we use conjugate priors and have fully observed data, we can
Daniel@0 1977 compute the posterior over the parameters in batch form as follows.
Daniel@0 1978 <pre>
Daniel@0 1979 cases = sample_bnet(bnet, nsamples);
Daniel@0 1980 bnet = bayes_update_params(bnet, cases);
Daniel@0 1981 LL = log_marg_lik_complete(bnet, cases);
Daniel@0 1982 </pre>
Daniel@0 1983 bnet.CPD{i}.prior contains the new Dirichlet pseudocounts,
Daniel@0 1984 and bnet.CPD{i}.CPT is set to the mean of the posterior (the
Daniel@0 1985 normalized counts).
Daniel@0 1986 (Hence if the initial pseudo counts are 0,
Daniel@0 1987 <tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the
Daniel@0 1988 same result.)
Daniel@0 1989
Daniel@0 1990
Daniel@0 1991
Daniel@0 1992
Daniel@0 1993 <p>
Daniel@0 1994 We can compute the same result sequentially (on-line) as follows.
Daniel@0 1995 <pre>
Daniel@0 1996 LL = 0;
Daniel@0 1997 for m=1:nsamples
Daniel@0 1998 LL = LL + log_marg_lik_complete(bnet, cases(:,m));
Daniel@0 1999 bnet = bayes_update_params(bnet, cases(:,m));
Daniel@0 2000 end
Daniel@0 2001 </pre>
Daniel@0 2002
Daniel@0 2003 The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of
Daniel@0 2004 sequential model selection which uses the same idea.
Daniel@0 2005 We generate data from the model A->B
Daniel@0 2006 and compute the posterior prob of all 3 dags on 2 nodes:
Daniel@0 2007 (1) A B, (2) A <- B , (3) A -> B
Daniel@0 2008 Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from
Daniel@0 2009 observational data alone, so we expect their posteriors to be the same
Daniel@0 2010 (assuming a prior which satisfies likelihood equivalence).
Daniel@0 2011 If we use random parameters, the "true" model only gets a higher posterior after 2000 trials!
Daniel@0 2012 However, if we make B a noisy NOT gate, the true model "wins" after 12
Daniel@0 2013 trials, as shown below (red = model 1, blue/green (superimposed)
Daniel@0 2014 represents models 2/3).
Daniel@0 2015 <p>
Daniel@0 2016 <img src="Figures/model_select.png">
Daniel@0 2017 <p>
Daniel@0 2018 The use of marginal likelihood for model selection is discussed in
Daniel@0 2019 greater detail in the
Daniel@0 2020 section on <a href="structure_learning">structure learning</a>.
Daniel@0 2021
Daniel@0 2022
Daniel@0 2023
Daniel@0 2024
Daniel@0 2025 <h2><a name="em">Maximum likelihood parameter estimation with missing values</h2>
Daniel@0 2026
Daniel@0 2027 Now we consider learning when some values are not observed.
Daniel@0 2028 Let us randomly hide half the values generated from the water
Daniel@0 2029 sprinkler example.
Daniel@0 2030 <pre>
Daniel@0 2031 samples2 = samples;
Daniel@0 2032 hide = rand(N, nsamples) > 0.5;
Daniel@0 2033 [I,J]=find(hide);
Daniel@0 2034 for k=1:length(I)
Daniel@0 2035 samples2{I(k), J(k)} = [];
Daniel@0 2036 end
Daniel@0 2037 </pre>
Daniel@0 2038 samples2{i,l} is the value of node i in training case l, or [] if unobserved.
Daniel@0 2039 <p>
Daniel@0 2040 Now we will compute the MLEs using the EM algorithm.
Daniel@0 2041 We need to use an inference algorithm to compute the expected
Daniel@0 2042 sufficient statistics in the E step; the M (maximization) step is as
Daniel@0 2043 above.
Daniel@0 2044 <pre>
Daniel@0 2045 engine2 = jtree_inf_engine(bnet2);
Daniel@0 2046 max_iter = 10;
Daniel@0 2047 [bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter);
Daniel@0 2048 </pre>
Daniel@0 2049 LLtrace(i) is the log-likelihood at iteration i. We can plot this as
Daniel@0 2050 follows:
Daniel@0 2051 <pre>
Daniel@0 2052 plot(LLtrace, 'x-')
Daniel@0 2053 </pre>
Daniel@0 2054 Let's display the results after 10 iterations of EM.
Daniel@0 2055 <pre>
Daniel@0 2056 celldisp(CPT4)
Daniel@0 2057 CPT4{1} =
Daniel@0 2058 0.6616
Daniel@0 2059 0.3384
Daniel@0 2060 CPT4{2} =
Daniel@0 2061 0.6510 0.3490
Daniel@0 2062 0.8751 0.1249
Daniel@0 2063 CPT4{3} =
Daniel@0 2064 0.8366 0.1634
Daniel@0 2065 0.0197 0.9803
Daniel@0 2066 CPT4{4} =
Daniel@0 2067 (:,:,1) =
Daniel@0 2068 0.8276 0.0546
Daniel@0 2069 0.5452 0.1658
Daniel@0 2070 (:,:,2) =
Daniel@0 2071 0.1724 0.9454
Daniel@0 2072 0.4548 0.8342
Daniel@0 2073 </pre>
Daniel@0 2074 We can get improved performance by using one or more of the following
Daniel@0 2075 methods:
Daniel@0 2076 <ul>
Daniel@0 2077 <li> Increasing the size of the training set.
Daniel@0 2078 <li> Decreasing the amount of hidden data.
Daniel@0 2079 <li> Running EM for longer.
Daniel@0 2080 <li> Using informative priors.
Daniel@0 2081 <li> Initialising EM from multiple starting points.
Daniel@0 2082 </ul>
Daniel@0 2083
Daniel@0 2084 Click <a href="#gaussian">here</a> for a discussion of learning
Daniel@0 2085 Gaussians, which can cause numerical problems.
Daniel@0 2086 <p>
Daniel@0 2087 For a more complete example of learning with EM,
Daniel@0 2088 see the script BNT/examples/static/learn1.m.
Daniel@0 2089
Daniel@0 2090 <h2><a name="tying">Parameter tying</h2>
Daniel@0 2091
Daniel@0 2092 In networks with repeated structure (e.g., chains and grids), it is
Daniel@0 2093 common to assume that the parameters are the same at every node. This
Daniel@0 2094 is called parameter tying, and reduces the amount of data needed for
Daniel@0 2095 learning.
Daniel@0 2096 <p>
Daniel@0 2097 When we have tied parameters, there is no longer a one-to-one
Daniel@0 2098 correspondence between nodes and CPDs.
Daniel@0 2099 Rather, each CPD species the parameters for a whole equivalence class
Daniel@0 2100 of nodes.
Daniel@0 2101 It is easiest to see this by example.
Daniel@0 2102 Consider the following <a href="usage_dbn.html#hmm">hidden Markov
Daniel@0 2103 model (HMM)</a>
Daniel@0 2104 <p>
Daniel@0 2105 <img src="Figures/hmm3.gif">
Daniel@0 2106 <p>
Daniel@0 2107 <!--
Daniel@0 2108 We can create this graph structure, assuming we have T time-slices,
Daniel@0 2109 as follows.
Daniel@0 2110 (We number the nodes as shown in the figure, but we could equally well
Daniel@0 2111 number the hidden nodes 1:T, and the observed nodes T+1:2T.)
Daniel@0 2112 <pre>
Daniel@0 2113 N = 2*T;
Daniel@0 2114 dag = zeros(N);
Daniel@0 2115 hnodes = 1:2:2*T;
Daniel@0 2116 for i=1:T-1
Daniel@0 2117 dag(hnodes(i), hnodes(i+1))=1;
Daniel@0 2118 end
Daniel@0 2119 onodes = 2:2:2*T;
Daniel@0 2120 for i=1:T
Daniel@0 2121 dag(hnodes(i), onodes(i)) = 1;
Daniel@0 2122 end
Daniel@0 2123 </pre>
Daniel@0 2124 <p>
Daniel@0 2125 The hidden nodes are always discrete, and have Q possible values each,
Daniel@0 2126 but the observed nodes can be discrete or continuous, and have O possible values/length.
Daniel@0 2127 <pre>
Daniel@0 2128 if cts_obs
Daniel@0 2129 dnodes = hnodes;
Daniel@0 2130 else
Daniel@0 2131 dnodes = 1:N;
Daniel@0 2132 end
Daniel@0 2133 ns = ones(1,N);
Daniel@0 2134 ns(hnodes) = Q;
Daniel@0 2135 ns(onodes) = O;
Daniel@0 2136 </pre>
Daniel@0 2137 -->
Daniel@0 2138 When HMMs are used for semi-infinite processes like speech recognition,
Daniel@0 2139 we assume the transition matrix
Daniel@0 2140 P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant
Daniel@0 2141 or homogenous Markov chain.
Daniel@0 2142 Hence hidden nodes 2, 3, ..., T
Daniel@0 2143 are all in the same equivalence class, say class Hclass.
Daniel@0 2144 Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the
Daniel@0 2145 same for all t, so the observed nodes are all in the same equivalence
Daniel@0 2146 class, say class Oclass.
Daniel@0 2147 Finally, the prior term P(H(1)) is in a class all by itself, say class
Daniel@0 2148 H1class.
Daniel@0 2149 This is illustrated below, where we explicitly represent the
Daniel@0 2150 parameters as random variables (dotted nodes).
Daniel@0 2151 <p>
Daniel@0 2152 <img src="Figures/hmm4_params.gif">
Daniel@0 2153 <p>
Daniel@0 2154 In BNT, we cannot represent parameters as random variables (nodes).
Daniel@0 2155 Instead, we "hide" the
Daniel@0 2156 parameters inside one CPD for each equivalence class,
Daniel@0 2157 and then specify that the other CPDs should share these parameters, as
Daniel@0 2158 follows.
Daniel@0 2159 <pre>
Daniel@0 2160 hnodes = 1:2:2*T;
Daniel@0 2161 onodes = 2:2:2*T;
Daniel@0 2162 H1class = 1; Hclass = 2; Oclass = 3;
Daniel@0 2163 eclass = ones(1,N);
Daniel@0 2164 eclass(hnodes(2:end)) = Hclass;
Daniel@0 2165 eclass(hnodes(1)) = H1class;
Daniel@0 2166 eclass(onodes) = Oclass;
Daniel@0 2167 % create dag and ns in the usual way
Daniel@0 2168 bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass);
Daniel@0 2169 </pre>
Daniel@0 2170 Finally, we define the parameters for each equivalence class:
Daniel@0 2171 <pre>
Daniel@0 2172 bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior
Daniel@0 2173 bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix
Daniel@0 2174 if cts_obs
Daniel@0 2175 bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1));
Daniel@0 2176 else
Daniel@0 2177 bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1));
Daniel@0 2178 end
Daniel@0 2179 </pre>
Daniel@0 2180 In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a
Daniel@0 2181 member of e's equivalence class; that is, it is not always the case
Daniel@0 2182 that e == j. You can use bnet.rep_of_eclass(e) to return the
Daniel@0 2183 representative of equivalence class e.
Daniel@0 2184 BNT will look up the parents of j to determine the size
Daniel@0 2185 of the CPT to use. It assumes that this is the same for all members of
Daniel@0 2186 the equivalence class.
Daniel@0 2187 Click <a href="param_tieing.html">here</a> for
Daniel@0 2188 a more complex example of parameter tying.
Daniel@0 2189 <p>
Daniel@0 2190 Note:
Daniel@0 2191 Normally one would define an HMM as a
Daniel@0 2192 <a href = "usage_dbn.html">Dynamic Bayes Net</a>
Daniel@0 2193 (see the function BNT/examples/dynamic/mk_chmm.m).
Daniel@0 2194 However, one can define an HMM as a static BN using the function
Daniel@0 2195 BNT/examples/static/Models/mk_hmm_bnet.m.
Daniel@0 2196
Daniel@0 2197
Daniel@0 2198
Daniel@0 2199 <h1><a name="structure_learning">Structure learning</h1>
Daniel@0 2200
Daniel@0 2201 Update (9/29/03):
Daniel@0 2202 Phillipe LeRay is developing some additional structure learning code
Daniel@0 2203 on top of BNT. Click
Daniel@0 2204 <a href="http://banquiseasi.insa-rouen.fr/projects/bnt-slp/">
Daniel@0 2205 here</a>
Daniel@0 2206 for details.
Daniel@0 2207
Daniel@0 2208 <p>
Daniel@0 2209
Daniel@0 2210 There are two very different approaches to structure learning:
Daniel@0 2211 constraint-based and search-and-score.
Daniel@0 2212 In the <a href="#constraint">constraint-based approach</a>,
Daniel@0 2213 we start with a fully connected graph, and remove edges if certain
Daniel@0 2214 conditional independencies are measured in the data.
Daniel@0 2215 This has the disadvantage that repeated independence tests lose
Daniel@0 2216 statistical power.
Daniel@0 2217 <p>
Daniel@0 2218 In the more popular search-and-score approach,
Daniel@0 2219 we perform a search through the space of possible DAGs, and either
Daniel@0 2220 return the best one found (a point estimate), or return a sample of the
Daniel@0 2221 models found (an approximation to the Bayesian posterior).
Daniel@0 2222 <p>
Daniel@0 2223 The number of DAGs as a function of the number of
Daniel@0 2224 nodes, G(n), is super-exponential in n,
Daniel@0 2225 and is given by the following recurrence
Daniel@0 2226 <!--(where R(i)=G(n)):-->
Daniel@0 2227 <p>
Daniel@0 2228 <center>
Daniel@0 2229 <IMG SRC="numDAGsEqn2.png">
Daniel@0 2230 </center>
Daniel@0 2231 <p>
Daniel@0 2232 The first few values
Daniel@0 2233 are shown below.
Daniel@0 2234
Daniel@0 2235 <table>
Daniel@0 2236 <tr> <th>n</th> <th align=left>G(n)</th> </tr>
Daniel@0 2237 <tr> <td>1</td> <td>1</td> </tr>
Daniel@0 2238 <tr> <td>2</td> <td>3</td> </tr>
Daniel@0 2239 <tr> <td>3</td> <td>25</td> </tr>
Daniel@0 2240 <tr> <td>4</td> <td>543</td> </tr>
Daniel@0 2241 <tr> <td>5</td> <td>29,281</td> </tr>
Daniel@0 2242 <tr> <td>6</td> <td>3,781,503</td> </tr>
Daniel@0 2243 <tr> <td>7</td> <td>1.1 x 10^9</td> </tr>
Daniel@0 2244 <tr> <td>8</td> <td>7.8 x 10^11</td> </tr>
Daniel@0 2245 <tr> <td>9</td> <td>1.2 x 10^15</td> </tr>
Daniel@0 2246 <tr> <td>10</td> <td>4.2 x 10^18</td> </tr>
Daniel@0 2247 </table>
Daniel@0 2248
Daniel@0 2249 Since the number of DAGs is super-exponential in the number of nodes,
Daniel@0 2250 we cannot exhaustively search the space, so we either use a local
Daniel@0 2251 search algorithm (e.g., greedy hill climbining, perhaps with multiple
Daniel@0 2252 restarts) or a global search algorithm (e.g., Markov Chain Monte
Daniel@0 2253 Carlo).
Daniel@0 2254 <p>
Daniel@0 2255 If we know a total ordering on the nodes,
Daniel@0 2256 finding the best structure amounts to picking the best set of parents
Daniel@0 2257 for each node independently.
Daniel@0 2258 This is what the K2 algorithm does.
Daniel@0 2259 If the ordering is unknown, we can search over orderings,
Daniel@0 2260 which is more efficient than searching over DAGs (Koller and Friedman, 2000).
Daniel@0 2261 <p>
Daniel@0 2262 In addition to the search procedure, we must specify the scoring
Daniel@0 2263 function. There are two popular choices. The Bayesian score integrates
Daniel@0 2264 out the parameters, i.e., it is the marginal likelihood of the model.
Daniel@0 2265 The BIC (Bayesian Information Criterion) is defined as
Daniel@0 2266 log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is
Daniel@0 2267 the ML estimate of the parameters, d is the number of parameters, and
Daniel@0 2268 N is the number of data cases.
Daniel@0 2269 The BIC method has the advantage of not requiring a prior.
Daniel@0 2270 <p>
Daniel@0 2271 BIC can be derived as a large sample
Daniel@0 2272 approximation to the marginal likelihood.
Daniel@0 2273 (It is also equal to the Minimum Description Length of a model.)
Daniel@0 2274 However, in practice, the sample size does not need to be very large
Daniel@0 2275 for the approximation to be good.
Daniel@0 2276 For example, in the figure below, we plot the ratio between the log marginal likelihood
Daniel@0 2277 and the BIC score against data-set size; we see that the ratio rapidly
Daniel@0 2278 approaches 1, especially for non-informative priors.
Daniel@0 2279 (This plot was generated by the file BNT/examples/static/bic1.m. It
Daniel@0 2280 uses the water sprinkler BN with BDeu Dirichlet priors with different
Daniel@0 2281 equivalent sample sizes.)
Daniel@0 2282
Daniel@0 2283 <p>
Daniel@0 2284 <center>
Daniel@0 2285 <IMG SRC="Figures/bic.png">
Daniel@0 2286 </center>
Daniel@0 2287 <p>
Daniel@0 2288
Daniel@0 2289 <p>
Daniel@0 2290 As with parameter learning, handling missing data/ hidden variables is
Daniel@0 2291 much harder than the fully observed case.
Daniel@0 2292 The structure learning routines in BNT can therefore be classified into 4
Daniel@0 2293 types, analogously to the parameter learning case.
Daniel@0 2294 <p>
Daniel@0 2295
Daniel@0 2296 <TABLE BORDER>
Daniel@0 2297 <tr>
Daniel@0 2298 <TH></TH>
Daniel@0 2299 <th>Full obs</th>
Daniel@0 2300 <th>Partial obs</th>
Daniel@0 2301 </tr>
Daniel@0 2302 <tr>
Daniel@0 2303 <th>Point</th>
Daniel@0 2304 <td><tt>learn_struct_K2</tt> <br>
Daniel@0 2305 <!-- <tt>learn_struct_hill_climb</tt></td> -->
Daniel@0 2306 <td><tt>not yet supported</tt></td>
Daniel@0 2307 </tr>
Daniel@0 2308 <tr>
Daniel@0 2309 <th>Bayes</th>
Daniel@0 2310 <td><tt>learn_struct_mcmc</tt></td>
Daniel@0 2311 <td>not yet supported</td>
Daniel@0 2312 </tr>
Daniel@0 2313 </table>
Daniel@0 2314
Daniel@0 2315
Daniel@0 2316 <h2><a name="markov_equiv">Markov equivalence</h2>
Daniel@0 2317
Daniel@0 2318 If two DAGs encode the same conditional independencies, they are
Daniel@0 2319 called Markov equivalent. The set of all DAGs can be paritioned into
Daniel@0 2320 Markov equivalence classes. Graphs within the same class can
Daniel@0 2321 have
Daniel@0 2322 the direction of some of their arcs reversed without changing any of
Daniel@0 2323 the CI relationships.
Daniel@0 2324 Each class can be represented by a PDAG
Daniel@0 2325 (partially directed acyclic graph) called an essential graph or
Daniel@0 2326 pattern. This specifies which edges must be oriented in a certain
Daniel@0 2327 direction, and which may be reversed.
Daniel@0 2328
Daniel@0 2329 <p>
Daniel@0 2330 When learning graph structure from observational data,
Daniel@0 2331 the best one can hope to do is to identify the model up to Markov
Daniel@0 2332 equivalence. To distinguish amongst graphs within the same equivalence
Daniel@0 2333 class, one needs interventional data: see the discussion on <a
Daniel@0 2334 href="#active">active learning</a> below.
Daniel@0 2335
Daniel@0 2336
Daniel@0 2337
Daniel@0 2338 <h2><a name="enumerate">Exhaustive search</h2>
Daniel@0 2339
Daniel@0 2340 The brute-force approach to structure learning is to enumerate all
Daniel@0 2341 possible DAGs, and score each one. This provides a "gold standard"
Daniel@0 2342 with which to compare other algorithms. We can do this as follows.
Daniel@0 2343 <pre>
Daniel@0 2344 dags = mk_all_dags(N);
Daniel@0 2345 score = score_dags(data, ns, dags);
Daniel@0 2346 </pre>
Daniel@0 2347 where data(i,m) is the value of node i in case m,
Daniel@0 2348 and ns(i) is the size of node i.
Daniel@0 2349 If the DAGs have a lot of families in common, we can cache the sufficient statistics,
Daniel@0 2350 making this potentially more efficient than scoring the DAGs one at a time.
Daniel@0 2351 (Caching is not currently implemented, however.)
Daniel@0 2352 <p>
Daniel@0 2353 By default, we use the Bayesian scoring metric, and assume CPDs are
Daniel@0 2354 represented by tables with BDeu(1) priors.
Daniel@0 2355 We can override these defaults as follows.
Daniel@0 2356 If we want to use uniform priors, we can say
Daniel@0 2357 <pre>
Daniel@0 2358 params = cell(1,N);
Daniel@0 2359 for i=1:N
Daniel@0 2360 params{i} = {'prior', 'unif'};
Daniel@0 2361 end
Daniel@0 2362 score = score_dags(data, ns, dags, 'params', params);
Daniel@0 2363 </pre>
Daniel@0 2364 params{i} is a cell-array, containing optional arguments that are
Daniel@0 2365 passed to the constructor for CPD i.
Daniel@0 2366 <p>
Daniel@0 2367 Now suppose we want to use different node types, e.g.,
Daniel@0 2368 Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both
Daniel@0 2369 these CPDs can support discrete and continuous parents, which is
Daniel@0 2370 necessary since all other nodes will be considered as parents).
Daniel@0 2371 The Bayesian scoring metric currently only works for tabular CPDs, so
Daniel@0 2372 we will use BIC:
Daniel@0 2373 <pre>
Daniel@0 2374 score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [],
Daniel@0 2375 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic')
Daniel@0 2376 </pre>
Daniel@0 2377 In practice, one can't enumerate all possible DAGs for N > 5,
Daniel@0 2378 but one can evaluate any reasonably-sized set of hypotheses in this
Daniel@0 2379 way (e.g., nearest neighbors of your current best guess).
Daniel@0 2380 Think of this as "computer assisted model refinement" as opposed to de
Daniel@0 2381 novo learning.
Daniel@0 2382
Daniel@0 2383
Daniel@0 2384 <h2><a name="K2">K2</h2>
Daniel@0 2385
Daniel@0 2386 The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows.
Daniel@0 2387 Initially each node has no parents. It then adds incrementally that parent whose addition most
Daniel@0 2388 increases the score of the resulting structure. When the addition of no single
Daniel@0 2389 parent can increase the score, it stops adding parents to the node.
Daniel@0 2390 Since we are using a fixed ordering, we do not need to check for
Daniel@0 2391 cycles, and can choose the parents for each node independently.
Daniel@0 2392 <p>
Daniel@0 2393 The original paper used the Bayesian scoring
Daniel@0 2394 metric with tabular CPDs and Dirichlet priors.
Daniel@0 2395 BNT generalizes this to allow any kind of CPD, and either the Bayesian
Daniel@0 2396 scoring metric or BIC, as in the example <a href="#enumerate">above</a>.
Daniel@0 2397 In addition, you can specify
Daniel@0 2398 an optional upper bound on the number of parents for each node.
Daniel@0 2399 The file BNT/examples/static/k2demo1.m gives an example of how to use K2.
Daniel@0 2400 We use the water sprinkler network and sample 100 cases from it as before.
Daniel@0 2401 Then we see how much data it takes to recover the generating structure:
Daniel@0 2402 <pre>
Daniel@0 2403 order = [C S R W];
Daniel@0 2404 max_fan_in = 2;
Daniel@0 2405 sz = 5:5:100;
Daniel@0 2406 for i=1:length(sz)
Daniel@0 2407 dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in);
Daniel@0 2408 correct(i) = isequal(dag, dag2);
Daniel@0 2409 end
Daniel@0 2410 </pre>
Daniel@0 2411 Here are the results.
Daniel@0 2412 <pre>
Daniel@0 2413 correct =
Daniel@0 2414 Columns 1 through 12
Daniel@0 2415 0 0 0 0 0 0 0 1 0 1 1 1
Daniel@0 2416 Columns 13 through 20
Daniel@0 2417 1 1 1 1 1 1 1 1
Daniel@0 2418 </pre>
Daniel@0 2419 So we see it takes about sz(10)=50 cases. (BIC behaves similarly,
Daniel@0 2420 showing that the prior doesn't matter too much.)
Daniel@0 2421 In general, we cannot hope to recover the "true" generating structure,
Daniel@0 2422 only one that is in its <a href="#markov_equiv">Markov equivalence
Daniel@0 2423 class</a>.
Daniel@0 2424
Daniel@0 2425
Daniel@0 2426 <h2><a name="hill_climb">Hill-climbing</h2>
Daniel@0 2427
Daniel@0 2428 Hill-climbing starts at a specific point in space,
Daniel@0 2429 considers all nearest neighbors, and moves to the neighbor
Daniel@0 2430 that has the highest score; if no neighbors have higher
Daniel@0 2431 score than the current point (i.e., we have reached a local maximum),
Daniel@0 2432 the algorithm stops. One can then restart in another part of the space.
Daniel@0 2433 <p>
Daniel@0 2434 A common definition of "neighbor" is all graphs that can be
Daniel@0 2435 generated from the current graph by adding, deleting or reversing a
Daniel@0 2436 single arc, subject to the acyclicity constraint.
Daniel@0 2437 Other neighborhoods are possible: see
Daniel@0 2438 <a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf">
Daniel@0 2439 Optimal Structure Identification with Greedy Search</a>, Max
Daniel@0 2440 Chickering, JMLR 2002.
Daniel@0 2441
Daniel@0 2442 <!--
Daniel@0 2443 Note: This algorithm is currently (Feb '02) being implemented by Qian
Daniel@0 2444 Diao.
Daniel@0 2445 -->
Daniel@0 2446
Daniel@0 2447
Daniel@0 2448 <h2><a name="mcmc">MCMC</h2>
Daniel@0 2449
Daniel@0 2450 We can use a Markov Chain Monte Carlo (MCMC) algorithm called
Daniel@0 2451 Metropolis-Hastings (MH) to search the space of all
Daniel@0 2452 DAGs.
Daniel@0 2453 The standard proposal distribution is to consider moving to all
Daniel@0 2454 nearest neighbors in the sense defined <a href="#hill_climb">above</a>.
Daniel@0 2455 <p>
Daniel@0 2456 The function can be called
Daniel@0 2457 as in the following example.
Daniel@0 2458 <pre>
Daniel@0 2459 [sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10);
Daniel@0 2460 </pre>
Daniel@0 2461 We can convert our set of sampled graphs to a histogram
Daniel@0 2462 (empirical posterior over all the DAGs) thus
Daniel@0 2463 <pre>
Daniel@0 2464 all_dags = mk_all_dags(N);
Daniel@0 2465 mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags);
Daniel@0 2466 </pre>
Daniel@0 2467 To see how well this performs, let us compute the exact posterior exhaustively.
Daniel@0 2468 <p>
Daniel@0 2469 <pre>
Daniel@0 2470 score = score_dags(data, ns, all_dags);
Daniel@0 2471 post = normalise(exp(score)); % assuming uniform structural prior
Daniel@0 2472 </pre>
Daniel@0 2473 We plot the results below.
Daniel@0 2474 (The data set was 100 samples drawn from a random 4 node bnet; see the
Daniel@0 2475 file BNT/examples/static/mcmc1.)
Daniel@0 2476 <pre>
Daniel@0 2477 subplot(2,1,1)
Daniel@0 2478 bar(post)
Daniel@0 2479 subplot(2,1,2)
Daniel@0 2480 bar(mcmc_post)
Daniel@0 2481 </pre>
Daniel@0 2482 <img src="Figures/mcmc_post.jpg" width="800" height="500">
Daniel@0 2483 <p>
Daniel@0 2484 We can also plot the acceptance ratio versus number of MCMC steps,
Daniel@0 2485 as a crude convergence diagnostic.
Daniel@0 2486 <pre>
Daniel@0 2487 clf
Daniel@0 2488 plot(accept_ratio)
Daniel@0 2489 </pre>
Daniel@0 2490 <img src="Figures/mcmc_accept.jpg" width="800" height="300">
Daniel@0 2491 <p>
Daniel@0 2492 Even though the number of samples needed by MCMC is theoretically
Daniel@0 2493 polynomial (not exponential) in the dimensionality of the search space, in practice it has been
Daniel@0 2494 found that MCMC does not converge in reasonable time for graphs with
Daniel@0 2495 more than about 10 nodes.
Daniel@0 2496
Daniel@0 2497
Daniel@0 2498
Daniel@0 2499
Daniel@0 2500 <h2><a name="active">Active structure learning</h2>
Daniel@0 2501
Daniel@0 2502 As was mentioned <a href="#markov_equiv">above</a>,
Daniel@0 2503 one can only learn a DAG up to Markov equivalence, even given infinite data.
Daniel@0 2504 If one is interested in learning the structure of a causal network,
Daniel@0 2505 one needs interventional data.
Daniel@0 2506 (By "intervention" we mean forcing a node to take on a specific value,
Daniel@0 2507 thereby effectively severing its incoming arcs.)
Daniel@0 2508 <p>
Daniel@0 2509 Most of the scoring functions accept an optional argument
Daniel@0 2510 that specifies whether a node was observed to have a certain value, or
Daniel@0 2511 was forced to have that value: we set clamped(i,m)=1 if node i was
Daniel@0 2512 forced in training case m. e.g., see the file
Daniel@0 2513 BNT/examples/static/cooper_yoo.
Daniel@0 2514 <p>
Daniel@0 2515 An interesting question is to decide which interventions to perform
Daniel@0 2516 (c.f., design of experiments). For details, see the following tech
Daniel@0 2517 report
Daniel@0 2518 <ul>
Daniel@0 2519 <li> <a href = "../../Papers/alearn.ps.gz">
Daniel@0 2520 Active learning of causal Bayes net structure</a>, Kevin Murphy, March
Daniel@0 2521 2001.
Daniel@0 2522 </ul>
Daniel@0 2523
Daniel@0 2524
Daniel@0 2525 <h2><a name="struct_em">Structural EM</h2>
Daniel@0 2526
Daniel@0 2527 Computing the Bayesian score when there is partial observability is
Daniel@0 2528 computationally challenging, because the parameter posterior becomes
Daniel@0 2529 multimodal (the hidden nodes induce a mixture distribution).
Daniel@0 2530 One therefore needs to use approximations such as BIC.
Daniel@0 2531 Unfortunately, search algorithms are still expensive, because we need
Daniel@0 2532 to run EM at each step to compute the MLE, which is needed to compute
Daniel@0 2533 the score of each model. An alternative approach is
Daniel@0 2534 to do the local search steps inside of the M step of EM, which is more
Daniel@0 2535 efficient since the data has been "filled in" - this is
Daniel@0 2536 called the structural EM algorithm (Friedman 1997), and provably
Daniel@0 2537 converges to a local maximum of the BIC score.
Daniel@0 2538 <p>
Daniel@0 2539 Wei Hu has implemented SEM for discrete nodes.
Daniel@0 2540 You can download his package from
Daniel@0 2541 <a href="../SEM.zip">here</a>.
Daniel@0 2542 Please address all questions about this code to
Daniel@0 2543 wei.hu@intel.com.
Daniel@0 2544 See also <a href="#phl">Phl's implementation of SEM</a>.
Daniel@0 2545
Daniel@0 2546 <!--
Daniel@0 2547 <h2><a name="reveal">REVEAL algorithm</h2>
Daniel@0 2548
Daniel@0 2549 A simple way to learn the structure of a fully observed, discrete,
Daniel@0 2550 factored DBN from a time series is described <a
Daniel@0 2551 href="usage_dbn.html#struct_learn">here</a>.
Daniel@0 2552 -->
Daniel@0 2553
Daniel@0 2554
Daniel@0 2555 <h2><a name="graphdraw">Visualizing the graph</h2>
Daniel@0 2556
Daniel@0 2557 Click <a href="graphviz.html">here</a> for more information
Daniel@0 2558 on graph visualization.
Daniel@0 2559
Daniel@0 2560 <h2><a name = "constraint">Constraint-based methods</h2>
Daniel@0 2561
Daniel@0 2562 The IC algorithm (Pearl and Verma, 1991),
Daniel@0 2563 and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993),
Daniel@0 2564 computes many conditional independence tests,
Daniel@0 2565 and combines these constraints into a
Daniel@0 2566 PDAG to represent the whole
Daniel@0 2567 <a href="#markov_equiv">Markov equivalence class</a>.
Daniel@0 2568 <p>
Daniel@0 2569 IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>.
Daniel@0 2570 (IC stands for inductive causation; PC stands for Peter and Clark,
Daniel@0 2571 the first names of Spirtes and Glymour; FCI stands for fast causal
Daniel@0 2572 inference.
Daniel@0 2573 What we, following Pearl (2000), call IC* was called
Daniel@0 2574 IC in the original Pearl and Verma paper.)
Daniel@0 2575 For details, see
Daniel@0 2576 <ul>
Daniel@0 2577 <li>
Daniel@0 2578 <a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation,
Daniel@0 2579 Prediction, and Search</a>, Spirtes, Glymour and
Daniel@0 2580 Scheines (SGS), 2001 (2nd edition), MIT Press.
Daniel@0 2581 <li>
Daniel@0 2582 <a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl,
Daniel@0 2583 2000, Cambridge University Press.
Daniel@0 2584 </ul>
Daniel@0 2585
Daniel@0 2586 <p>
Daniel@0 2587
Daniel@0 2588 The PC algorithm takes as arguments a function f, the number of nodes N,
Daniel@0 2589 the maximum fan in K, and additional arguments A which are passed to f.
Daniel@0 2590 The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0
Daniel@0 2591 otherwise.
Daniel@0 2592 For example, suppose we cheat by
Daniel@0 2593 passing in a CI "oracle" which has access to the true DAG; the oracle
Daniel@0 2594 tests for d-separation in this DAG, i.e.,
Daniel@0 2595 f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows.
Daniel@0 2596 <pre>
Daniel@0 2597 pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag);
Daniel@0 2598 </pre>
Daniel@0 2599 pdag(i,j) = -1 if there is definitely an i->j arc,
Daniel@0 2600 and pdag(i,j) = 1 if there is either an i->j or and i<-j arc.
Daniel@0 2601 <p>
Daniel@0 2602 Applied to the sprinkler network, this returns
Daniel@0 2603 <pre>
Daniel@0 2604 pdag =
Daniel@0 2605 0 1 1 0
Daniel@0 2606 1 0 0 -1
Daniel@0 2607 1 0 0 -1
Daniel@0 2608 0 0 0 0
Daniel@0 2609 </pre>
Daniel@0 2610 So as expected, we see that the V-structure at the W node is uniquely identified,
Daniel@0 2611 but the other arcs have ambiguous orientation.
Daniel@0 2612 <p>
Daniel@0 2613 We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS
Daniel@0 2614 book.
Daniel@0 2615 This example concerns the female orgasm.
Daniel@0 2616 We are given a correlation matrix C between 7 measured factors (such
Daniel@0 2617 as subjective experiences of coital and masturbatory experiences),
Daniel@0 2618 derived from 281 samples, and want to learn a causal model of the
Daniel@0 2619 data. We will not discuss the merits of this type of work here, but
Daniel@0 2620 merely show how to reproduce the results in the SGS book.
Daniel@0 2621 Their program,
Daniel@0 2622 <a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>,
Daniel@0 2623 makes use of the Fisher Z-test for conditional
Daniel@0 2624 independence, so we do the same:
Daniel@0 2625 <pre>
Daniel@0 2626 max_fan_in = 4;
Daniel@0 2627 nsamples = 281;
Daniel@0 2628 alpha = 0.05;
Daniel@0 2629 pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha);
Daniel@0 2630 </pre>
Daniel@0 2631 In this case, the CI test is
Daniel@0 2632 <pre>
Daniel@0 2633 f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha)
Daniel@0 2634 </pre>
Daniel@0 2635 The results match those of Fig 12a of SGS apart from two edge
Daniel@0 2636 differences; presumably this is due to rounding error (although it
Daniel@0 2637 could be a bug, either in BNT or in Tetrad).
Daniel@0 2638 This example can be found in the file BNT/examples/static/pc2.m.
Daniel@0 2639
Daniel@0 2640 <p>
Daniel@0 2641
Daniel@0 2642 The IC* algorithm (Pearl and Verma, 1991),
Daniel@0 2643 and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993),
Daniel@0 2644 are like the IC/PC algorithm, except that they can detect the presence
Daniel@0 2645 of latent variables.
Daniel@0 2646 See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar
Daniel@0 2647 Kushnir. The output is a matrix P, defined as follows
Daniel@0 2648 (see Pearl (2000), p52 for details):
Daniel@0 2649 <pre>
Daniel@0 2650 % 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.
Daniel@0 2651 % P(i,j) = -2 if there is a marked directed i-*>j edge.
Daniel@0 2652 % P(i,j) = P(j,i) = 1 if there is and undirected edge i--j
Daniel@0 2653 % P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j.
Daniel@0 2654 </pre>
Daniel@0 2655
Daniel@0 2656
Daniel@0 2657 <h2><a name="phl">Philippe Leray's structure learning package</h2>
Daniel@0 2658
Daniel@0 2659 Philippe Leray has written a
Daniel@0 2660 <a href="http://bnt.insa-rouen.fr/ajouts.html">
Daniel@0 2661 structure learning package</a> that uses BNT.
Daniel@0 2662
Daniel@0 2663 It currently (Juen 2003) has the following features:
Daniel@0 2664 <ul>
Daniel@0 2665 <li>PC with Chi2 statistical test
Daniel@0 2666 <li> MWST : Maximum weighted Spanning Tree
Daniel@0 2667 <li> Hill Climbing
Daniel@0 2668 <li> Greedy Search
Daniel@0 2669 <li> Structural EM
Daniel@0 2670 <li> hist_ic : optimal Histogram based on IC information criterion
Daniel@0 2671 <li> cpdag_to_dag
Daniel@0 2672 <li> dag_to_cpdag
Daniel@0 2673 <li> ...
Daniel@0 2674 </ul>
Daniel@0 2675
Daniel@0 2676
Daniel@0 2677 </a>
Daniel@0 2678
Daniel@0 2679
Daniel@0 2680 <!--
Daniel@0 2681 <h2><a name="read_learning">Further reading on learning</h2>
Daniel@0 2682
Daniel@0 2683 I recommend the following tutorials for more details on learning.
Daniel@0 2684 <ul>
Daniel@0 2685 <li> <a
Daniel@0 2686 href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short
Daniel@0 2687 tutorial</a> on graphical models, which contains an overview of learning.
Daniel@0 2688
Daniel@0 2689 <li>
Daniel@0 2690 <A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS">
Daniel@0 2691 A tutorial on learning with Bayesian networks</a>, D. Heckerman,
Daniel@0 2692 Microsoft Research Tech Report, 1995.
Daniel@0 2693
Daniel@0 2694 <li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja">
Daniel@0 2695 Operations for Learning with Graphical Models</a>,
Daniel@0 2696 W. L. Buntine, JAIR'94, 159--225.
Daniel@0 2697 </ul>
Daniel@0 2698 <p>
Daniel@0 2699 -->
Daniel@0 2700
Daniel@0 2701
Daniel@0 2702
Daniel@0 2703
Daniel@0 2704
Daniel@0 2705 <h1><a name="engines">Inference engines</h1>
Daniel@0 2706
Daniel@0 2707 Up until now, we have used the junction tree algorithm for inference.
Daniel@0 2708 However, sometimes this is too slow, or not even applicable.
Daniel@0 2709 In general, there are many inference algorithms each of which make
Daniel@0 2710 different tradeoffs between speed, accuracy, complexity and
Daniel@0 2711 generality. Furthermore, there might be many implementations of the
Daniel@0 2712 same algorithm; for instance, a general purpose, readable version,
Daniel@0 2713 and a highly-optimized, specialized one.
Daniel@0 2714 To cope with this variety, we treat each inference algorithm as an
Daniel@0 2715 object, which we call an inference engine.
Daniel@0 2716
Daniel@0 2717 <p>
Daniel@0 2718 An inference engine is an object that contains a bnet and supports the
Daniel@0 2719 'enter_evidence' and 'marginal_nodes' methods. The engine constructor
Daniel@0 2720 takes the bnet as argument and may do some model-specific processing.
Daniel@0 2721 When 'enter_evidence' is called, the engine may do some
Daniel@0 2722 evidence-specific processing. Finally, when 'marginal_nodes' is
Daniel@0 2723 called, the engine may do some query-specific processing.
Daniel@0 2724
Daniel@0 2725 <p>
Daniel@0 2726 The amount of work done when each stage is specified -- structure,
Daniel@0 2727 parameters, evidence, and query -- depends on the engine. The cost of
Daniel@0 2728 work done early in this sequence can be amortized. On the other hand,
Daniel@0 2729 one can make better optimizations if one waits until later in the
Daniel@0 2730 sequence.
Daniel@0 2731 For example, the parameters might imply
Daniel@0 2732 conditional indpendencies that are not evident in the graph structure,
Daniel@0 2733 but can nevertheless be exploited; the evidence indicates which nodes
Daniel@0 2734 are observed and hence can effectively be disconnected from the
Daniel@0 2735 graph; and the query might indicate that large parts of the network
Daniel@0 2736 are d-separated from the query nodes. (Since it is not the actual
Daniel@0 2737 <em>values</em> of the evidence that matters, just which nodes are observed,
Daniel@0 2738 many engines allow you to specify which nodes will be observed when they are constructed,
Daniel@0 2739 i.e., before calling 'enter_evidence'. Some engines can still cope if
Daniel@0 2740 the actual pattern of evidence is different, e.g., if there is missing
Daniel@0 2741 data.)
Daniel@0 2742 <p>
Daniel@0 2743
Daniel@0 2744 Although being maximally lazy (i.e., only doing work when a query is
Daniel@0 2745 issued) may seem desirable,
Daniel@0 2746 this is not always the most efficient.
Daniel@0 2747 For example,
Daniel@0 2748 when learning using EM, we need to call marginal_nodes N times, where N is the
Daniel@0 2749 number of nodes. <a href="varelim">Variable elimination</a> would end
Daniel@0 2750 up repeating a lot of work
Daniel@0 2751 each time marginal_nodes is called, making it inefficient for
Daniel@0 2752 learning. The junction tree algorithm, by contrast, uses dynamic
Daniel@0 2753 programming to avoid this redundant computation --- it calculates all
Daniel@0 2754 marginals in two passes during 'enter_evidence', so calling
Daniel@0 2755 'marginal_nodes' takes constant time.
Daniel@0 2756 <p>
Daniel@0 2757 We will discuss some of the inference algorithms implemented in BNT
Daniel@0 2758 below, and finish with a <a href="#engine_summary">summary</a> of all
Daniel@0 2759 of them.
Daniel@0 2760
Daniel@0 2761
Daniel@0 2762
Daniel@0 2763
Daniel@0 2764
Daniel@0 2765
Daniel@0 2766
Daniel@0 2767 <h2><a name="varelim">Variable elimination</h2>
Daniel@0 2768
Daniel@0 2769 The variable elimination algorithm, also known as bucket elimination
Daniel@0 2770 or peeling, is one of the simplest inference algorithms.
Daniel@0 2771 The basic idea is to "push sums inside of products"; this is explained
Daniel@0 2772 in more detail
Daniel@0 2773 <a
Daniel@0 2774 href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>.
Daniel@0 2775 <p>
Daniel@0 2776 The principle of distributing sums over products can be generalized
Daniel@0 2777 greatly to apply to any commutative semiring.
Daniel@0 2778 This forms the basis of many common algorithms, such as Viterbi
Daniel@0 2779 decoding and the Fast Fourier Transform. For details, see
Daniel@0 2780
Daniel@0 2781 <ul>
Daniel@0 2782 <li> R. McEliece and S. M. Aji, 2000.
Daniel@0 2783 <!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">-->
Daniel@0 2784 <a href="GDL.pdf">
Daniel@0 2785 The Generalized Distributive Law</a>,
Daniel@0 2786 IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000),
Daniel@0 2787 pp. 325--343.
Daniel@0 2788
Daniel@0 2789
Daniel@0 2790 <li>
Daniel@0 2791 F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001.
Daniel@0 2792 <a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html">
Daniel@0 2793 Factor graphs and the sum-product algorithm</a>
Daniel@0 2794 IEEE Transactions on Information Theory, February, 2001.
Daniel@0 2795
Daniel@0 2796 </ul>
Daniel@0 2797
Daniel@0 2798 <p>
Daniel@0 2799 Choosing an order in which to sum out the variables so as to minimize
Daniel@0 2800 computational cost is known to be NP-hard.
Daniel@0 2801 The implementation of this algorithm in
Daniel@0 2802 <tt>var_elim_inf_engine</tt> makes no attempt to optimize this
Daniel@0 2803 ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a
Daniel@0 2804 greedy search procedure to find a good ordering).
Daniel@0 2805 <p>
Daniel@0 2806 Note: unlike most algorithms, var_elim does all its computational work
Daniel@0 2807 inside of <tt>marginal_nodes</tt>, not inside of
Daniel@0 2808 <tt>enter_evidence</tt>.
Daniel@0 2809
Daniel@0 2810
Daniel@0 2811
Daniel@0 2812
Daniel@0 2813 <h2><a name="global">Global inference methods</h2>
Daniel@0 2814
Daniel@0 2815 The simplest inference algorithm of all is to explicitely construct
Daniel@0 2816 the joint distribution over all the nodes, and then to marginalize it.
Daniel@0 2817 This is implemented in <tt>global_joint_inf_engine</tt>.
Daniel@0 2818 Since the size of the joint is exponential in the
Daniel@0 2819 number of discrete (hidden) nodes, this is not a very practical algorithm.
Daniel@0 2820 It is included merely for pedagogical and debugging purposes.
Daniel@0 2821 <p>
Daniel@0 2822 Three specialized versions of this algorithm have also been implemented,
Daniel@0 2823 corresponding to the cases where all the nodes are discrete (D), all
Daniel@0 2824 are Gaussian (G), and some are discrete and some Gaussian (CG).
Daniel@0 2825 They are called <tt>enumerative_inf_engine</tt>,
Daniel@0 2826 <tt>gaussian_inf_engine</tt>,
Daniel@0 2827 and <tt>cond_gauss_inf_engine</tt> respectively.
Daniel@0 2828 <p>
Daniel@0 2829 Note: unlike most algorithms, these global inference algorithms do all their computational work
Daniel@0 2830 inside of <tt>marginal_nodes</tt>, not inside of
Daniel@0 2831 <tt>enter_evidence</tt>.
Daniel@0 2832
Daniel@0 2833
Daniel@0 2834 <h2><a name="quickscore">Quickscore</h2>
Daniel@0 2835
Daniel@0 2836 The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network,
Daniel@0 2837 since the cliques are so big.
Daniel@0 2838 One simple trick we can use is to notice that hidden leaves do not
Daniel@0 2839 affect the posteriors on the roots, and hence do not need to be
Daniel@0 2840 included in the network.
Daniel@0 2841 A second trick is to notice that the negative findings can be
Daniel@0 2842 "absorbed" into the prior:
Daniel@0 2843 see the file
Daniel@0 2844 BNT/examples/static/mk_minimal_qmr_bnet for details.
Daniel@0 2845 <p>
Daniel@0 2846
Daniel@0 2847 A much more significant speedup is obtained by exploiting special
Daniel@0 2848 properties of the noisy-or node, as done by the quickscore
Daniel@0 2849 algorithm. For details, see
Daniel@0 2850 <ul>
Daniel@0 2851 <li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89.
Daniel@0 2852 <li> Rish and Dechter, "On the impact of causal independence", UCI
Daniel@0 2853 tech report, 1998.
Daniel@0 2854 </ul>
Daniel@0 2855
Daniel@0 2856 This has been implemented in BNT as a special-purpose inference
Daniel@0 2857 engine, which can be created and used as follows:
Daniel@0 2858 <pre>
Daniel@0 2859 engine = quickscore_inf_engine(inhibit, leak, prior);
Daniel@0 2860 engine = enter_evidence(engine, pos, neg);
Daniel@0 2861 m = marginal_nodes(engine, i);
Daniel@0 2862 </pre>
Daniel@0 2863
Daniel@0 2864
Daniel@0 2865 <h2><a name="belprop">Belief propagation</h2>
Daniel@0 2866
Daniel@0 2867 Even using quickscore, exact inference takes time that is exponential
Daniel@0 2868 in the number of positive findings.
Daniel@0 2869 Hence for large networks we need to resort to approximate inference techniques.
Daniel@0 2870 See for example
Daniel@0 2871 <ul>
Daniel@0 2872 <li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the
Daniel@0 2873 QMR-DT network", JAIR 10, 1999.
Daniel@0 2874
Daniel@0 2875 <li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study",
Daniel@0 2876 UAI 99.
Daniel@0 2877 </ul>
Daniel@0 2878 The latter approximation
Daniel@0 2879 entails applying Pearl's belief propagation algorithm to a model even
Daniel@0 2880 if it has loops (hence the name loopy belief propagation).
Daniel@0 2881 Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives
Daniel@0 2882 exact results when applied to singly-connected graphs
Daniel@0 2883 (a.k.a. polytrees, since
Daniel@0 2884 the underlying undirected topology is a tree, but a node may have
Daniel@0 2885 multiple parents).
Daniel@0 2886 To apply this algorithm to a graph with loops,
Daniel@0 2887 use <tt>pearl_inf_engine</tt>.
Daniel@0 2888 This can use a centralized or distributed message passing protocol.
Daniel@0 2889 You can use it as in the following example.
Daniel@0 2890 <pre>
Daniel@0 2891 engine = pearl_inf_engine(bnet, 'max_iter', 30);
Daniel@0 2892 engine = enter_evidence(engine, evidence);
Daniel@0 2893 m = marginal_nodes(engine, i);
Daniel@0 2894 </pre>
Daniel@0 2895 We found that this algorithm often converges, and when it does, often
Daniel@0 2896 is very accurate, but it depends on the precise setting of the
Daniel@0 2897 parameter values of the network.
Daniel@0 2898 (See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.)
Daniel@0 2899 Understanding when and why belief propagation converges/ works
Daniel@0 2900 is a topic of ongoing research.
Daniel@0 2901 <p>
Daniel@0 2902 <tt>pearl_inf_engine</tt> can exploit special structure in noisy-or
Daniel@0 2903 and gmux nodes to compute messages efficiently.
Daniel@0 2904 <p>
Daniel@0 2905 <tt>belprop_inf_engine</tt> is like pearl, but uses potentials to
Daniel@0 2906 represent messages. Hence this is slower.
Daniel@0 2907 <p>
Daniel@0 2908 <tt>belprop_fg_inf_engine</tt> is like belprop,
Daniel@0 2909 but is designed for factor graphs.
Daniel@0 2910
Daniel@0 2911
Daniel@0 2912
Daniel@0 2913 <h2><a name="sampling">Sampling</h2>
Daniel@0 2914
Daniel@0 2915 BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms:
Daniel@0 2916 <ul>
Daniel@0 2917 <li> <tt>likelihood_weighting_inf_engine</tt> which does importance
Daniel@0 2918 sampling and can handle any node type.
Daniel@0 2919 <li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi.
Daniel@0 2920 Currently this can only handle tabular CPDs.
Daniel@0 2921 For a much faster and more powerful Gibbs sampling program, see
Daniel@0 2922 <a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>.
Daniel@0 2923 </ul>
Daniel@0 2924 Note: To generate samples from a network (which is not the same as inference!),
Daniel@0 2925 use <tt>sample_bnet</tt>.
Daniel@0 2926
Daniel@0 2927
Daniel@0 2928
Daniel@0 2929 <h2><a name="engine_summary">Summary of inference engines</h2>
Daniel@0 2930
Daniel@0 2931
Daniel@0 2932 The inference engines differ in many ways. Here are
Daniel@0 2933 some of the major "axes":
Daniel@0 2934 <ul>
Daniel@0 2935 <li> Works for all topologies or makes restrictions?
Daniel@0 2936 <li> Works for all node types or makes restrictions?
Daniel@0 2937 <li> Exact or approximate inference?
Daniel@0 2938 </ul>
Daniel@0 2939
Daniel@0 2940 <p>
Daniel@0 2941 In terms of topology, most engines handle any kind of DAG.
Daniel@0 2942 <tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which
Daniel@0 2943 can be used to represent directed, undirected, and mixed (chain)
Daniel@0 2944 graphs.
Daniel@0 2945 (In the future, we plan to support exact inference on chain graphs.)
Daniel@0 2946 <tt>quickscore</tt> only works on QMR-like models.
Daniel@0 2947 <p>
Daniel@0 2948 In terms of node types: algorithms that use potentials can handle
Daniel@0 2949 discrete (D), Gaussian (G) or conditional Gaussian (CG) models.
Daniel@0 2950 Sampling algorithms can essentially handle any kind of node (distribution).
Daniel@0 2951 Other algorithms make more restrictive assumptions in exchange for
Daniel@0 2952 speed.
Daniel@0 2953 <p>
Daniel@0 2954 Finally, most algorithms are designed to give the exact answer.
Daniel@0 2955 The belief propagation algorithms are exact if applied to trees, and
Daniel@0 2956 in some other cases.
Daniel@0 2957 Sampling is considered approximate, even though, in the limit of an
Daniel@0 2958 infinite number of samples, it gives the exact answer.
Daniel@0 2959
Daniel@0 2960 <p>
Daniel@0 2961
Daniel@0 2962 Here is a summary of the properties
Daniel@0 2963 of all the engines in BNT which work on static networks.
Daniel@0 2964 <p>
Daniel@0 2965 <table>
Daniel@0 2966 <table border units = pixels><tr>
Daniel@0 2967 <td align=left width=0>Name
Daniel@0 2968 <td align=left width=0>Exact?
Daniel@0 2969 <td align=left width=0>Node type?
Daniel@0 2970 <td align=left width=0>topology
Daniel@0 2971 <tr>
Daniel@0 2972 <tr>
Daniel@0 2973 <td align=left> belprop
Daniel@0 2974 <td align=left> approx
Daniel@0 2975 <td align=left> D
Daniel@0 2976 <td align=left> DAG
Daniel@0 2977 <tr>
Daniel@0 2978 <td align=left> belprop_fg
Daniel@0 2979 <td align=left> approx
Daniel@0 2980 <td align=left> D
Daniel@0 2981 <td align=left> factor graph
Daniel@0 2982 <tr>
Daniel@0 2983 <td align=left> cond_gauss
Daniel@0 2984 <td align=left> exact
Daniel@0 2985 <td align=left> CG
Daniel@0 2986 <td align=left> DAG
Daniel@0 2987 <tr>
Daniel@0 2988 <td align=left> enumerative
Daniel@0 2989 <td align=left> exact
Daniel@0 2990 <td align=left> D
Daniel@0 2991 <td align=left> DAG
Daniel@0 2992 <tr>
Daniel@0 2993 <td align=left> gaussian
Daniel@0 2994 <td align=left> exact
Daniel@0 2995 <td align=left> G
Daniel@0 2996 <td align=left> DAG
Daniel@0 2997 <tr>
Daniel@0 2998 <td align=left> gibbs
Daniel@0 2999 <td align=left> approx
Daniel@0 3000 <td align=left> D
Daniel@0 3001 <td align=left> DAG
Daniel@0 3002 <tr>
Daniel@0 3003 <td align=left> global_joint
Daniel@0 3004 <td align=left> exact
Daniel@0 3005 <td align=left> D,G,CG
Daniel@0 3006 <td align=left> DAG
Daniel@0 3007 <tr>
Daniel@0 3008 <td align=left> jtree
Daniel@0 3009 <td align=left> exact
Daniel@0 3010 <td align=left> D,G,CG
Daniel@0 3011 <td align=left> DAG
Daniel@0 3012 b<tr>
Daniel@0 3013 <td align=left> likelihood_weighting
Daniel@0 3014 <td align=left> approx
Daniel@0 3015 <td align=left> any
Daniel@0 3016 <td align=left> DAG
Daniel@0 3017 <tr>
Daniel@0 3018 <td align=left> pearl
Daniel@0 3019 <td align=left> approx
Daniel@0 3020 <td align=left> D,G
Daniel@0 3021 <td align=left> DAG
Daniel@0 3022 <tr>
Daniel@0 3023 <td align=left> pearl
Daniel@0 3024 <td align=left> exact
Daniel@0 3025 <td align=left> D,G
Daniel@0 3026 <td align=left> polytree
Daniel@0 3027 <tr>
Daniel@0 3028 <td align=left> quickscore
Daniel@0 3029 <td align=left> exact
Daniel@0 3030 <td align=left> noisy-or
Daniel@0 3031 <td align=left> QMR
Daniel@0 3032 <tr>
Daniel@0 3033 <td align=left> stab_cond_gauss
Daniel@0 3034 <td align=left> exact
Daniel@0 3035 <td align=left> CG
Daniel@0 3036 <td align=left> DAG
Daniel@0 3037 <tr>
Daniel@0 3038 <td align=left> var_elim
Daniel@0 3039 <td align=left> exact
Daniel@0 3040 <td align=left> D,G,CG
Daniel@0 3041 <td align=left> DAG
Daniel@0 3042 </table>
Daniel@0 3043
Daniel@0 3044
Daniel@0 3045
Daniel@0 3046 <h1><a name="influence">Influence diagrams/ decision making</h1>
Daniel@0 3047
Daniel@0 3048 BNT implements an exact algorithm for solving LIMIDs (limited memory
Daniel@0 3049 influence diagrams), described in
Daniel@0 3050 <ul>
Daniel@0 3051 <li> S. L. Lauritzen and D. Nilsson.
Daniel@0 3052 <a href="http://www.math.auc.dk/~steffen/papers/limids.pdf">
Daniel@0 3053 Representing and solving decision problems with limited
Daniel@0 3054 information</a>
Daniel@0 3055 Management Science, 47, 1238 - 1251. September 2001.
Daniel@0 3056 </ul>
Daniel@0 3057 LIMIDs explicitely show all information arcs, rather than implicitely
Daniel@0 3058 assuming no forgetting. This allows them to model forgetful
Daniel@0 3059 controllers.
Daniel@0 3060 <p>
Daniel@0 3061 See the examples in <tt>BNT/examples/limids</tt> for details.
Daniel@0 3062
Daniel@0 3063
Daniel@0 3064
Daniel@0 3065
Daniel@0 3066 <h1>DBNs, HMMs, Kalman filters and all that</h1>
Daniel@0 3067
Daniel@0 3068 Click <a href="usage_dbn.html">here</a> for documentation about how to
Daniel@0 3069 use BNT for dynamical systems and sequence data.
Daniel@0 3070
Daniel@0 3071
Daniel@0 3072 </BODY>