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| date | Tue, 10 Feb 2015 15:05:51 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolboxes/FullBNT-1.0.7/docs/usage.html Tue Feb 10 15:05:51 2015 +0000 @@ -0,0 +1,3072 @@ +<HEAD> +<TITLE>How to use the Bayes Net Toolbox</TITLE> +</HEAD> + +<BODY BGCOLOR="#FFFFFF"> +<!-- white background is better for the pictures and equations --> + +<h1>How to use the Bayes Net Toolbox</h1> + +This documentation was last updated on 29 October 2007. +<br> +Click +<a href="http://bnt.insa-rouen.fr/">here</a> +for a French version of this documentation (last updated in 2005). +<p> + +<ul> +<li> <a href="install.html">Installation</a> + +<li> <a href="#basics">Creating your first Bayes net</a> + <ul> + <li> <a href="#basics">Creating a model by hand</a> + <li> <a href="#file">Loading a model from a file</a> + <li> <a href="#GUI">Creating a model using a GUI</a> + <li> <a href="graphviz.html">Graph visualization</a> + </ul> + +<li> <a href="#inference">Inference</a> + <ul> + <li> <a href="#marginal">Computing marginal distributions</a> + <li> <a href="#joint">Computing joint distributions</a> + <li> <a href="#soft">Soft/virtual evidence</a> + <li> <a href="#mpe">Most probable explanation</a> + </ul> + +<li> <a href="#cpd">Conditional Probability Distributions</a> + <ul> + <li> <a href="#tabular">Tabular (multinomial) nodes</a> + <li> <a href="#noisyor">Noisy-or nodes</a> + <li> <a href="#deterministic">Other (noisy) deterministic nodes</a> + <li> <a href="#softmax">Softmax (multinomial logit) nodes</a> + <li> <a href="#mlp">Neural network nodes</a> + <li> <a href="#root">Root nodes</a> + <li> <a href="#gaussian">Gaussian nodes</a> + <li> <a href="#glm">Generalized linear model nodes</a> + <li> <a href="#dtree">Classification/regression tree nodes</a> + <li> <a href="#nongauss">Other continuous distributions</a> + <li> <a href="#cpd_summary">Summary of CPD types</a> + </ul> + +<li> <a href="#examples">Example models</a> + <ul> + <li> <a + href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt"> +Gaussian mixture models</a> + <li> <a href="#pca">PCA, ICA, and all that</a> + <li> <a href="#mixep">Mixtures of experts</a> + <li> <a href="#hme">Hierarchical mixtures of experts</a> + <li> <a href="#qmr">QMR</a> + <li> <a href="#cg_model">Conditional Gaussian models</a> + <li> <a href="#hybrid">Other hybrid models</a> + </ul> + +<li> <a href="#param_learning">Parameter learning</a> + <ul> + <li> <a href="#load_data">Loading data from a file</a> + <li> <a href="#mle_complete">Maximum likelihood parameter estimation from complete data</a> + <li> <a href="#prior">Parameter priors</a> + <li> <a href="#bayes_learn">(Sequential) Bayesian parameter updating from complete data</a> + <li> <a href="#em">Maximum likelihood parameter estimation with missing values (EM)</a> + <li> <a href="#tying">Parameter tying</a> + </ul> + +<li> <a href="#structure_learning">Structure learning</a> + <ul> + <li> <a href="#enumerate">Exhaustive search</a> + <li> <a href="#K2">K2</a> + <li> <a href="#hill_climb">Hill-climbing</a> + <li> <a href="#mcmc">MCMC</a> + <li> <a href="#active">Active learning</a> + <li> <a href="#struct_em">Structural EM</a> + <li> <a href="#graphdraw">Visualizing the learned graph structure</a> + <li> <a href="#constraint">Constraint-based methods</a> + </ul> + + +<li> <a href="#engines">Inference engines</a> + <ul> + <li> <a href="#jtree">Junction tree</a> + <li> <a href="#varelim">Variable elimination</a> + <li> <a href="#global">Global inference methods</a> + <li> <a href="#quickscore">Quickscore</a> + <li> <a href="#belprop">Belief propagation</a> + <li> <a href="#sampling">Sampling (Monte Carlo)</a> + <li> <a href="#engine_summary">Summary of inference engines</a> + </ul> + + +<li> <a href="#influence">Influence diagrams/ decision making</a> + + +<li> <a href="usage_dbn.html">DBNs, HMMs, Kalman filters and all that</a> +</ul> + +</ul> + + + +<h1><a name="basics">Creating your first Bayes net</h1> + +To define a Bayes net, you must specify the graph structure and then +the parameters. We look at each in turn, using a simple example +(adapted from Russell and +Norvig, "Artificial Intelligence: a Modern Approach", Prentice Hall, +1995, p454). + + +<h2>Graph structure</h2> + + +Consider the following network. + +<p> +<center> +<IMG SRC="Figures/sprinkler.gif"> +</center> +<p> + +<P> +To specify this directed acyclic graph (dag), we create an adjacency matrix: +<PRE> +N = 4; +dag = zeros(N,N); +C = 1; S = 2; R = 3; W = 4; +dag(C,[R S]) = 1; +dag(R,W) = 1; +dag(S,W)=1; +</PRE> +<P> +We have numbered the nodes as follows: +Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. +<b>The nodes must always be numbered in topological order, i.e., +ancestors before descendants.</b> +For a more complicated graph, this is a little inconvenient: we will +see how to get around this <a href="usage_dbn.html#bat">below</a>. +<p> +In Matlab 6, you can use logical arrays instead of double arrays, +which are 4 times smaller: +<pre> +dag = false(N,N); +dag(C,[R S]) = true; +... +</pre> +However, <b>some graph functions (eg acyclic) do not work on +logical arrays</b>! +<p> +You can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>. +For details on GUIs, +click <a href="#GUI">here</a>. + +<h2>Creating the Bayes net shell</h2> + +In addition to specifying the graph structure, +we must specify the size and type of each node. +If a node is discrete, its size is the +number of possible values +each node can take on; if a node is continuous, +it can be a vector, and its size is the length of this vector. +In this case, we will assume all nodes are discrete and binary. +<PRE> +discrete_nodes = 1:N; +node_sizes = 2*ones(1,N); +</pre> +If the nodes were not binary, you could type e.g., +<pre> +node_sizes = [4 2 3 5]; +</pre> +meaning that Cloudy has 4 possible values, +Sprinkler has 2 possible values, etc. +Note that these are cardinal values, not ordinal, i.e., +they are not ordered in any way, like 'low', 'medium', 'high'. +<p> +We are now ready to make the Bayes net: +<pre> +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes); +</PRE> +By default, all nodes are assumed to be discrete, so we can also just +write +<pre> +bnet = mk_bnet(dag, node_sizes); +</PRE> +You may also specify which nodes will be observed. +If you don't know, or if this not fixed in advance, +just use the empty list (the default). +<pre> +onodes = []; +bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes, 'observed', onodes); +</PRE> +Note that optional arguments are specified using a name/value syntax. +This is common for many BNT functions. +In general, to find out more about a function (e.g., which optional +arguments it takes), please see its +documentation string by typing +<pre> +help mk_bnet +</pre> +See also other <a href="matlab_tips.html">useful Matlab tips</a>. +<p> +It is possible to associate names with nodes, as follows: +<pre> +bnet = mk_bnet(dag, node_sizes, 'names', {'cloudy','S','R','W'}, 'discrete', 1:4); +</pre> +You can then refer to a node by its name: +<pre> +C = bnet.names('cloudy'); % bnet.names is an associative array +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +</pre> +This feature uses my own associative array class. + + +<h2><a name="cpt">Parameters</h2> + +A model consists of the graph structure and the parameters. +The parameters are represented by CPD objects (CPD = Conditional +Probability Distribution), which define the probability distribution +of a node given its parents. +(We will use the terms "node" and "random variable" interchangeably.) +The simplest kind of CPD is a table (multi-dimensional array), which +is suitable when all the nodes are discrete-valued. Note that the discrete +values are not assumed to be ordered in any way; that is, they +represent categorical quantities, like male and female, rather than +ordinal quantities, like low, medium and high. +(We will discuss CPDs in more detail <a href="#cpd">below</a>.) +<p> +Tabular CPDs, also called CPTs (conditional probability tables), +are stored as multidimensional arrays, where the dimensions +are arranged in the same order as the nodes, e.g., the CPT for node 4 +(WetGrass) is indexed by Sprinkler (2), Rain (3) and then WetGrass (4) itself. +Hence the child is always the last dimension. +If a node has no parents, its CPT is a column vector representing its +prior. +Note that in Matlab (unlike C), arrays are indexed +from 1, and are layed out in memory such that the first index toggles +fastest, e.g., the CPT for node 4 (WetGrass) is as follows +<P> +<P><IMG ALIGN=BOTTOM SRC="Figures/CPTgrass.gif"><P> +<P> +where we have used the convention that false==1, true==2. +We can create this CPT in Matlab as follows +<PRE> +CPT = zeros(2,2,2); +CPT(1,1,1) = 1.0; +CPT(2,1,1) = 0.1; +... +</PRE> +Here is an easier way: +<PRE> +CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]); +</PRE> +In fact, we don't need to reshape the array, since the CPD constructor +will do that for us. So we can just write +<pre> +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</pre> +The other nodes are created similarly (using the old syntax for +optional parameters) +<PRE> +bnet.CPD{C} = tabular_CPD(bnet, C, [0.5 0.5]); +bnet.CPD{R} = tabular_CPD(bnet, R, [0.8 0.2 0.2 0.8]); +bnet.CPD{S} = tabular_CPD(bnet, S, [0.5 0.9 0.5 0.1]); +bnet.CPD{W} = tabular_CPD(bnet, W, [1 0.1 0.1 0.01 0 0.9 0.9 0.99]); +</PRE> + + +<h2><a name="rnd_cpt">Random Parameters</h2> + +If we do not specify the CPT, random parameters will be +created, i.e., each "row" of the CPT will be drawn from the uniform distribution. +To ensure repeatable results, use +<pre> +rand('state', seed); +randn('state', seed); +</pre> +To control the degree of randomness (entropy), +you can sample each row of the CPT from a Dirichlet(p,p,...) distribution. +If p << 1, this encourages "deterministic" CPTs (one entry near 1, the rest near 0). +If p = 1, each entry is drawn from U[0,1]. +If p >> 1, the entries will all be near 1/k, where k is the arity of +this node, i.e., each row will be nearly uniform. +You can do this as follows, assuming this node +is number i, and ns is the node_sizes. +<pre> +k = ns(i); +ps = parents(dag, i); +psz = prod(ns(ps)); +CPT = sample_dirichlet(p*ones(1,k), psz); +bnet.CPD{i} = tabular_CPD(bnet, i, 'CPT', CPT); +</pre> + + +<h2><a name="file">Loading a network from a file</h2> + +If you already have a Bayes net represented in the XML-based +<a href="http://www.cs.cmu.edu/afs/cs/user/fgcozman/www/Research/InterchangeFormat/"> +Bayes Net Interchange Format (BNIF)</a> (e.g., downloaded from the +<a +href="http://www.cs.huji.ac.il/labs/compbio/Repository"> +Bayes Net repository</a>), +you can convert it to BNT format using +the +<a href="http://www.digitas.harvard.edu/~ken/bif2bnt/">BIF-BNT Java +program</a> written by Ken Shan. +(This is not necessarily up-to-date.) +<p> +<b>It is currently not possible to save/load a BNT matlab object to +file</b>, but this is easily fixed if you modify all the constructors +for all the classes (see matlab documentation). + +<h2><a name="GUI">Creating a model using a GUI</h2> + +<ul> +<li>Senthil Nachimuthu +has started (Oct 07) an open source +GUI for BNT called +<a href="http://projeny.sourceforge.net">projeny</a> +using Java. This is a successor to BNJ. + +<li> +Philippe LeRay has written (Sep 05) +a +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-editor/"> +BNT GUI</a> in matlab. + +<li> +<a +href="http://www.dataonstage.com/BNT/PACKAGES/LinkStrength/index.html"> +LinkStrength</a>, + a package by Imme Ebert-Uphoff for visualizing the strength of +dependencies between nodes. + +</ul> + +<h2>Graph visualization</h2> + +Click <a href="graphviz.html">here</a> for more information +on graph visualization. + + +<h1><a name="inference">Inference</h1> + +Having created the BN, we can now use it for inference. +There are many different algorithms for doing inference in Bayes nets, +that make different tradeoffs between speed, +complexity, generality, and accuracy. +BNT therefore offers a variety of different inference +"engines". We will discuss these +in more detail <a href="#engines">below</a>. +For now, we will use the junction tree +engine, which is the mother of all exact inference algorithms. +This can be created as follows. +<pre> +engine = jtree_inf_engine(bnet); +</pre> +The other engines have similar constructors, but might take +additional, algorithm-specific parameters. +All engines are used in the same way, once they have been created. +We illustrate this in the following sections. + + +<h2><a name="marginal">Computing marginal distributions</h2> + +Suppose we want to compute the probability that the sprinker was on +given that the grass is wet. +The evidence consists of the fact that W=2. All the other nodes +are hidden (unobserved). We can specify this as follows. +<pre> +evidence = cell(1,N); +evidence{W} = 2; +</pre> +We use a 1D cell array instead of a vector to +cope with the fact that nodes can be vectors of different lengths. +In addition, the value [] can be used +to denote 'no evidence', instead of having to specify the observation +pattern as a separate argument. +(Click <a href="cellarray.html">here</a> for a quick tutorial on cell +arrays in matlab.) +<p> +We are now ready to add the evidence to the engine. +<pre> +[engine, loglik] = enter_evidence(engine, evidence); +</pre> +The behavior of this function is algorithm-specific, and is discussed +in more detail <a href="#engines">below</a>. +In the case of the jtree engine, +enter_evidence implements a two-pass message-passing scheme. +The first return argument contains the modified engine, which +incorporates the evidence. The second return argument contains the +log-likelihood of the evidence. (Not all engines are capable of +computing the log-likelihood.) +<p> +Finally, we can compute p=P(S=2|W=2) as follows. +<PRE> +marg = marginal_nodes(engine, S); +marg.T +ans = + 0.57024 + 0.42976 +p = marg.T(2); +</PRE> +We see that p = 0.4298. +<p> +Now let us add the evidence that it was raining, and see what +difference it makes. +<PRE> +evidence{R} = 2; +[engine, loglik] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, S); +p = marg.T(2); +</PRE> +We find that p = P(S=2|W=2,R=2) = 0.1945, +which is lower than +before, because the rain can ``explain away'' the +fact that the grass is wet. +<p> +You can plot a marginal distribution over a discrete variable +as a barchart using the built 'bar' function: +<pre> +bar(marg.T) +</pre> +This is what it looks like + +<p> +<center> +<IMG SRC="Figures/sprinkler_bar.gif"> +</center> +<p> + +<h2><a name="observed">Observed nodes</h2> + +What happens if we ask for the marginal on an observed node, e.g. P(W|W=2)? +An observed discrete node effectively only has 1 value (the observed + one) --- all other values would result in 0 probability. +For efficiency, BNT treats observed (discrete) nodes as if they were + set to 1, as we see below: +<pre> +evidence = cell(1,N); +evidence{W} = 2; +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, W); +m.T +ans = + 1 +</pre> +This can get a little confusing, since we assigned W=2. +So we can ask BNT to add the evidence back in by passing in an optional argument: +<pre> +m = marginal_nodes(engine, W, 1); +m.T +ans = + 0 + 1 +</pre> +This shows that P(W=1|W=2) = 0 and P(W=2|W=2) = 1. + + + +<h2><a name="joint">Computing joint distributions</h2> + +We can compute the joint probability on a set of nodes as in the +following example. +<pre> +evidence = cell(1,N); +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]); +</pre> +m is a structure. The 'T' field is a multi-dimensional array (in +this case, 3-dimensional) that contains the joint probability +distribution on the specified nodes. +<pre> +>> m.T +ans(:,:,1) = + 0.2900 0.0410 + 0.0210 0.0009 +ans(:,:,2) = + 0 0.3690 + 0.1890 0.0891 +</pre> +We see that P(S=1,R=1,W=2) = 0, since it is impossible for the grass +to be wet if both the rain and sprinkler are off. +<p> +Let us now add some evidence to R. +<pre> +evidence{R} = 2; +[engine, ll] = enter_evidence(engine, evidence); +m = marginal_nodes(engine, [S R W]) +m = + domain: [2 3 4] + T: [2x1x2 double] +>> m.T +m.T +ans(:,:,1) = + 0.0820 + 0.0018 +ans(:,:,2) = + 0.7380 + 0.1782 +</pre> +The joint T(i,j,k) = P(S=i,R=j,W=k|evidence) +should have T(i,1,k) = 0 for all i,k, since R=1 is incompatible +with the evidence that R=2. +Instead of creating large tables with many 0s, BNT sets the effective +size of observed (discrete) nodes to 1, as explained above. +This is why m.T has size 2x1x2. +To get a 2x2x2 table, type +<pre> +m = marginal_nodes(engine, [S R W], 1) +m = + domain: [2 3 4] + T: [2x2x2 double] +>> m.T +m.T +ans(:,:,1) = + 0 0.082 + 0 0.0018 +ans(:,:,2) = + 0 0.738 + 0 0.1782 +</pre> + +<p> +Note: It is not always possible to compute the joint on arbitrary +sets of nodes: it depends on which inference engine you use, as discussed +in more detail <a href="#engines">below</a>. + + +<h2><a name="soft">Soft/virtual evidence</h2> + +Sometimes a node is not observed, but we have some distribution over +its possible values; this is often called "soft" or "virtual" +evidence. +One can use this as follows +<pre> +[engine, loglik] = enter_evidence(engine, evidence, 'soft', soft_evidence); +</pre> +where soft_evidence{i} is either [] (if node i has no soft evidence) +or is a vector representing the probability distribution over i's +possible values. +For example, if we don't know i's exact value, but we know its +likelihood ratio is 60/40, we can write evidence{i} = [] and +soft_evidence{i} = [0.6 0.4]. +<p> +Currently only jtree_inf_engine supports this option. +It assumes that all hidden nodes, and all nodes for +which we have soft evidence, are discrete. +For a longer example, see BNT/examples/static/softev1.m. + + +<h2><a name="mpe">Most probable explanation</h2> + +To compute the most probable explanation (MPE) of the evidence (i.e., +the most probable assignment, or a mode of the joint), use +<pre> +[mpe, ll] = calc_mpe(engine, evidence); +</pre> +mpe{i} is the most likely value of node i. +This calls enter_evidence with the 'maximize' flag set to 1, which +causes the engine to do max-product instead of sum-product. +The resulting max-marginals are then thresholded. +If there is more than one maximum probability assignment, we must take + care to break ties in a consistent manner (thresholding the + max-marginals may give the wrong result). To force this behavior, + type +<pre> +[mpe, ll] = calc_mpe(engine, evidence, 1); +</pre> +Note that computing the MPE is someties called abductive reasoning. + +<p> +You can also use <tt>calc_mpe_bucket</tt> written by Ron Zohar, +that does a forwards max-product pass, and then a backwards traceback +pass, which is how Viterbi is traditionally implemented. + + + +<h1><a name="cpd">Conditional Probability Distributions</h1> + +A Conditional Probability Distributions (CPD) +defines P(X(i) | X(Pa(i))), where X(i) is the i'th node, and X(Pa(i)) +are the parents of node i. There are many ways to represent this +distribution, which depend in part on whether X(i) and X(Pa(i)) are +discrete, continuous, or a combination. +We will discuss various representations below. + + +<h2><a name="tabular">Tabular nodes</h2> + +If the CPD is represented as a table (i.e., if it is a multinomial +distribution), it has a number of parameters that is exponential in +the number of parents. See the example <a href="#cpt">above</a>. + + +<h2><a name="noisyor">Noisy-or nodes</h2> + +A noisy-OR node is like a regular logical OR gate except that +sometimes the effects of parents that are on get inhibited. +Let the prob. that parent i gets inhibited be q(i). +Then a node, C, with 2 parents, A and B, has the following CPD, where +we use F and T to represent off and on (1 and 2 in BNT). +<pre> +A B P(C=off) P(C=on) +--------------------------- +F F 1.0 0.0 +T F q(A) 1-q(A) +F T q(B) 1-q(B) +T T q(A)q(B) 1-q(A)q(B) +</pre> +Thus we see that the causes get inhibited independently. +It is common to associate a "leak" node with a noisy-or CPD, which is +like a parent that is always on. This can account for all other unmodelled +causes which might turn the node on. +<p> +The noisy-or distribution is similar to the logistic distribution. +To see this, let the nodes, S(i), have values in {0,1}, and let q(i,j) +be the prob. that j inhibits i. Then +<pre> +Pr(S(i)=1 | parents(S(i))) = 1 - prod_{j} q(i,j)^S(j) +</pre> +Now define w(i,j) = -ln q(i,j) and rho(x) = 1-exp(-x). Then +<pre> +Pr(S(i)=1 | parents(S(i))) = rho(sum_j w(i,j) S(j)) +</pre> +For a sigmoid node, we have +<pre> +Pr(S(i)=1 | parents(S(i))) = sigma(-sum_j w(i,j) S(j)) +</pre> +where sigma(x) = 1/(1+exp(-x)). Hence they differ in the choice of +the activation function (although both are monotonically increasing). +In addition, in the case of a noisy-or, the weights are constrained to be +positive, since they derive from probabilities q(i,j). +In both cases, the number of parameters is <em>linear</em> in the +number of parents, unlike the case of a multinomial distribution, +where the number of parameters is exponential in the number of parents. +We will see an example of noisy-OR nodes <a href="#qmr">below</a>. + + +<h2><a name="deterministic">Other (noisy) deterministic nodes</h2> + +Deterministic CPDs for discrete random variables can be created using +the deterministic_CPD class. It is also possible to 'flip' the output +of the function with some probability, to simulate noise. +The boolean_CPD class is just a special case of a +deterministic CPD, where the parents and child are all binary. +<p> +Both of these classes are just "syntactic sugar" for the tabular_CPD +class. + + + +<h2><a name="softmax">Softmax nodes</h2> + +If we have a discrete node with a continuous parent, +we can define its CPD using a softmax function +(also known as the multinomial logit function). +This acts like a soft thresholding operator, and is defined as follows: +<pre> + exp(w(:,i)'*x + b(i)) +Pr(Q=i | X=x) = ----------------------------- + sum_j exp(w(:,j)'*x + b(j)) + +</pre> +The parameters of a softmax node, w(:,i) and b(i), i=1..|Q|, have the +following interpretation: w(:,i)-w(:,j) is the normal vector to the +decision boundary between classes i and j, +and b(i)-b(j) is its offset (bias). For example, suppose +X is a 2-vector, and Q is binary. Then +<pre> +w = [1 -1; + 0 0]; + +b = [0 0]; +</pre> +means class 1 are points in the 2D plane with positive x coordinate, +and class 2 are points in the 2D plane with negative x coordinate. +If w has large magnitude, the decision boundary is sharp, otherwise it +is soft. +In the special case that Q is binary (0/1), the softmax function reduces to the logistic +(sigmoid) function. +<p> +Fitting a softmax function can be done using the iteratively reweighted +least squares (IRLS) algorithm. +We use the implementation from +<a href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +Note that since +the softmax distribution is not in the exponential family, it does not +have finite sufficient statistics, and hence we must store all the +training data in uncompressed form. +If this takes too much space, one should use online (stochastic) gradient +descent (not implemented in BNT). +<p> +If a softmax node also has discrete parents, +we use a different set of w/b parameters for each combination of +parent values, as in the <a href="#gaussian">conditional linear +Gaussian CPD</a>. +This feature was implemented by Pierpaolo Brutti. +He is currently extending it so that discrete parents can be treated +as if they were continuous, by adding indicator variables to the X +vector. +<p> +We will see an example of softmax nodes <a href="#mixexp">below</a>. + + +<h2><a name="mlp">Neural network nodes</h2> + +Pierpaolo Brutti has implemented the mlp_CPD class, which uses a multi layer perceptron +to implement a mapping from continuous parents to discrete children, +similar to the softmax function. +(If there are also discrete parents, it creates a mixture of MLPs.) +It uses code from <a +href="http://www.ncrg.aston.ac.uk/netlab/">Netlab</a>. +This is work in progress. + +<h2><a name="root">Root nodes</h2> + +A root node has no parents and no parameters; it can be used to model +an observed, exogeneous input variable, i.e., one which is "outside" +the model. +This is useful for conditional density models. +We will see an example of root nodes <a href="#mixexp">below</a>. + + +<h2><a name="gaussian">Gaussian nodes</h2> + +We now consider a distribution suitable for the continuous-valued nodes. +Suppose the node is called Y, its continuous parents (if any) are +called X, and its discrete parents (if any) are called Q. +The distribution on Y is defined as follows: +<pre> +- no parents: Y ~ N(mu, Sigma) +- cts parents : Y|X=x ~ N(mu + W x, Sigma) +- discrete parents: Y|Q=i ~ N(mu(:,i), Sigma(:,:,i)) +- cts and discrete parents: Y|X=x,Q=i ~ N(mu(:,i) + W(:,:,i) * x, Sigma(:,:,i)) +</pre> +where N(mu, Sigma) denotes a Normal distribution with mean mu and +covariance Sigma. Let |X|, |Y| and |Q| denote the sizes of X, Y and Q +respectively. +If there are no discrete parents, |Q|=1; if there is +more than one, then |Q| = a vector of the sizes of each discrete parent. +If there are no continuous parents, |X|=0; if there is more than one, +then |X| = the sum of their sizes. +Then mu is a |Y|*|Q| vector, Sigma is a |Y|*|Y|*|Q| positive +semi-definite matrix, and W is a |Y|*|X|*|Q| regression (weight) +matrix. +<p> +We can create a Gaussian node with random parameters as follows. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i); +</pre> +We can specify the value of one or more of the parameters as in the +following example, in which |Y|=2, and |Q|=1. +<pre> +bnet.CPD{i} = gaussian_CPD(bnet, i, 'mean', [0; 0], 'weights', randn(Y,X), 'cov', eye(Y)); +</pre> +<p> +We will see an example of conditional linear Gaussian nodes <a +href="#cg_model">below</a>. +<p> +<b>When learning Gaussians from data</b>, it is helpful to ensure the +data has a small magnitde +(see e.g., KPMstats/standardize) to prevent numerical problems. +Unless you have a lot of data, it is also a very good idea to use +diagonal instead of full covariance matrices. +(BNT does not currently support spherical covariances, although it +would be easy to add, since KPMstats/clg_Mstep supports this option; +you would just need to modify gaussian_CPD/update_ess to accumulate +weighted inner products.) + + + +<h2><a name="nongauss">Other continuous distributions</h2> + +Currently BNT does not support any CPDs for continuous nodes other +than the Gaussian. +However, you can use a mixture of Gaussians to +approximate other continuous distributions. We will see some an example +of this with the IFA model <a href="#pca">below</a>. + + +<h2><a name="glm">Generalized linear model nodes</h2> + +In the future, we may incorporate some of the functionality of +<a href = +"http://www.sci.usq.edu.au/staff/dunn/glmlab/glmlab.html">glmlab</a> +into BNT. + + +<h2><a name="dtree">Classification/regression tree nodes</h2> + +We plan to add classification and regression trees to define CPDs for +discrete and continuous nodes, respectively. +Trees have many advantages: they are easy to interpret, they can do +feature selection, they can +handle discrete and continuous inputs, they do not make strong +assumptions about the form of the distribution, the number of +parameters can grow in a data-dependent way (i.e., they are +semi-parametric), they can handle missing data, etc. +However, they are not yet implemented. +<!-- +Yimin Zhang is currently (Feb '02) implementing this. +--> + + +<h2><a name="cpd_summary">Summary of CPD types</h2> + +We list all the different types of CPDs supported by BNT. +For each CPD, we specify if the child and parents can be discrete (D) or +continuous (C) (Binary (B) nodes are a special case). +We also specify which methods each class supports. +If a method is inherited, the name of the parent class is mentioned. +If a parent class calls a child method, this is mentioned. +<p> +The <tt>CPD_to_CPT</tt> method converts a CPD to a table; this +requires that the child and all parents are discrete. +The CPT might be exponentially big... +<tt>convert_to_table</tt> evaluates a CPD with evidence, and +represents the the resulting potential as an array. +This requires that the child is discrete, and any continuous parents +are observed. +<tt>convert_to_pot</tt> evaluates a CPD with evidence, and +represents the resulting potential as a dpot, gpot, cgpot or upot, as +requested. (d=discrete, g=Gaussian, cg = conditional Gaussian, u = +utility). + +<p> +When we sample a node, all the parents are observed. +When we compute the (log) probability of a node, all the parents and +the child are observed. +<p> +We also specify if the parameters are learnable. +For learning with EM, we require +the methods <tt>reset_ess</tt>, <tt>update_ess</tt> and +<tt>maximize_params</tt>. +For learning from fully observed data, we require +the method <tt>learn_params</tt>. +By default, all classes inherit this from generic_CPD, which simply +calls <tt>update_ess</tt> N times, once for each data case, followed +by <tt>maximize_params</tt>, i.e., it is like EM, without the E step. +Some classes implement a batch formula, which is quicker. +<p> +Bayesian learning means computing a posterior over the parameters +given fully observed data. +<p> +Pearl means we implement the methods <tt>compute_pi</tt> and +<tt>compute_lambda_msg</tt>, used by +<tt>pearl_inf_engine</tt>, which runs on directed graphs. +<tt>belprop_inf_engine</tt> only needs <tt>convert_to_pot</tt>.H +The pearl methods can exploit special properties of the CPDs for +computing the messages efficiently, whereas belprop does not. +<p> +The only method implemented by generic_CPD is <tt>adjustable_CPD</tt>, +which is not shown, since it is not very interesting. + + +<p> + + +<table> +<table border units = pixels><tr> +<td align=center>Name +<td align=center>Child +<td align=center>Parents +<td align=center>Comments +<td align=center>CPD_to_CPT +<td align=center>conv_to_table +<td align=center>conv_to_pot +<td align=center>sample +<td align=center>prob +<td align=center>learn +<td align=center>Bayes +<td align=center>Pearl + + +<tr> +<!-- Name--><td> +<!-- Child--><td> +<!-- Parents--><td> +<!-- Comments--><td> +<!-- CPD_to_CPT--><td> +<!-- conv_to_table--><td> +<!-- conv_to_pot--><td> +<!-- sample--><td> +<!-- prob--><td> +<!-- learn--><td> +<!-- Bayes--><td> +<!-- Pearl--><td> + +<tr> +<!-- Name--><td>boolean +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>deterministic +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>Syntactic sugar for tabular +<!-- CPD_to_CPT--><td>- +<!-- conv_to_table--><td>- +<!-- conv_to_pot--><td>- +<!-- sample--><td>- +<!-- prob--><td>- +<!-- learn--><td>- +<!-- Bayes--><td>- +<!-- Pearl--><td>- + +<tr> +<!-- Name--><td>Discrete +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Calls CPD_to_CPT +<!-- conv_to_pot--><td>Calls conv_to_table +<!-- sample--><td>Calls conv_to_table +<!-- prob--><td>Calls conv_to_table +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>Gaussian +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + +<tr> +<!-- Name--><td>gmux +<!-- Child--><td>C +<!-- Parents--><td>C/D +<!-- Comments--><td>multiplexer +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>MLP +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>multi layer perceptron +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>noisy-or +<!-- Child--><td>B +<!-- Parents--><td>B +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>Y + + +<tr> +<!-- Name--><td>root +<!-- Child--><td>C/D +<!-- Parents--><td>none +<!-- Comments--><td>no params +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>Y +<!-- sample--><td>Y +<!-- prob--><td>Y +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>softmax +<!-- Child--><td>D +<!-- Parents--><td>C/D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>Y +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>generic +<!-- Child--><td>C/D +<!-- Parents--><td>C/D +<!-- Comments--><td>Virtual class +<!-- CPD_to_CPT--><td>N +<!-- conv_to_table--><td>N +<!-- conv_to_pot--><td>N +<!-- sample--><td>N +<!-- prob--><td>N +<!-- learn--><td>N +<!-- Bayes--><td>N +<!-- Pearl--><td>N + + +<tr> +<!-- Name--><td>Tabular +<!-- Child--><td>D +<!-- Parents--><td>D +<!-- Comments--><td>- +<!-- CPD_to_CPT--><td>Y +<!-- conv_to_table--><td>Inherits from discrete +<!-- conv_to_pot--><td>Inherits from discrete +<!-- sample--><td>Inherits from discrete +<!-- prob--><td>Inherits from discrete +<!-- learn--><td>Y +<!-- Bayes--><td>Y +<!-- Pearl--><td>Y + +</table> + + + +<h1><a name="examples">Example models</h1> + + +<h2>Gaussian mixture models</h2> + +Richard W. DeVaul has made a detailed tutorial on how to fit mixtures +of Gaussians using BNT. Available +<a href="http://www.media.mit.edu/wearables/mithril/BNT/mixtureBNT.txt">here</a>. + + +<h2><a name="pca">PCA, ICA, and all that </h2> + +In Figure (a) below, we show how Factor Analysis can be thought of as a +graphical model. Here, X has an N(0,I) prior, and +Y|X=x ~ N(mu + Wx, Psi), +where Psi is diagonal and W is called the "factor loading matrix". +Since the noise on both X and Y is diagonal, the components of these +vectors are uncorrelated, and hence can be represented as individual +scalar nodes, as we show in (b). +(This is useful if parts of the observations on the Y vector are occasionally missing.) +We usually take k=|X| << |Y|=D, so the model tries to explain +many observations using a low-dimensional subspace. + + +<center> +<table> +<tr> +<td><img src="Figures/fa.gif"> +<td><img src="Figures/fa_scalar.gif"> +<td><img src="Figures/mfa.gif"> +<td><img src="Figures/ifa.gif"> +<tr> +<td align=center> (a) +<td align=center> (b) +<td align=center> (c) +<td align=center> (d) +</table> +</center> + +<p> +We can create this model in BNT as follows. +<pre> +ns = [k D]; +dag = zeros(2,2); +dag(1,2) = 1; +bnet = mk_bnet(dag, ns, 'discrete', []); +bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', zeros(k,1), 'cov', eye(k), ... + 'cov_type', 'diag', 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(D,1), 'cov', diag(Psi0), 'weights', W0, ... + 'cov_type', 'diag', 'clamp_mean', 1); +</pre> + +The root node is clamped to the N(0,I) distribution, so that we will +not update these parameters during learning. +The mean of the leaf node is clamped to 0, +since we assume the data has been centered (had its mean subtracted +off); this is just for simplicity. +Finally, the covariance of the leaf node is constrained to be +diagonal. W0 and Psi0 are the initial parameter guesses. + +<p> +We can fit this model (i.e., estimate its parameters in a maximum +likelihood (ML) sense) using EM, as we +explain <a href="#em">below</a>. +Not surprisingly, the ML estimates for mu and Psi turn out to be +identical to the +sample mean and variance, which can be computed directly as +<pre> +mu_ML = mean(data); +Psi_ML = diag(cov(data)); +</pre> +Note that W can only be identified up to a rotation matrix, because of +the spherical symmetry of the source. + +<p> +If we restrict Psi to be spherical, i.e., Psi = sigma*I, +there is a closed-form solution for W as well, +i.e., we do not need to use EM. +In particular, W contains the first |X| eigenvectors of the sample covariance +matrix, with scalings determined by the eigenvalues and sigma. +Classical PCA can be obtained by taking the sigma->0 limit. +For details, see + +<ul> +<li> <a href="ftp://hope.caltech.edu/pub/roweis/Empca/empca.ps"> +"EM algorithms for PCA and SPCA"</a>, Sam Roweis, NIPS 97. +(<a href="ftp://hope.caltech.edu/pub/roweis/Code/empca.tar.gz"> +Matlab software</a>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +By adding a hidden discrete variable, we can create mixtures of FA +models, as shown in (c). +Now we can explain the data using a set of subspaces. +We can create this model in BNT as follows. +<pre> +ns = [M k D]; +dag = zeros(3); +dag(1,3) = 1; +dag(2,3) = 1; +bnet = mk_bnet(dag, ns, 'discrete', 1); +bnet.CPD{1} = tabular_CPD(bnet, 1, Pi0); +bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(k, 1), 'cov', eye(k), 'cov_type', 'diag', ... + 'clamp_mean', 1, 'clamp_cov', 1); +bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', Mu0', 'cov', repmat(diag(Psi0), [1 1 M]), ... + 'weights', W0, 'cov_type', 'diag', 'tied_cov', 1); +</pre> +Notice how the covariance matrix for Y is the same for all values of +Q; that is, the noise level in each sub-space is assumed the same. +However, we allow the offset, mu, to vary. +For details, see +<ul> + +<LI> +<a HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/tr-96-1.ps.gz"> The EM +Algorithm for Mixtures of Factor Analyzers </A>, +Ghahramani, Z. and Hinton, G.E. (1996), +University of Toronto +Technical Report CRG-TR-96-1. +(<A HREF="ftp://ftp.cs.toronto.edu/pub/zoubin/mfa.tar.gz">Matlab software</A>) + +<p> +<li> +<a +href=http://neural-server.aston.ac.uk/cgi-bin/tr_avail.pl?trnumber=NCRG/97/003> +"Mixtures of probabilistic principal component analyzers"</a>, +Tipping and Bishop, Neural Computation 11(2):443--482, 1999. +</ul> + +<p> +I have included Zoubin's specialized MFA code (with his permission) +with the toolbox, so you can check that BNT gives the same results: +see 'BNT/examples/static/mfa1.m'. + +<p> +Independent Factor Analysis (IFA) generalizes FA by allowing a +non-Gaussian prior on each component of X. +(Note that we can approximate a non-Gaussian prior using a mixture of +Gaussians.) +This means that the likelihood function is no longer rotationally +invariant, so we can uniquely identify W and the hidden +sources X. +IFA also allows a non-diagonal Psi (i.e. correlations between the components of Y). +We recover classical Independent Components Analysis (ICA) +in the Psi -> 0 limit, and by assuming that |X|=|Y|, so that the +weight matrix W is square and invertible. +For details, see +<ul> +<li> +<a href="http://www.gatsby.ucl.ac.uk/~hagai/ifa.ps">Independent Factor +Analysis</a>, H. Attias, Neural Computation 11: 803--851, 1998. +</ul> + + + +<h2><a name="mixexp">Mixtures of experts</h2> + +As an example of the use of the softmax function, +we introduce the Mixture of Experts model. +<!-- +We also show +the Hierarchical Mixture of Experts model, where the hierarchy has two +levels. +(This is essentially a probabilistic decision tree of height two.) +--> +As before, +circles denote continuous-valued nodes, +squares denote discrete nodes, clear +means hidden, and shaded means observed. +<p> +<center> +<table> +<tr> +<td><img src="Figures/mixexp.gif"> +<!-- +<td><img src="Figures/hme.gif"> +--> +</table> +</center> +<p> +X is the observed +input, Y is the output, and +the Q nodes are hidden "gating" nodes, which select the appropriate +set of parameters for Y. During training, Y is assumed observed, +but for testing, the goal is to predict Y given X. +Note that this is a <em>conditional</em> density model, so we don't +associate any parameters with X. +Hence X's CPD will be a root CPD, which is a way of modelling +exogenous nodes. +If the output is a continuous-valued quantity, +we assume the "experts" are linear-regression units, +and set Y's CPD to linear-Gaussian. +If the output is discrete, we set Y's CPD to a softmax function. +The Q CPDs will always be softmax functions. + +<p> +As a concrete example, consider the mixture of experts model where X and Y are +scalars, and Q is binary. +This is just piecewise linear regression, where +we have two line segments, i.e., +<P> +<IMG ALIGN=BOTTOM SRC="Eqns/lin_reg_eqn.gif"> +<P> +We can create this model with random parameters as follows. +(This code is bundled in BNT/examples/static/mixexp2.m.) +<PRE> +X = 1; +Q = 2; +Y = 3; +dag = zeros(3,3); +dag(X,[Q Y]) = 1 +dag(Q,Y) = 1; +ns = [1 2 1]; % make X and Y scalars, and have 2 experts +onodes = [1 3]; +bnet = mk_bnet(dag, ns, 'discrete', 2, 'observed', onodes); + +rand('state', 0); +randn('state', 0); +bnet.CPD{1} = root_CPD(bnet, 1); +bnet.CPD{2} = softmax_CPD(bnet, 2); +bnet.CPD{3} = gaussian_CPD(bnet, 3); +</PRE> +Now let us fit this model using <a href="#em">EM</a>. +First we <a href="#load_data">load the data</a> (1000 training cases) and plot them. +<P> +<PRE> +data = load('/examples/static/Misc/mixexp_data.txt', '-ascii'); +plot(data(:,1), data(:,2), '.'); +</PRE> +<p> +<center> +<IMG SRC="Figures/mixexp_data.gif"> +</center> +<p> +This is what the model looks like before training. +(Thanks to Thomas Hofman for writing this plotting routine.) +<p> +<center> +<IMG SRC="Figures/mixexp_before.gif"> +</center> +<p> +Now let's train the model, and plot the final performance. +(We will discuss how to train models in more detail <a href="#param_learning">below</a>.) +<P> +<PRE> +ncases = size(data, 1); % each row of data is a training case +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); % each column of cases is a training case +engine = jtree_inf_engine(bnet); +max_iter = 20; +[bnet2, LLtrace] = learn_params_em(engine, cases, max_iter); +</PRE> +(We specify which nodes will be observed when we create the engine. +Hence BNT knows that the hidden nodes are all discrete. +For complex models, this can lead to a significant speedup.) +Below we show what the model looks like after 16 iterations of EM +(with 100 IRLS iterations per M step), when it converged +using the default convergence tolerance (that the +fractional change in the log-likelihood be less than 1e-3). +Before learning, the log-likelihood was +-322.927442; afterwards, it was -13.728778. +<p> +<center> +<IMG SRC="Figures/mixexp_after.gif"> +</center> +(See BNT/examples/static/mixexp2.m for details of the code.) + + + +<h2><a name="hme">Hierarchical mixtures of experts</h2> + +A hierarchical mixture of experts (HME) extends the mixture of experts +model by having more than one hidden node. A two-level example is shown below, along +with its more traditional representation as a neural network. +This is like a (balanced) probabilistic decision tree of height 2. +<p> +<center> +<IMG SRC="Figures/HMEforMatlab.jpg"> +</center> +<p> +<a href="mailto:pbrutti@stat.cmu.edu">Pierpaolo Brutti</a> +has written an extensive set of routines for HMEs, +which are bundled with BNT: see the examples/static/HME directory. +These routines allow you to choose the number of hidden (gating) +layers, and the form of the experts (softmax or MLP). +See the file hmemenu, which provides a demo. +For example, the figure below shows the decision boundaries learned +for a ternary classification problem, using a 2 level HME with softmax +gates and softmax experts; the training set is on the left, the +testing set on the right. +<p> +<center> +<!--<IMG SRC="Figures/hme_dec_boundary.gif">--> +<IMG SRC="Figures/hme_dec_boundary.png"> +</center> +<p> + + +<p> +For more details, see the following: +<ul> + +<li> <a href="http://www.cs.berkeley.edu/~jordan/papers/hierarchies.ps.Z"> +Hierarchical mixtures of experts and the EM algorithm</a> +M. I. Jordan and R. A. Jacobs. Neural Computation, 6, 181-214, 1994. + +<li> <a href = +"http://www.cs.berkeley.edu/~dmartin/software">David Martin's +matlab code for HME</a> + +<li> <a +href="http://www.cs.berkeley.edu/~jordan/papers/uai.ps.Z">Why the +logistic function? A tutorial discussion on +probabilities and neural networks.</a> M. I. Jordan. MIT Computational +Cognitive Science Report 9503, August 1995. + +<li> "Generalized Linear Models", McCullagh and Nelder, Chapman and +Halll, 1983. + +<li> +"Improved learning algorithms for mixtures of experts in multiclass +classification". +K. Chen, L. Xu, H. Chi. +Neural Networks (1999) 12: 1229-1252. + +<li> <a href="http://www.oigeeza.com/steve/"> +Classification Using Hierarchical Mixtures of Experts</a> +S.R. Waterhouse and A.J. Robinson. +In Proc. IEEE Workshop on Neural Network for Signal Processing IV (1994), pp. 177-186 + +<li> <a href="http://www.idiap.ch/~perry/"> +Localized mixtures of experts</a>, +P. Moerland, 1998. + +<li> "Nonlinear gated experts for time series", +A.S. Weigend and M. Mangeas, 1995. + +</ul> + + +<h2><a name="qmr">QMR</h2> + +Bayes nets originally arose out of an attempt to add probabilities to +expert systems, and this is still the most common use for BNs. +A famous example is +QMR-DT, a decision-theoretic reformulation of the Quick Medical +Reference (QMR) model. +<p> +<center> +<IMG ALIGN=BOTTOM SRC="Figures/qmr.gif"> +</center> +Here, the top layer represents hidden disease nodes, and the bottom +layer represents observed symptom nodes. +The goal is to infer the posterior probability of each disease given +all the symptoms (which can be present, absent or unknown). +Each node in the top layer has a Bernoulli prior (with a low prior +probability that the disease is present). +Since each node in the bottom layer has a high fan-in, we use a +noisy-OR parameterization; each disease has an independent chance of +causing each symptom. +The real QMR-DT model is copyright, but +we can create a random QMR-like model as follows. +<pre> +function bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% MK_QMR_BNET Make a QMR model +% bnet = mk_qmr_bnet(G, inhibit, leak, prior) +% +% G(i,j) = 1 iff there is an arc from disease i to finding j +% inhibit(i,j) = inhibition probability on i->j arc +% leak(j) = inhibition prob. on leak->j arc +% prior(i) = prob. disease i is on + +[Ndiseases Nfindings] = size(inhibit); +N = Ndiseases + Nfindings; +finding_node = Ndiseases+1:N; +ns = 2*ones(1,N); +dag = zeros(N,N); +dag(1:Ndiseases, finding_node) = G; +bnet = mk_bnet(dag, ns, 'observed', finding_node); + +for d=1:Ndiseases + CPT = [1-prior(d) prior(d)]; + bnet.CPD{d} = tabular_CPD(bnet, d, CPT'); +end + +for i=1:Nfindings + fnode = finding_node(i); + ps = parents(G, i); + bnet.CPD{fnode} = noisyor_CPD(bnet, fnode, leak(i), inhibit(ps, i)); +end +</pre> +In the file BNT/examples/static/qmr1, we create a random bipartite +graph G, with 5 diseases and 10 findings, and random parameters. +(In general, to create a random dag, use 'mk_random_dag'.) +We can visualize the resulting graph structure using +the methods discussed <a href="#graphdraw">below</a>, with the +following results: +<p> +<img src="Figures/qmr.rnd.jpg"> + +<p> +Now let us put some random evidence on all the leaves except the very +first and very last, and compute the disease posteriors. +<pre> +pos = 2:floor(Nfindings/2); +neg = (pos(end)+1):(Nfindings-1); +onodes = myunion(pos, neg); +evidence = cell(1, N); +evidence(findings(pos)) = num2cell(repmat(2, 1, length(pos))); +evidence(findings(neg)) = num2cell(repmat(1, 1, length(neg))); + +engine = jtree_inf_engine(bnet); +[engine, ll] = enter_evidence(engine, evidence); +post = zeros(1, Ndiseases); +for i=diseases(:)' + m = marginal_nodes(engine, i); + post(i) = m.T(2); +end +</pre> +Junction tree can be quite slow on large QMR models. +Fortunately, it is possible to exploit properties of the noisy-OR +function to speed up exact inference using an algorithm called +<a href="#quickscore">quickscore</a>, discussed below. + + + + + +<h2><a name="cg_model">Conditional Gaussian models</h2> + +A conditional Gaussian model is one in which, conditioned on all the discrete +nodes, the distribution over the remaining (continuous) nodes is +multivariate Gaussian. This means we can have arcs from discrete (D) +to continuous (C) nodes, but not vice versa. +(We <em>are</em> allowed C->D arcs if the continuous nodes are observed, +as in the <a href="#mixexp">mixture of experts</a> model, +since this distribution can be represented with a discrete potential.) +<p> +We now give an example of a CG model, from +the paper "Propagation of Probabilities, Means amd +Variances in Mixed Graphical Association Models", Steffen Lauritzen, +JASA 87(420):1098--1108, 1992 (reprinted in the book "Probabilistic Networks and Expert +Systems", R. G. Cowell, A. P. Dawid, S. L. Lauritzen and +D. J. Spiegelhalter, Springer, 1999.) + +<h3>Specifying the graph</h3> + +Consider the model of waste emissions from an incinerator plant shown below. +We follow the standard convention that shaded nodes are observed, +clear nodes are hidden. +We also use the non-standard convention that +square nodes are discrete (tabular) and round nodes are +Gaussian. + +<p> +<center> +<IMG SRC="Figures/cg1.gif"> +</center> +<p> + +We can create this model as follows. +<pre> +F = 1; W = 2; E = 3; B = 4; C = 5; D = 6; Min = 7; Mout = 8; L = 9; +n = 9; + +dag = zeros(n); +dag(F,E)=1; +dag(W,[E Min D]) = 1; +dag(E,D)=1; +dag(B,[C D])=1; +dag(D,[L Mout])=1; +dag(Min,Mout)=1; + +% node sizes - all cts nodes are scalar, all discrete nodes are binary +ns = ones(1, n); +dnodes = [F W B]; +cnodes = mysetdiff(1:n, dnodes); +ns(dnodes) = 2; + +bnet = mk_bnet(dag, ns, 'discrete', dnodes); +</pre> +'dnodes' is a list of the discrete nodes; 'cnodes' is the continuous +nodes. 'mysetdiff' is a faster version of the built-in 'setdiff'. +<p> + + +<h3>Specifying the parameters</h3> + +The parameters of the discrete nodes can be specified as follows. +<pre> +bnet.CPD{B} = tabular_CPD(bnet, B, 'CPT', [0.85 0.15]); % 1=stable, 2=unstable +bnet.CPD{F} = tabular_CPD(bnet, F, 'CPT', [0.95 0.05]); % 1=intact, 2=defect +bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [2/7 5/7]); % 1=industrial, 2=household +</pre> + +<p> +The parameters of the continuous nodes can be specified as follows. +<pre> +bnet.CPD{E} = gaussian_CPD(bnet, E, 'mean', [-3.9 -0.4 -3.2 -0.5], ... + 'cov', [0.00002 0.0001 0.00002 0.0001]); +bnet.CPD{D} = gaussian_CPD(bnet, D, 'mean', [6.5 6.0 7.5 7.0], ... + 'cov', [0.03 0.04 0.1 0.1], 'weights', [1 1 1 1]); +bnet.CPD{C} = gaussian_CPD(bnet, C, 'mean', [-2 -1], 'cov', [0.1 0.3]); +bnet.CPD{L} = gaussian_CPD(bnet, L, 'mean', 3, 'cov', 0.25, 'weights', -0.5); +bnet.CPD{Min} = gaussian_CPD(bnet, Min, 'mean', [0.5 -0.5], 'cov', [0.01 0.005]); +bnet.CPD{Mout} = gaussian_CPD(bnet, Mout, 'mean', 0, 'cov', 0.002, 'weights', [1 1]); +</pre> + + +<h3><a name="cg_infer">Inference</h3> + +<!--Let us perform inference in the <a href="#cg_model">waste incinerator example</a>.--> +First we compute the unconditional marginals. +<pre> +engine = jtree_inf_engine(bnet); +evidence = cell(1,n); +[engine, ll] = enter_evidence(engine, evidence); +marg = marginal_nodes(engine, E); +</pre> +<!--(Of course, we could use <tt>cond_gauss_inf_engine</tt> instead of jtree.)--> +'marg' is a structure that contains the fields 'mu' and 'Sigma', which +contain the mean and (co)variance of the marginal on E. +In this case, they are both scalars. +Let us check they match the published figures (to 2 decimal places). +<!--(We can't expect +more precision than this in general because I have implemented the algorithm of +Lauritzen (1992), which can be numerically unstable.)--> +<pre> +tol = 1e-2; +assert(approxeq(marg.mu, -3.25, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.709, tol)); +</pre> +We can compute the other posteriors similarly. +Now let us add some evidence. +<pre> +evidence = cell(1,n); +evidence{W} = 1; % industrial +evidence{L} = 1.1; +evidence{C} = -0.9; +[engine, ll] = enter_evidence(engine, evidence); +</pre> +Now we find +<pre> +marg = marginal_nodes(engine, E); +assert(approxeq(marg.mu, -3.8983, tol)); +assert(approxeq(sqrt(marg.Sigma), 0.0763, tol)); +</pre> + + +We can also compute the joint probability on a set of nodes. +For example, P(D, Mout | evidence) is a 2D Gaussian: +<pre> +marg = marginal_nodes(engine, [D Mout]) +marg = + domain: [6 8] + mu: [2x1 double] + Sigma: [2x2 double] + T: 1.0000 +</pre> +The mean is +<pre> +marg.mu +ans = + 3.6077 + 4.1077 +</pre> +and the covariance matrix is +<pre> +marg.Sigma +ans = + 0.1062 0.1062 + 0.1062 0.1182 +</pre> +It is easy to visualize this posterior using standard Matlab plotting +functions, e.g., +<pre> +gaussplot2d(marg.mu, marg.Sigma); +</pre> +produces the following picture. + +<p> +<center> +<IMG SRC="Figures/gaussplot.png"> +</center> +<p> + + +The T field indicates that the mixing weight of this Gaussian +component is 1.0. +If the joint contains discrete and continuous variables, the result +will be a mixture of Gaussians, e.g., +<pre> +marg = marginal_nodes(engine, [F E]) + domain: [1 3] + mu: [-3.9000 -0.4003] + Sigma: [1x1x2 double] + T: [0.9995 4.7373e-04] +</pre> +The interpretation is +Sigma(i,j,k) = Cov[ E(i) E(j) | F=k ]. +In this case, E is a scalar, so i=j=1; k specifies the mixture component. +<p> +We saw in the sprinkler network that BNT sets the effective size of +observed discrete nodes to 1, since they only have one legal value. +For continuous nodes, BNT sets their length to 0, +since they have been reduced to a point. +For example, +<pre> +marg = marginal_nodes(engine, [B C]) + domain: [4 5] + mu: [] + Sigma: [] + T: [0.0123 0.9877] +</pre> +It is simple to post-process the output of marginal_nodes. +For example, the file BNT/examples/static/cg1 sets the mu term of +observed nodes to their observed value, and the Sigma term to 0 (since +observed nodes have no variance). + +<p> +Note that the implemented version of the junction tree is numerically +unstable when using CG potentials +(which is why, in the example above, we only required our answers to agree with +the published ones to 2dp.) +This is why you might want to use <tt>stab_cond_gauss_inf_engine</tt>, +implemented by Shan Huang. This is described in + +<ul> +<li> "Stable Local Computation with Conditional Gaussian Distributions", +S. Lauritzen and F. Jensen, Tech Report R-99-2014, +Dept. Math. Sciences, Allborg Univ., 1999. +</ul> + +However, even the numerically stable version +can be computationally intractable if there are many hidden discrete +nodes, because the number of mixture components grows exponentially e.g., in a +<a href="usage_dbn.html#lds">switching linear dynamical system</a>. +In general, one must resort to approximate inference techniques: see +the discussion on <a href="#engines">inference engines</a> below. + + +<h2><a name="hybrid">Other hybrid models</h2> + +When we have C->D arcs, where C is hidden, we need to use +approximate inference. +One approach (not implemented in BNT) is described in +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/hybrid_uai99.ps.gz">A +Variational Approximation for Bayesian Networks with +Discrete and Continuous Latent Variables</a>, +K. Murphy, UAI 99. +</ul> +Of course, one can always use <a href="#sampling">sampling</a> methods +for approximate inference in such models. + + + +<h1><a name="param_learning">Parameter Learning</h1> + +The parameter estimation routines in BNT can be classified into 4 +types, depending on whether the goal is to compute +a full (Bayesian) posterior over the parameters or just a point +estimate (e.g., Maximum Likelihood or Maximum A Posteriori), +and whether all the variables are fully observed or there is missing +data/ hidden variables (partial observability). +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_params</tt></td> + <td><tt>learn_params_em</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>bayes_update_params</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="load_data">Loading data from a file</h2> + +To load numeric data from an ASCII text file called 'dat.txt', where each row is a +case and columns are separated by white-space, such as +<pre> +011979 1626.5 0.0 +021979 1367.0 0.0 +... +</pre> +you can use +<pre> +data = load('dat.txt'); +</pre> +or +<pre> +load dat.txt -ascii +</pre> +In the latter case, the data is stored in a variable called 'dat' (the +filename minus the extension). +Alternatively, suppose the data is stored in a .csv file (has commas +separating the columns, and contains a header line), such as +<pre> +header info goes here +ORD,011979,1626.5,0.0 +DSM,021979,1367.0,0.0 +... +</pre> +You can load this using +<pre> +[a,b,c,d] = textread('dat.txt', '%s %d %f %f', 'delimiter', ',', 'headerlines', 1); +</pre> +If your file is not in either of these formats, you can either use Perl to convert +it to this format, or use the Matlab scanf command. +Type +<tt> +help iofun +</tt> +for more information on Matlab's file functions. +<!-- +<p> +To load data directly from Excel, +you should buy the +<a href="http://www.mathworks.com/products/excellink/">Excel Link</a>. +To load data directly from a relational database, +you should buy the +<a href="http://www.mathworks.com/products/database">Database +toolbox</a>. +--> +<p> +BNT learning routines require data to be stored in a cell array. +data{i,m} is the value of node i in case (example) m, i.e., each +<em>column</em> is a case. +If node i is not observed in case m (missing value), set +data{i,m} = []. +(Not all the learning routines can cope with such missing values, however.) +In the special case that all the nodes are observed and are +scalar-valued (as opposed to vector-valued), the data can be +stored in a matrix (as opposed to a cell-array). +<p> +Suppose, as in the <a href="#mixexp">mixture of experts example</a>, +that we have 3 nodes in the graph: X(1) is the observed input, X(3) is +the observed output, and X(2) is a hidden (gating) node. We can +create the dataset as follows. +<pre> +data = load('dat.txt'); +ncases = size(data, 1); +cases = cell(3, ncases); +cases([1 3], :) = num2cell(data'); +</pre> +Notice how we transposed the data, to convert rows into columns. +Also, cases{2,m} = [] for all m, since X(2) is always hidden. + + +<h2><a name="mle_complete">Maximum likelihood parameter estimation from complete data</h2> + +As an example, let's generate some data from the sprinkler network, randomize the parameters, +and then try to recover the original model. +First we create some training data using forwards sampling. +<pre> +samples = cell(N, nsamples); +for i=1:nsamples + samples(:,i) = sample_bnet(bnet); +end +</pre> +samples{j,i} contains the value of the j'th node in case i. +sample_bnet returns a cell array because, in general, each node might +be a vector of different length. +In this case, all nodes are discrete (and hence scalars), so we +could have used a regular array instead (which can be quicker): +<pre> +data = cell2num(samples); +</pre +So now data(j,i) = samples{j,i}. +<p> +Now we create a network with random parameters. +(The initial values of bnet2 don't matter in this case, since we can find the +globally optimal MLE independent of where we start.) +<pre> +% Make a tabula rasa +bnet2 = mk_bnet(dag, node_sizes); +seed = 0; +rand('state', seed); +bnet2.CPD{C} = tabular_CPD(bnet2, C); +bnet2.CPD{R} = tabular_CPD(bnet2, R); +bnet2.CPD{S} = tabular_CPD(bnet2, S); +bnet2.CPD{W} = tabular_CPD(bnet2, W); +</pre> +Finally, we find the maximum likelihood estimates of the parameters. +<pre> +bnet3 = learn_params(bnet2, samples); +</pre> +To view the learned parameters, we use a little Matlab hackery. +<pre> +CPT3 = cell(1,N); +for i=1:N + s=struct(bnet3.CPD{i}); % violate object privacy + CPT3{i}=s.CPT; +end +</pre> +Here are the parameters learned for node 4. +<pre> +dispcpt(CPT3{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.2000 0.8000 +1 2 : 0.2273 0.7727 +2 2 : 0.0000 1.0000 +</pre> +So we see that the learned parameters are fairly close to the "true" +ones, which we display below. +<pre> +dispcpt(CPT{4}) +1 1 : 1.0000 0.0000 +2 1 : 0.1000 0.9000 +1 2 : 0.1000 0.9000 +2 2 : 0.0100 0.9900 +</pre> +We can get better results by using a larger training set, or using +informative priors (see <a href="#prior">below</a>). + + + +<h2><a name="prior">Parameter priors</h2> + +Currently, only tabular CPDs can have priors on their parameters. +The conjugate prior for a multinomial is the Dirichlet. +(For binary random variables, the multinomial is the same as the +Bernoulli, and the Dirichlet is the same as the Beta.) +<p> +The Dirichlet has a simple interpretation in terms of pseudo counts. +If we let N_ijk = the num. times X_i=k and Pa_i=j occurs in the +training set, where Pa_i are the parents of X_i, +then the maximum likelihood (ML) estimate is +T_ijk = N_ijk / N_ij (where N_ij = sum_k' N_ijk'), which will be 0 if N_ijk=0. +To prevent us from declaring that (X_i=k, Pa_i=j) is impossible just because this +event was not seen in the training set, +we can pretend we saw value k of X_i, for each value j of Pa_i some number (alpha_ijk) +of times in the past. +The MAP (maximum a posterior) estimate is then +<pre> +T_ijk = (N_ijk + alpha_ijk) / (N_ij + alpha_ij) +</pre> +and is never 0 if all alpha_ijk > 0. +For example, consider the network A->B, where A is binary and B has 3 +values. +A uniform prior for B has the form +<pre> + B=1 B=2 B=3 +A=1 1 1 1 +A=2 1 1 1 +</pre> +which can be created using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'unif'); +</pre> +This prior does not satisfy the likelihood equivalence principle, +which says that <a href="#markov_equiv">Markov equivalent</a> models +should have the same marginal likelihood. +A prior that does satisfy this principle is shown below. +Heckerman (1995) calls this the +BDeu prior (likelihood equivalent uniform Bayesian Dirichlet). +<pre> + B=1 B=2 B=3 +A=1 1/6 1/6 1/6 +A=2 1/6 1/6 1/6 +</pre> +where we put N/(q*r) in each bin; N is the equivalent sample size, +r=|A|, q = |B|. +This can be created as follows +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', 'BDeu'); +</pre> +Here, 1 is the equivalent sample size, and is the strength of the +prior. +You can change this using +<pre> +tabular_CPD(bnet, i, 'prior_type', 'dirichlet', 'dirichlet_type', ... + 'BDeu', 'dirichlet_weight', 10); +</pre> +<!--where counts is an array of pseudo-counts of the same size as the +CPT.--> +<!-- +<p> +When you specify a prior, you should set row i of the CPT to the +normalized version of row i of the pseudo-count matrix, i.e., to the +expected values of the parameters. This will ensure that computing the +marginal likelihood sequentially (see <a +href="#bayes_learn">below</a>) and in batch form gives the same +results. +To do this, proceed as follows. +<pre> +tabular_CPD(bnet, i, 'prior', counts, 'CPT', mk_stochastic(counts)); +</pre> +For a non-informative prior, you can just write +<pre> +tabular_CPD(bnet, i, 'prior', 'unif', 'CPT', 'unif'); +</pre> +--> + + +<h2><a name="bayes_learn">(Sequential) Bayesian parameter updating from complete data</h2> + +If we use conjugate priors and have fully observed data, we can +compute the posterior over the parameters in batch form as follows. +<pre> +cases = sample_bnet(bnet, nsamples); +bnet = bayes_update_params(bnet, cases); +LL = log_marg_lik_complete(bnet, cases); +</pre> +bnet.CPD{i}.prior contains the new Dirichlet pseudocounts, +and bnet.CPD{i}.CPT is set to the mean of the posterior (the +normalized counts). +(Hence if the initial pseudo counts are 0, +<tt>bayes_update_params</tt> and <tt>learn_params</tt> will give the +same result.) + + + + +<p> +We can compute the same result sequentially (on-line) as follows. +<pre> +LL = 0; +for m=1:nsamples + LL = LL + log_marg_lik_complete(bnet, cases(:,m)); + bnet = bayes_update_params(bnet, cases(:,m)); +end +</pre> + +The file <tt>BNT/examples/static/StructLearn/model_select1</tt> has an example of +sequential model selection which uses the same idea. +We generate data from the model A->B +and compute the posterior prob of all 3 dags on 2 nodes: + (1) A B, (2) A <- B , (3) A -> B +Models 2 and 3 are <a href="#markov_equiv">Markov equivalent</a>, and therefore indistinguishable from +observational data alone, so we expect their posteriors to be the same +(assuming a prior which satisfies likelihood equivalence). +If we use random parameters, the "true" model only gets a higher posterior after 2000 trials! +However, if we make B a noisy NOT gate, the true model "wins" after 12 +trials, as shown below (red = model 1, blue/green (superimposed) +represents models 2/3). +<p> +<img src="Figures/model_select.png"> +<p> +The use of marginal likelihood for model selection is discussed in +greater detail in the +section on <a href="structure_learning">structure learning</a>. + + + + +<h2><a name="em">Maximum likelihood parameter estimation with missing values</h2> + +Now we consider learning when some values are not observed. +Let us randomly hide half the values generated from the water +sprinkler example. +<pre> +samples2 = samples; +hide = rand(N, nsamples) > 0.5; +[I,J]=find(hide); +for k=1:length(I) + samples2{I(k), J(k)} = []; +end +</pre> +samples2{i,l} is the value of node i in training case l, or [] if unobserved. +<p> +Now we will compute the MLEs using the EM algorithm. +We need to use an inference algorithm to compute the expected +sufficient statistics in the E step; the M (maximization) step is as +above. +<pre> +engine2 = jtree_inf_engine(bnet2); +max_iter = 10; +[bnet4, LLtrace] = learn_params_em(engine2, samples2, max_iter); +</pre> +LLtrace(i) is the log-likelihood at iteration i. We can plot this as +follows: +<pre> +plot(LLtrace, 'x-') +</pre> +Let's display the results after 10 iterations of EM. +<pre> +celldisp(CPT4) +CPT4{1} = + 0.6616 + 0.3384 +CPT4{2} = + 0.6510 0.3490 + 0.8751 0.1249 +CPT4{3} = + 0.8366 0.1634 + 0.0197 0.9803 +CPT4{4} = +(:,:,1) = + 0.8276 0.0546 + 0.5452 0.1658 +(:,:,2) = + 0.1724 0.9454 + 0.4548 0.8342 +</pre> +We can get improved performance by using one or more of the following +methods: +<ul> +<li> Increasing the size of the training set. +<li> Decreasing the amount of hidden data. +<li> Running EM for longer. +<li> Using informative priors. +<li> Initialising EM from multiple starting points. +</ul> + +Click <a href="#gaussian">here</a> for a discussion of learning +Gaussians, which can cause numerical problems. +<p> +For a more complete example of learning with EM, +see the script BNT/examples/static/learn1.m. + +<h2><a name="tying">Parameter tying</h2> + +In networks with repeated structure (e.g., chains and grids), it is +common to assume that the parameters are the same at every node. This +is called parameter tying, and reduces the amount of data needed for +learning. +<p> +When we have tied parameters, there is no longer a one-to-one +correspondence between nodes and CPDs. +Rather, each CPD species the parameters for a whole equivalence class +of nodes. +It is easiest to see this by example. +Consider the following <a href="usage_dbn.html#hmm">hidden Markov +model (HMM)</a> +<p> +<img src="Figures/hmm3.gif"> +<p> +<!-- +We can create this graph structure, assuming we have T time-slices, +as follows. +(We number the nodes as shown in the figure, but we could equally well +number the hidden nodes 1:T, and the observed nodes T+1:2T.) +<pre> +N = 2*T; +dag = zeros(N); +hnodes = 1:2:2*T; +for i=1:T-1 + dag(hnodes(i), hnodes(i+1))=1; +end +onodes = 2:2:2*T; +for i=1:T + dag(hnodes(i), onodes(i)) = 1; +end +</pre> +<p> +The hidden nodes are always discrete, and have Q possible values each, +but the observed nodes can be discrete or continuous, and have O possible values/length. +<pre> +if cts_obs + dnodes = hnodes; +else + dnodes = 1:N; +end +ns = ones(1,N); +ns(hnodes) = Q; +ns(onodes) = O; +</pre> +--> +When HMMs are used for semi-infinite processes like speech recognition, +we assume the transition matrix +P(H(t+1)|H(t)) is the same for all t; this is called a time-invariant +or homogenous Markov chain. +Hence hidden nodes 2, 3, ..., T +are all in the same equivalence class, say class Hclass. +Similarly, the observation matrix P(O(t)|H(t)) is assumed to be the +same for all t, so the observed nodes are all in the same equivalence +class, say class Oclass. +Finally, the prior term P(H(1)) is in a class all by itself, say class +H1class. +This is illustrated below, where we explicitly represent the +parameters as random variables (dotted nodes). +<p> +<img src="Figures/hmm4_params.gif"> +<p> +In BNT, we cannot represent parameters as random variables (nodes). +Instead, we "hide" the +parameters inside one CPD for each equivalence class, +and then specify that the other CPDs should share these parameters, as +follows. +<pre> +hnodes = 1:2:2*T; +onodes = 2:2:2*T; +H1class = 1; Hclass = 2; Oclass = 3; +eclass = ones(1,N); +eclass(hnodes(2:end)) = Hclass; +eclass(hnodes(1)) = H1class; +eclass(onodes) = Oclass; +% create dag and ns in the usual way +bnet = mk_bnet(dag, ns, 'discrete', dnodes, 'equiv_class', eclass); +</pre> +Finally, we define the parameters for each equivalence class: +<pre> +bnet.CPD{H1class} = tabular_CPD(bnet, hnodes(1)); % prior +bnet.CPD{Hclass} = tabular_CPD(bnet, hnodes(2)); % transition matrix +if cts_obs + bnet.CPD{Oclass} = gaussian_CPD(bnet, onodes(1)); +else + bnet.CPD{Oclass} = tabular_CPD(bnet, onodes(1)); +end +</pre> +In general, if bnet.CPD{e} = xxx_CPD(bnet, j), then j should be a +member of e's equivalence class; that is, it is not always the case +that e == j. You can use bnet.rep_of_eclass(e) to return the +representative of equivalence class e. +BNT will look up the parents of j to determine the size +of the CPT to use. It assumes that this is the same for all members of +the equivalence class. +Click <a href="param_tieing.html">here</a> for +a more complex example of parameter tying. +<p> +Note: +Normally one would define an HMM as a +<a href = "usage_dbn.html">Dynamic Bayes Net</a> +(see the function BNT/examples/dynamic/mk_chmm.m). +However, one can define an HMM as a static BN using the function +BNT/examples/static/Models/mk_hmm_bnet.m. + + + +<h1><a name="structure_learning">Structure learning</h1> + +Update (9/29/03): +Phillipe LeRay is developing some additional structure learning code +on top of BNT. Click +<a href="http://banquiseasi.insa-rouen.fr/projects/bnt-slp/"> +here</a> +for details. + +<p> + +There are two very different approaches to structure learning: +constraint-based and search-and-score. +In the <a href="#constraint">constraint-based approach</a>, +we start with a fully connected graph, and remove edges if certain +conditional independencies are measured in the data. +This has the disadvantage that repeated independence tests lose +statistical power. +<p> +In the more popular search-and-score approach, +we perform a search through the space of possible DAGs, and either +return the best one found (a point estimate), or return a sample of the +models found (an approximation to the Bayesian posterior). +<p> +The number of DAGs as a function of the number of +nodes, G(n), is super-exponential in n, +and is given by the following recurrence +<!--(where R(i)=G(n)):--> +<p> +<center> +<IMG SRC="numDAGsEqn2.png"> +</center> +<p> +The first few values +are shown below. + +<table> +<tr> <th>n</th> <th align=left>G(n)</th> </tr> +<tr> <td>1</td> <td>1</td> </tr> +<tr> <td>2</td> <td>3</td> </tr> +<tr> <td>3</td> <td>25</td> </tr> +<tr> <td>4</td> <td>543</td> </tr> +<tr> <td>5</td> <td>29,281</td> </tr> +<tr> <td>6</td> <td>3,781,503</td> </tr> +<tr> <td>7</td> <td>1.1 x 10^9</td> </tr> +<tr> <td>8</td> <td>7.8 x 10^11</td> </tr> +<tr> <td>9</td> <td>1.2 x 10^15</td> </tr> +<tr> <td>10</td> <td>4.2 x 10^18</td> </tr> +</table> + +Since the number of DAGs is super-exponential in the number of nodes, +we cannot exhaustively search the space, so we either use a local +search algorithm (e.g., greedy hill climbining, perhaps with multiple +restarts) or a global search algorithm (e.g., Markov Chain Monte +Carlo). +<p> +If we know a total ordering on the nodes, +finding the best structure amounts to picking the best set of parents +for each node independently. +This is what the K2 algorithm does. +If the ordering is unknown, we can search over orderings, +which is more efficient than searching over DAGs (Koller and Friedman, 2000). +<p> +In addition to the search procedure, we must specify the scoring +function. There are two popular choices. The Bayesian score integrates +out the parameters, i.e., it is the marginal likelihood of the model. +The BIC (Bayesian Information Criterion) is defined as +log P(D|theta_hat) - 0.5*d*log(N), where D is the data, theta_hat is +the ML estimate of the parameters, d is the number of parameters, and +N is the number of data cases. +The BIC method has the advantage of not requiring a prior. +<p> +BIC can be derived as a large sample +approximation to the marginal likelihood. +(It is also equal to the Minimum Description Length of a model.) +However, in practice, the sample size does not need to be very large +for the approximation to be good. +For example, in the figure below, we plot the ratio between the log marginal likelihood +and the BIC score against data-set size; we see that the ratio rapidly +approaches 1, especially for non-informative priors. +(This plot was generated by the file BNT/examples/static/bic1.m. It +uses the water sprinkler BN with BDeu Dirichlet priors with different +equivalent sample sizes.) + +<p> +<center> +<IMG SRC="Figures/bic.png"> +</center> +<p> + +<p> +As with parameter learning, handling missing data/ hidden variables is +much harder than the fully observed case. +The structure learning routines in BNT can therefore be classified into 4 +types, analogously to the parameter learning case. +<p> + +<TABLE BORDER> +<tr> + <TH></TH> + <th>Full obs</th> + <th>Partial obs</th> +</tr> +<tr> + <th>Point</th> + <td><tt>learn_struct_K2</tt> <br> +<!-- <tt>learn_struct_hill_climb</tt></td> --> + <td><tt>not yet supported</tt></td> +</tr> +<tr> + <th>Bayes</th> + <td><tt>learn_struct_mcmc</tt></td> + <td>not yet supported</td> +</tr> +</table> + + +<h2><a name="markov_equiv">Markov equivalence</h2> + +If two DAGs encode the same conditional independencies, they are +called Markov equivalent. The set of all DAGs can be paritioned into +Markov equivalence classes. Graphs within the same class can +have +the direction of some of their arcs reversed without changing any of +the CI relationships. +Each class can be represented by a PDAG +(partially directed acyclic graph) called an essential graph or +pattern. This specifies which edges must be oriented in a certain +direction, and which may be reversed. + +<p> +When learning graph structure from observational data, +the best one can hope to do is to identify the model up to Markov +equivalence. To distinguish amongst graphs within the same equivalence +class, one needs interventional data: see the discussion on <a +href="#active">active learning</a> below. + + + +<h2><a name="enumerate">Exhaustive search</h2> + +The brute-force approach to structure learning is to enumerate all +possible DAGs, and score each one. This provides a "gold standard" +with which to compare other algorithms. We can do this as follows. +<pre> +dags = mk_all_dags(N); +score = score_dags(data, ns, dags); +</pre> +where data(i,m) is the value of node i in case m, +and ns(i) is the size of node i. +If the DAGs have a lot of families in common, we can cache the sufficient statistics, +making this potentially more efficient than scoring the DAGs one at a time. +(Caching is not currently implemented, however.) +<p> +By default, we use the Bayesian scoring metric, and assume CPDs are +represented by tables with BDeu(1) priors. +We can override these defaults as follows. +If we want to use uniform priors, we can say +<pre> +params = cell(1,N); +for i=1:N + params{i} = {'prior', 'unif'}; +end +score = score_dags(data, ns, dags, 'params', params); +</pre> +params{i} is a cell-array, containing optional arguments that are +passed to the constructor for CPD i. +<p> +Now suppose we want to use different node types, e.g., +Suppose nodes 1 and 2 are Gaussian, and nodes 3 and 4 softmax (both +these CPDs can support discrete and continuous parents, which is +necessary since all other nodes will be considered as parents). +The Bayesian scoring metric currently only works for tabular CPDs, so +we will use BIC: +<pre> +score = score_dags(data, ns, dags, 'discrete', [3 4], 'params', [], + 'type', {'gaussian', 'gaussian', 'softmax', softmax'}, 'scoring_fn', 'bic') +</pre> +In practice, one can't enumerate all possible DAGs for N > 5, +but one can evaluate any reasonably-sized set of hypotheses in this +way (e.g., nearest neighbors of your current best guess). +Think of this as "computer assisted model refinement" as opposed to de +novo learning. + + +<h2><a name="K2">K2</h2> + +The K2 algorithm (Cooper and Herskovits, 1992) is a greedy search algorithm that works as follows. +Initially each node has no parents. It then adds incrementally that parent whose addition most +increases the score of the resulting structure. When the addition of no single +parent can increase the score, it stops adding parents to the node. +Since we are using a fixed ordering, we do not need to check for +cycles, and can choose the parents for each node independently. +<p> +The original paper used the Bayesian scoring +metric with tabular CPDs and Dirichlet priors. +BNT generalizes this to allow any kind of CPD, and either the Bayesian +scoring metric or BIC, as in the example <a href="#enumerate">above</a>. +In addition, you can specify +an optional upper bound on the number of parents for each node. +The file BNT/examples/static/k2demo1.m gives an example of how to use K2. +We use the water sprinkler network and sample 100 cases from it as before. +Then we see how much data it takes to recover the generating structure: +<pre> +order = [C S R W]; +max_fan_in = 2; +sz = 5:5:100; +for i=1:length(sz) + dag2 = learn_struct_K2(data(:,1:sz(i)), node_sizes, order, 'max_fan_in', max_fan_in); + correct(i) = isequal(dag, dag2); +end +</pre> +Here are the results. +<pre> +correct = + Columns 1 through 12 + 0 0 0 0 0 0 0 1 0 1 1 1 + Columns 13 through 20 + 1 1 1 1 1 1 1 1 +</pre> +So we see it takes about sz(10)=50 cases. (BIC behaves similarly, +showing that the prior doesn't matter too much.) +In general, we cannot hope to recover the "true" generating structure, +only one that is in its <a href="#markov_equiv">Markov equivalence +class</a>. + + +<h2><a name="hill_climb">Hill-climbing</h2> + +Hill-climbing starts at a specific point in space, +considers all nearest neighbors, and moves to the neighbor +that has the highest score; if no neighbors have higher +score than the current point (i.e., we have reached a local maximum), +the algorithm stops. One can then restart in another part of the space. +<p> +A common definition of "neighbor" is all graphs that can be +generated from the current graph by adding, deleting or reversing a +single arc, subject to the acyclicity constraint. +Other neighborhoods are possible: see +<a href="http://research.microsoft.com/~dmax/publications/jmlr02.pdf"> +Optimal Structure Identification with Greedy Search</a>, Max +Chickering, JMLR 2002. + +<!-- +Note: This algorithm is currently (Feb '02) being implemented by Qian +Diao. +--> + + +<h2><a name="mcmc">MCMC</h2> + +We can use a Markov Chain Monte Carlo (MCMC) algorithm called +Metropolis-Hastings (MH) to search the space of all +DAGs. +The standard proposal distribution is to consider moving to all +nearest neighbors in the sense defined <a href="#hill_climb">above</a>. +<p> +The function can be called +as in the following example. +<pre> +[sampled_graphs, accept_ratio] = learn_struct_mcmc(data, ns, 'nsamples', 100, 'burnin', 10); +</pre> +We can convert our set of sampled graphs to a histogram +(empirical posterior over all the DAGs) thus +<pre> +all_dags = mk_all_dags(N); +mcmc_post = mcmc_sample_to_hist(sampled_graphs, all_dags); +</pre> +To see how well this performs, let us compute the exact posterior exhaustively. +<p> +<pre> +score = score_dags(data, ns, all_dags); +post = normalise(exp(score)); % assuming uniform structural prior +</pre> +We plot the results below. +(The data set was 100 samples drawn from a random 4 node bnet; see the +file BNT/examples/static/mcmc1.) +<pre> +subplot(2,1,1) +bar(post) +subplot(2,1,2) +bar(mcmc_post) +</pre> +<img src="Figures/mcmc_post.jpg" width="800" height="500"> +<p> +We can also plot the acceptance ratio versus number of MCMC steps, +as a crude convergence diagnostic. +<pre> +clf +plot(accept_ratio) +</pre> +<img src="Figures/mcmc_accept.jpg" width="800" height="300"> +<p> +Even though the number of samples needed by MCMC is theoretically +polynomial (not exponential) in the dimensionality of the search space, in practice it has been +found that MCMC does not converge in reasonable time for graphs with +more than about 10 nodes. + + + + +<h2><a name="active">Active structure learning</h2> + +As was mentioned <a href="#markov_equiv">above</a>, +one can only learn a DAG up to Markov equivalence, even given infinite data. +If one is interested in learning the structure of a causal network, +one needs interventional data. +(By "intervention" we mean forcing a node to take on a specific value, +thereby effectively severing its incoming arcs.) +<p> +Most of the scoring functions accept an optional argument +that specifies whether a node was observed to have a certain value, or +was forced to have that value: we set clamped(i,m)=1 if node i was +forced in training case m. e.g., see the file +BNT/examples/static/cooper_yoo. +<p> +An interesting question is to decide which interventions to perform +(c.f., design of experiments). For details, see the following tech +report +<ul> +<li> <a href = "../../Papers/alearn.ps.gz"> +Active learning of causal Bayes net structure</a>, Kevin Murphy, March +2001. +</ul> + + +<h2><a name="struct_em">Structural EM</h2> + +Computing the Bayesian score when there is partial observability is +computationally challenging, because the parameter posterior becomes +multimodal (the hidden nodes induce a mixture distribution). +One therefore needs to use approximations such as BIC. +Unfortunately, search algorithms are still expensive, because we need +to run EM at each step to compute the MLE, which is needed to compute +the score of each model. An alternative approach is +to do the local search steps inside of the M step of EM, which is more +efficient since the data has been "filled in" - this is +called the structural EM algorithm (Friedman 1997), and provably +converges to a local maximum of the BIC score. +<p> +Wei Hu has implemented SEM for discrete nodes. +You can download his package from +<a href="../SEM.zip">here</a>. +Please address all questions about this code to +wei.hu@intel.com. +See also <a href="#phl">Phl's implementation of SEM</a>. + +<!-- +<h2><a name="reveal">REVEAL algorithm</h2> + +A simple way to learn the structure of a fully observed, discrete, +factored DBN from a time series is described <a +href="usage_dbn.html#struct_learn">here</a>. +--> + + +<h2><a name="graphdraw">Visualizing the graph</h2> + +Click <a href="graphviz.html">here</a> for more information +on graph visualization. + +<h2><a name = "constraint">Constraint-based methods</h2> + +The IC algorithm (Pearl and Verma, 1991), +and the faster, but otherwise equivalent, PC algorithm (Spirtes, Glymour, and Scheines 1993), +computes many conditional independence tests, +and combines these constraints into a +PDAG to represent the whole +<a href="#markov_equiv">Markov equivalence class</a>. +<p> +IC*/FCI extend IC/PC to handle latent variables: see <a href="#ic_star">below</a>. +(IC stands for inductive causation; PC stands for Peter and Clark, +the first names of Spirtes and Glymour; FCI stands for fast causal +inference. +What we, following Pearl (2000), call IC* was called +IC in the original Pearl and Verma paper.) +For details, see +<ul> +<li> +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Causation, +Prediction, and Search</a>, Spirtes, Glymour and +Scheines (SGS), 2001 (2nd edition), MIT Press. +<li> +<a href="http://bayes.cs.ucla.edu/BOOK-2K/index.html">Causality: Models, Reasoning and Inference</a>, J. Pearl, +2000, Cambridge University Press. +</ul> + +<p> + +The PC algorithm takes as arguments a function f, the number of nodes N, +the maximum fan in K, and additional arguments A which are passed to f. +The function f(X,Y,S,A) returns 1 if X is conditionally independent of Y given S, and 0 +otherwise. +For example, suppose we cheat by +passing in a CI "oracle" which has access to the true DAG; the oracle +tests for d-separation in this DAG, i.e., +f(X,Y,S) calls dsep(X,Y,S,dag). We can to this as follows. +<pre> +pdag = learn_struct_pdag_pc('dsep', N, max_fan_in, dag); +</pre> +pdag(i,j) = -1 if there is definitely an i->j arc, +and pdag(i,j) = 1 if there is either an i->j or and i<-j arc. +<p> +Applied to the sprinkler network, this returns +<pre> +pdag = + 0 1 1 0 + 1 0 0 -1 + 1 0 0 -1 + 0 0 0 0 +</pre> +So as expected, we see that the V-structure at the W node is uniquely identified, +but the other arcs have ambiguous orientation. +<p> +We now give an example from p141 (1st edn) / p103 (2nd end) of the SGS +book. +This example concerns the female orgasm. +We are given a correlation matrix C between 7 measured factors (such +as subjective experiences of coital and masturbatory experiences), +derived from 281 samples, and want to learn a causal model of the +data. We will not discuss the merits of this type of work here, but +merely show how to reproduce the results in the SGS book. +Their program, +<a href="http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html">Tetrad</a>, +makes use of the Fisher Z-test for conditional +independence, so we do the same: +<pre> +max_fan_in = 4; +nsamples = 281; +alpha = 0.05; +pdag = learn_struct_pdag_pc('cond_indep_fisher_z', n, max_fan_in, C, nsamples, alpha); +</pre> +In this case, the CI test is +<pre> +f(X,Y,S) = cond_indep_fisher_z(X,Y,S, C,nsamples,alpha) +</pre> +The results match those of Fig 12a of SGS apart from two edge +differences; presumably this is due to rounding error (although it +could be a bug, either in BNT or in Tetrad). +This example can be found in the file BNT/examples/static/pc2.m. + +<p> + +The IC* algorithm (Pearl and Verma, 1991), +and the faster FCI algorithm (Spirtes, Glymour, and Scheines 1993), +are like the IC/PC algorithm, except that they can detect the presence +of latent variables. +See the file <tt>learn_struct_pdag_ic_star</tt> written by Tamar +Kushnir. The output is a matrix P, defined as follows +(see Pearl (2000), p52 for details): +<pre> +% 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. +% P(i,j) = -2 if there is a marked directed i-*>j edge. +% P(i,j) = P(j,i) = 1 if there is and undirected edge i--j +% P(i,j) = P(j,i) = 2 if there is a latent variable L such that i<-L->j. +</pre> + + +<h2><a name="phl">Philippe Leray's structure learning package</h2> + +Philippe Leray has written a +<a href="http://bnt.insa-rouen.fr/ajouts.html"> +structure learning package</a> that uses BNT. + +It currently (Juen 2003) has the following features: +<ul> +<li>PC with Chi2 statistical test +<li> MWST : Maximum weighted Spanning Tree +<li> Hill Climbing +<li> Greedy Search +<li> Structural EM +<li> hist_ic : optimal Histogram based on IC information criterion +<li> cpdag_to_dag +<li> dag_to_cpdag +<li> ... +</ul> + + +</a> + + +<!-- +<h2><a name="read_learning">Further reading on learning</h2> + +I recommend the following tutorials for more details on learning. +<ul> +<li> <a +href="http://www.cs.berkeley.edu/~murphyk/Papers/intel.ps.gz">My short +tutorial</a> on graphical models, which contains an overview of learning. + +<li> +<A HREF="ftp://ftp.research.microsoft.com/pub/tr/TR-95-06.PS"> +A tutorial on learning with Bayesian networks</a>, D. Heckerman, +Microsoft Research Tech Report, 1995. + +<li> <A HREF="http://www-cad.eecs.berkeley.edu/~wray/Mirror/lwgmja"> +Operations for Learning with Graphical Models</a>, +W. L. Buntine, JAIR'94, 159--225. +</ul> +<p> +--> + + + + + +<h1><a name="engines">Inference engines</h1> + +Up until now, we have used the junction tree algorithm for inference. +However, sometimes this is too slow, or not even applicable. +In general, there are many inference algorithms each of which make +different tradeoffs between speed, accuracy, complexity and +generality. Furthermore, there might be many implementations of the +same algorithm; for instance, a general purpose, readable version, +and a highly-optimized, specialized one. +To cope with this variety, we treat each inference algorithm as an +object, which we call an inference engine. + +<p> +An inference engine is an object that contains a bnet and supports the +'enter_evidence' and 'marginal_nodes' methods. The engine constructor +takes the bnet as argument and may do some model-specific processing. +When 'enter_evidence' is called, the engine may do some +evidence-specific processing. Finally, when 'marginal_nodes' is +called, the engine may do some query-specific processing. + +<p> +The amount of work done when each stage is specified -- structure, +parameters, evidence, and query -- depends on the engine. The cost of +work done early in this sequence can be amortized. On the other hand, +one can make better optimizations if one waits until later in the +sequence. +For example, the parameters might imply +conditional indpendencies that are not evident in the graph structure, +but can nevertheless be exploited; the evidence indicates which nodes +are observed and hence can effectively be disconnected from the +graph; and the query might indicate that large parts of the network +are d-separated from the query nodes. (Since it is not the actual +<em>values</em> of the evidence that matters, just which nodes are observed, +many engines allow you to specify which nodes will be observed when they are constructed, +i.e., before calling 'enter_evidence'. Some engines can still cope if +the actual pattern of evidence is different, e.g., if there is missing +data.) +<p> + +Although being maximally lazy (i.e., only doing work when a query is +issued) may seem desirable, +this is not always the most efficient. +For example, +when learning using EM, we need to call marginal_nodes N times, where N is the +number of nodes. <a href="varelim">Variable elimination</a> would end +up repeating a lot of work +each time marginal_nodes is called, making it inefficient for +learning. The junction tree algorithm, by contrast, uses dynamic +programming to avoid this redundant computation --- it calculates all +marginals in two passes during 'enter_evidence', so calling +'marginal_nodes' takes constant time. +<p> +We will discuss some of the inference algorithms implemented in BNT +below, and finish with a <a href="#engine_summary">summary</a> of all +of them. + + + + + + + +<h2><a name="varelim">Variable elimination</h2> + +The variable elimination algorithm, also known as bucket elimination +or peeling, is one of the simplest inference algorithms. +The basic idea is to "push sums inside of products"; this is explained +in more detail +<a +href="http://HTTP.CS.Berkeley.EDU/~murphyk/Bayes/bayes.html#infer">here</a>. +<p> +The principle of distributing sums over products can be generalized +greatly to apply to any commutative semiring. +This forms the basis of many common algorithms, such as Viterbi +decoding and the Fast Fourier Transform. For details, see + +<ul> +<li> R. McEliece and S. M. Aji, 2000. +<!--<a href="http://www.systems.caltech.edu/EE/Faculty/rjm/papers/GDL.ps">--> +<a href="GDL.pdf"> +The Generalized Distributive Law</a>, +IEEE Trans. Inform. Theory, vol. 46, no. 2 (March 2000), +pp. 325--343. + + +<li> +F. R. Kschischang, B. J. Frey and H.-A. Loeliger, 2001. +<a href="http://www.cs.toronto.edu/~frey/papers/fgspa.abs.html"> +Factor graphs and the sum-product algorithm</a> +IEEE Transactions on Information Theory, February, 2001. + +</ul> + +<p> +Choosing an order in which to sum out the variables so as to minimize +computational cost is known to be NP-hard. +The implementation of this algorithm in +<tt>var_elim_inf_engine</tt> makes no attempt to optimize this +ordering (in contrast, say, to <tt>jtree_inf_engine</tt>, which uses a +greedy search procedure to find a good ordering). +<p> +Note: unlike most algorithms, var_elim does all its computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + + + +<h2><a name="global">Global inference methods</h2> + +The simplest inference algorithm of all is to explicitely construct +the joint distribution over all the nodes, and then to marginalize it. +This is implemented in <tt>global_joint_inf_engine</tt>. +Since the size of the joint is exponential in the +number of discrete (hidden) nodes, this is not a very practical algorithm. +It is included merely for pedagogical and debugging purposes. +<p> +Three specialized versions of this algorithm have also been implemented, +corresponding to the cases where all the nodes are discrete (D), all +are Gaussian (G), and some are discrete and some Gaussian (CG). +They are called <tt>enumerative_inf_engine</tt>, +<tt>gaussian_inf_engine</tt>, +and <tt>cond_gauss_inf_engine</tt> respectively. +<p> +Note: unlike most algorithms, these global inference algorithms do all their computational work +inside of <tt>marginal_nodes</tt>, not inside of +<tt>enter_evidence</tt>. + + +<h2><a name="quickscore">Quickscore</h2> + +The junction tree algorithm is quite slow on the <a href="#qmr">QMR</a> network, +since the cliques are so big. +One simple trick we can use is to notice that hidden leaves do not +affect the posteriors on the roots, and hence do not need to be +included in the network. +A second trick is to notice that the negative findings can be +"absorbed" into the prior: +see the file +BNT/examples/static/mk_minimal_qmr_bnet for details. +<p> + +A much more significant speedup is obtained by exploiting special +properties of the noisy-or node, as done by the quickscore +algorithm. For details, see +<ul> +<li> Heckerman, "A tractable inference algorithm for diagnosing multiple diseases", UAI 89. +<li> Rish and Dechter, "On the impact of causal independence", UCI +tech report, 1998. +</ul> + +This has been implemented in BNT as a special-purpose inference +engine, which can be created and used as follows: +<pre> +engine = quickscore_inf_engine(inhibit, leak, prior); +engine = enter_evidence(engine, pos, neg); +m = marginal_nodes(engine, i); +</pre> + + +<h2><a name="belprop">Belief propagation</h2> + +Even using quickscore, exact inference takes time that is exponential +in the number of positive findings. +Hence for large networks we need to resort to approximate inference techniques. +See for example +<ul> +<li> T. Jaakkola and M. Jordan, "Variational probabilistic inference and the +QMR-DT network", JAIR 10, 1999. + +<li> K. Murphy, Y. Weiss and M. Jordan, "Loopy belief propagation for approximate inference: an empirical study", + UAI 99. +</ul> +The latter approximation +entails applying Pearl's belief propagation algorithm to a model even +if it has loops (hence the name loopy belief propagation). +Pearl's algorithm, implemented as <tt>pearl_inf_engine</tt>, gives +exact results when applied to singly-connected graphs +(a.k.a. polytrees, since +the underlying undirected topology is a tree, but a node may have +multiple parents). +To apply this algorithm to a graph with loops, +use <tt>pearl_inf_engine</tt>. +This can use a centralized or distributed message passing protocol. +You can use it as in the following example. +<pre> +engine = pearl_inf_engine(bnet, 'max_iter', 30); +engine = enter_evidence(engine, evidence); +m = marginal_nodes(engine, i); +</pre> +We found that this algorithm often converges, and when it does, often +is very accurate, but it depends on the precise setting of the +parameter values of the network. +(See the file BNT/examples/static/qmr1 to repeat the experiment for yourself.) +Understanding when and why belief propagation converges/ works +is a topic of ongoing research. +<p> +<tt>pearl_inf_engine</tt> can exploit special structure in noisy-or +and gmux nodes to compute messages efficiently. +<p> +<tt>belprop_inf_engine</tt> is like pearl, but uses potentials to +represent messages. Hence this is slower. +<p> +<tt>belprop_fg_inf_engine</tt> is like belprop, +but is designed for factor graphs. + + + +<h2><a name="sampling">Sampling</h2> + +BNT now (Mar '02) has two sampling (Monte Carlo) inference algorithms: +<ul> +<li> <tt>likelihood_weighting_inf_engine</tt> which does importance +sampling and can handle any node type. +<li> <tt>gibbs_sampling_inf_engine</tt>, written by Bhaskara Marthi. +Currently this can only handle tabular CPDs. +For a much faster and more powerful Gibbs sampling program, see +<a href="http://www.mrc-bsu.cam.ac.uk/bugs">BUGS</a>. +</ul> +Note: To generate samples from a network (which is not the same as inference!), +use <tt>sample_bnet</tt>. + + + +<h2><a name="engine_summary">Summary of inference engines</h2> + + +The inference engines differ in many ways. Here are +some of the major "axes": +<ul> +<li> Works for all topologies or makes restrictions? +<li> Works for all node types or makes restrictions? +<li> Exact or approximate inference? +</ul> + +<p> +In terms of topology, most engines handle any kind of DAG. +<tt>belprop_fg</tt> does approximate inference on factor graphs (FG), which +can be used to represent directed, undirected, and mixed (chain) +graphs. +(In the future, we plan to support exact inference on chain graphs.) +<tt>quickscore</tt> only works on QMR-like models. +<p> +In terms of node types: algorithms that use potentials can handle +discrete (D), Gaussian (G) or conditional Gaussian (CG) models. +Sampling algorithms can essentially handle any kind of node (distribution). +Other algorithms make more restrictive assumptions in exchange for +speed. +<p> +Finally, most algorithms are designed to give the exact answer. +The belief propagation algorithms are exact if applied to trees, and +in some other cases. +Sampling is considered approximate, even though, in the limit of an +infinite number of samples, it gives the exact answer. + +<p> + +Here is a summary of the properties +of all the engines in BNT which work on static networks. +<p> +<table> +<table border units = pixels><tr> +<td align=left width=0>Name +<td align=left width=0>Exact? +<td align=left width=0>Node type? +<td align=left width=0>topology +<tr> +<tr> +<td align=left> belprop +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> belprop_fg +<td align=left> approx +<td align=left> D +<td align=left> factor graph +<tr> +<td align=left> cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> enumerative +<td align=left> exact +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> gaussian +<td align=left> exact +<td align=left> G +<td align=left> DAG +<tr> +<td align=left> gibbs +<td align=left> approx +<td align=left> D +<td align=left> DAG +<tr> +<td align=left> global_joint +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +<tr> +<td align=left> jtree +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +b<tr> +<td align=left> likelihood_weighting +<td align=left> approx +<td align=left> any +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> approx +<td align=left> D,G +<td align=left> DAG +<tr> +<td align=left> pearl +<td align=left> exact +<td align=left> D,G +<td align=left> polytree +<tr> +<td align=left> quickscore +<td align=left> exact +<td align=left> noisy-or +<td align=left> QMR +<tr> +<td align=left> stab_cond_gauss +<td align=left> exact +<td align=left> CG +<td align=left> DAG +<tr> +<td align=left> var_elim +<td align=left> exact +<td align=left> D,G,CG +<td align=left> DAG +</table> + + + +<h1><a name="influence">Influence diagrams/ decision making</h1> + +BNT implements an exact algorithm for solving LIMIDs (limited memory +influence diagrams), described in +<ul> +<li> S. L. Lauritzen and D. Nilsson. +<a href="http://www.math.auc.dk/~steffen/papers/limids.pdf"> +Representing and solving decision problems with limited +information</a> +Management Science, 47, 1238 - 1251. September 2001. +</ul> +LIMIDs explicitely show all information arcs, rather than implicitely +assuming no forgetting. This allows them to model forgetful +controllers. +<p> +See the examples in <tt>BNT/examples/limids</tt> for details. + + + + +<h1>DBNs, HMMs, Kalman filters and all that</h1> + +Click <a href="usage_dbn.html">here</a> for documentation about how to +use BNT for dynamical systems and sequence data. + + +</BODY>