Please go through the example and exercise sections below, and submit a single sheet of paper with the result of the exercise section. Contact me (kaess@cc) or post to the newsgroup if you need any help installing the toolbox or using Matlab.
This document is copied in parts from How to use the Bayes Net Toolbox by Kevin Murphy.
For an introduction to Matlab and links to other tutorials, see my Another Brief Matlab Tutorial page.
You need Matlab version 5.2 or newer to run BNT. (Versions 5.0 and 5.1 have a memory leak which seems to sometimes crash BNT.)
>> cd /net/hzr1/users/kaess/FullBNT/BNT
To specify this directed acyclic graph (DAG), create an adjacency matrix:
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;
We have numbered the nodes as follows: Cloudy = 1, Sprinkler = 2, Rain = 3, WetGrass = 4. The nodes must always be numbered in topological order, i.e., ancestors before descendants.
In addition to specifying the DAG, 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.
discrete_nodes = 1:N; node_sizes = 2*ones(1,N);
We are now ready to make the Bayes net:
bnet = mk_bnet(dag, node_sizes, 'discrete', discrete_nodes);By default, all nodes are assumed to be discrete, so we can also just write
bnet = mk_bnet(dag, node_sizes);For more details on the parameters of mk_bnet, please see its documentation string by typing
help mk_bnet
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
where we have used the convention that false==1, true==2. We can create this CPT in Matlab as follows
CPT = zeros(2,2,2); CPT(1,1,1) = 1.0; CPT(2,1,1) = 0.1; ...Here is an easier way:
CPT = reshape([1 0.1 0.1 0.01 0 0.9 0.9 0.99], [2 2 2]);In fact, we don't need to reshape the array, since the CPD constructor will do that for us. So we can just write
bnet.CPD{W} = tabular_CPD(bnet, W, 'CPT', [1 0.1 0.1 0.01 0 0.9 0.9 0.99]);
The other nodes are created similarly (using the old syntax for
optional parameters)
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]);
engine = jtree_inf_engine(bnet);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.
Suppose we want to compute the probability that the sprinkler 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.
evidence = cell(1,N);
evidence{W} = 2;
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.
We are now ready to add the evidence to the engine.
[engine, loglik] = enter_evidence(engine, evidence);The behavior of this function is algorithm-specific, and is discussed in more detail below. 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.)
Finally, we can compute p=P(S=2|W=2) as follows.
marg = marginal_nodes(engine, S);
marg.T
ans =
0.57024
0.42976
p = marg.T(2);
We see that p = 0.4298.
Write down or print the commands you used to make this change to the net given in the example section above. Write down or print the probability that the sprinkler was on given that the grass is wet (use the following commands):
engine = jtree_inf_engine(bnet);
evidence = cell(1,N);
evidence{W} = 2;
[engine, loglik] = enter_evidence(engine, evidence);
marg = marginal_nodes(engine, S);
marg.T(2)
For a more detailed introduction to the toolbox, see How to use the Bayes Net Toolbox by Kevin Murphy.