Inference in Belief Networks

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Abstract

Belief network is a very powerful tool for probabilistic reasoning. In this article I will demonstrate a C# implementation of bucket elimination algorithm for making inference in belief networks.

Introduction

Belief network, also known as Bayesian network or graphical model, is a graph in which nodes with conditional probability table (CPT) represent random variables, and links or arrows that connect nodes represent influence. See Fig.1 for example

Fig.1 WetGrass belief network. P(X=T) can be obtained by 1-P(X=F)

Mathematically belief network is equivalent to the joint probability distribution. It provides us a natural way to model probabilistic relationships among a set of variables of interests. Because of its probabilistic nature, belief network is very useful in encoding uncertain knowledge. Once we have the network in hand, we can make inference from it, no matter if it is diagnostic (from effects to causes), or if it is causal (from causes to effects), or mixed. We can then make decisions or recommendations based on the results of inference. That is how the belief network finds its way in artificial intelligence.

Belief network is a very rich field to explore both in theory and application. However, I will not discuss any detail in this article because of limited space. Instead, I will walk you through a simple example without providing proofs. Hopefully this article can give you a feeling of how the belief network works. If you are interested, you may pick up a textbook like [1], or search for tutorials in the Internet.

Inference

Conditional probability, written as, say, P(A|B,C), tells us the probability of the event A given the event B and C. A joint probability such as P(A, B) indicates the probability of both event A and B. According to product rule of probability calculus, a joint probability can be separated into product of prior probability and conditional probability,

P(A, B) = P(A|B)P(B),          (1)

where P(B) is called prior or unconditional probability.

The task of inference is to compute conditional probability P (query | evidence ) for a set of query variables with some evidence variables given. Consider the belief network given in Fig.1, if we want to know the chance of not cloudy when grass is wet, we would calculate P( cloudy=0 | wetGrass=1 ). By rearranging eq.(1), we have:

P(c=0 | w=1) = P(c=0, w=1)/P(w=1).        (2)

To calculate P(w=1), we must include influence from W node’s parent, which is represented as arrows in the belief network. The W node has two parents S and R. W’s parents are also influenced by their parents. Repeating this process until reach prior probability, it gives us

P(w=1) = ∑<SUB>s,r</SUB>P(s,r)P(w=1|s,r) = ∑<SUB>c,s,r</SUB>P(c)P(s|c)P(r|c)P(w=1|s,r).     (3)

Calculate P(c=0, w=1) in similar way, we have

P(c=0 | w=1) = P(c=0)∑<SUB>s,r</SUB>P(s|c=0)P(r|c=0)P(w=1|s,r)/P(w=1).    (4)

Now all right side probability is explicitly defined in the CPT of the belief network.

Bucket elimination algorithm

From example above we can see that making an inference is about calculating joint probability (P(w=1) can be viewed as a special case of joint). If we have an algorithm that computes joint probability P(c,s,r,w), we can make any inference in the belief network.

Let us rewrite (3) as

P(w=1) = ∑<SUB>c</SUB>P(c)∑<SUB>s</SUB>P(s|c)∑<SUB>r</SUB>P(r|c)P(w=1|s,r).   (5)

The idea for bucket elimination algorithm [2], or in general, variable elimination algorithm, is distributing sums over products. Start from the innermost (the rightmost) of (5),

P(w=1) = ∑<SUB>c</SUB>P(c)∑<SUB>s</SUB>P(s|c)∑<SUB>r</SUB>P(r|c)λ<SUB>w</SUB>(s,r)    where λ<SUB>w</SUB>(s,r)=P(w=1|s,r)
       = ∑<SUB>c</SUB>P(c)∑<SUB>s</SUB>P(s|c)λ<SUB>r</SUB>(c,s)    where λ<SUB>r</SUB>(c,s)=∑<SUB>r</SUB>P(r|c)λ<SUB>w</SUB>(s,r)
       = ∑<SUB>c</SUB>P(c)λ<SUB>s</SUB>(c)    where λ<SUB>s</SUB>(c)=∑<SUB>s</SUB>P(s|c)λ<SUB>r</SUB>(c,s)
       = λ<SUB>c,</SUB>         (6)

Now the node is replaced by bucket λ, which contains corresponding node’s CPT and child buckets. Calculating process works in reverse order. It starts from outmost node, i.e. root node. If a node is evidence or query, we use the supplied value, otherwise we sum over all possible values. The process can then be implemented recursively.

Implementation

The implementation has a static structure shown in Fig.2. The source code for download is a simplified version. It only works for binary node, i.e. the node only takes two values of 0 and 1. I also removed all exception controls such as try-catches and asserts, and removed all optimizations though some of them are are obvious and straightforward. The purpose is to make the code easier to read. It should not be difficult for you to tweak the code once you understand it.

Fig.2 Static structure of belief network classes

The nodes are defined in XML file. There are two files, BN_WetGrass.xml and BN_Alarm.xml, included in the download, both are taken from [1]. They have completely different topologic structure. You may design your own nodes. Just keep in mind that the order of nodes is important, any parent node of a node must be defined before the node. The method BNet::Build("nodes.xml") loads nodes from XML file and then creates a belief network with these nodes.

To make an inference, you may do the following

BNet net = new BNet();
net.Build("bn_wetGrass.xml");
BNInference infer = new BElim(net)
double pr = infer.GetBelief("sprinkler=1; cloudy=0", "WetGrass=0; Rain=1");

The bucket is initialized in BElim::PrepareBucket(). After reading example eq. (6) carefully, you will see that each bucket needs input not only from the parent nodes of its corresponding node but also from the nodes asked by its child bucket.

The core of the algorithm is in BElim::Sum(int id, int para), as explained in the following fragment:

protected double Sum(int node_id, int para)
{
    // skipped ... 
    double pr = 0.0;

    foreach(int value in theNode.Range)
    {
        if( theNode.IsEvidence && theNode.Evidence != value )
            continue;

        double tmpPr = theNode.CPT;

        foreach(Bucket nxtBuck in theBuck.childBuckets)
        {
            int next_para = 0;
            foreach(BNode node in nxtBuck.parentNodes)
            {
                //calculate next_para; 
            }
            tmpPr *= Sum(nxtBuck.id, next_para);
        }
        pr += tmpPr;
    }
    return pr;
}

Here we have taken advantage of binary node. The integer para is used as a bit array of size 32. It allows us to play bit operation on it and so make the code more efficient and compact. The sum goes through all possible values unless the node has a supplied value. Then it takes into account the contributions from its child buckets. To call the child’s contribution, the sum must provide parameters and let the next bucket know which node has what value, etc. I will not go any further, because these things are better explained by code than by words.

Use Belief Network

The WetGrass network is pretty simple, however it already shows some subtlety. For instance, we may calculate the probability of the sprinkler working given the grass is wet. We have P( sprinkler=1 | wetGrass=1 ) = 0.43. This inference is diagnostic, because it tells us the cause from the effect. On the other hand, if we know it is raining or not raining, then P( sprinkler=1 | wetGrass=1; rain=1 ) = 0.19, or P(sprinkler=1 | wetGrass=1; rain=0 ) = 1.0. You see the extra information of raining changes the probability of sprinkler=1, although sprinkler and rain does not directly influence each other. This effect is known as explaining away. That is the existence of one makes the other less likely, even though they are independent.

You can play with the code and try different inference with both WetGrass and Alarm network. You may or may not have surprise. The reasoning is not always straightforward. Your common sense may not always be correct.

Coding notes

The reason of using C# is because I want to learn it. I have used C# as a better C++, but not a better better C :). As a long time C++ programmer it took me a while to get used to use new without thinking where to put delete. The only thing that bothers me is the ArrayList. It seems to me that there is no way to specify type in declaration; the type is decided on the fly when you add an item to it. This can be a nice feature. However it may cause a problem if we want to separate design from implementation, we must rely on a comment to remind implementer the item type the list contains. Maybe I missed something here.

The code is edited with Emacs editor and complied by csc.exe from C# SDK downloaded from Microsoft site. The Emacs editor restricts me to write a better UI.

References

1. S. Russell and P. Norvig. "Artificial Intelligence: A Modern Approach ". Prentice Hall. 1995
2. R. Dechter. “Bucket Elimination: A Unifying Framework for Probabilistic Inference” in "Learning in Graphical Models". MIT Press. 1998. Ed. M. I. Jordan.