The Bayesian belief network, once trained, can be used for classification. Based on the Bayes' theorem, which is defined in the The Bayes classification section of Chapter 3, Classification, it is defined with two parts, one directed acyclic graph and conditional probability tables (CPT) for each variable; this is in turn represented by one node in the graph and models the uncertainty by graphically representing the conditional dependencies between distinct components. The arcs in the image give a representation of causal knowledge. The interaction among the diverse sources of uncertainty is also graphically illustrated.

The uncertainty comes from various sources:

Here is an example of the Bayesian belief network with four Boolean variables and the corresponding arcs. Whether the grass is wet is influenced by the work results of sprinkler and whether it has just rained, and so on. Each arc has a certain probability.

Let us have a look at the CPT representation of :

In the network, each variable is conditionally independent of its non-descendants. Here is the definition of the joint probability distribution:

The Bayesian belief network (BBN) algorithm

Before the application of the BBN algorithm to classification, we need to train it first. In the process of training the network, the expert knowledge, that is, the prior knowledge, can be used in the training process to help the design of the network. For the variables that participated in direct dependency, experts must specify their conditional probability. There are many algorithms to learn the network from the training dataset; we will introduce an adaptive probabilistic networks algorithm.

The input parameters for the BBN algorithm are as follows:

• T, denotes a total order of the variables
• CPT

The output of the algorithm is the topology structure of BBN, which is as follows:

T denotes a total order of the variables in line 2. In line 5, denotes the set of variables preceding .

> source("ch_04_bnn.R")