Bayes Networks in AI

BAYES NETWORKS 

It is a structure that is used to represent conditional independence conventionally. 

It is also called as belief network 

It is directed, acyclic graph. The nodes in the graph are labeled by random variables.

The node A_i in the acyclic graph is conditionally independent of any subset of nodes except the descendants of A_i if the parents of A_i are given. 

If the parents of the node A_i is .. represented by P(A_i) nd it non-7clescdndents are denoted by a set of node A(A_i). Then,

        I(A_i, A(Ai) | P(A_i)) 

which means that, 

        P(A_i | A(A_i), P(A_i)) = P(A_i | P(A_i)) 

The joint probability of all the nodes in the network can be represented as,

        

where this is got by applying conditional independencies to chain expression. 

These networks are also called as casual networks.

Example:

    Suppose that there are two events which could cause grass to be wet, that is by either the sprinkler or if it's raining. Also, suppose that the rain has a direct influence on sprinkler because the sprinkler is usually not turned on when it rains. Then the situation can be modelled using Bayes network. All the three variables have two possible values, T (for true) and F (for false). 

    The situation can be depicted by the Figure. 3.4.1, 

Fig: A Bayes Network 

The joint probability function is,

    P(G, S, R) P(G | S, P(S | R) P(R) from the chain rule. Here the names of the variables have been abbreviated to G = Grass wet, S = sprinkler turned on and R = Raining. 

It is to be noted that, inorder, to calculate joint probabilities we have to know conditional probability functions of every node which is present in the network and conditioned on its parents. For the nodes without parents prior probabilities of the variables are present which are not conditioned on other nodes. These are shown in the above Figure.