Question

For n Boolean variables, the full joint probability distribution contains 2^n numbers. A Bayes net can reduce this to n 2^k, where k is:

A. the number of nodes in the network.

B. the maximum number of nodes that directly influence any node.

C. the number of parameters in each conditional probability table.

EXPLAIN, THANKS

Answer 1

Here note that in Bayesian network, we need to only take account of probability distribution of any node x with conditioning upon only those nodes that directly influence node x, because other nodes which does not directly influence in probability distribution of a particular node x are irrelevant and hence their value does not play any role in determining the conditional probability distribution of node x.

Hence if k is the  maximum number of nodes that directly influence node x, then probability distribution of node x needs to be determined with conditioning upon all possible 2^k value of other k nodes.

Hence option B is correct.

Bayesian network reduces the number of parameters in determining full joint distribution by breaking this probability into product of conditional probability over subset of nodes where the number of conditional variable is not greater than k.

Please comment for any clarification.