## Monday, October 31, 2011

### Bayes network, variable independence and AI

In one of the Stanford AI class homework (now closed), the following question was asked. Given the following Bayes network, are B and C independent knowing A and D?

Following the usual rules, we are presented with a dilemna: A being known would imply that B and C are independent, but D being known implies that B and C are dependent.

From a probabilistic point of view, two variables A and B are independent if

Pr[A=v]=Pr[A=v|B=u]

So, to check the independence of two variables, one has to compute the conditional probability and compare to the probability without the added condition.

A quick solution is to model the Bayes network with a python script you may find on my github. The result shows that B and C are dependent if A and D are known, but independent if only A is known.

P(B)= 0.590436
P(B|C)= 0.464383811907
P(C)= 0.629475
P(C|B)= 0.49508837537

P(C,(D))= 0.650509052716
P(C|B,(D))= 0.554403493324
P(B,(D))= 0.704196855276
P(B|C,(D))= 0.600159513419

P(C,(A))= 0.448997199205
P(C|B,(A))= 0.449069612702
P(B,(A))= 0.949959858907
P(B|C,(A))= 0.9501130668

P(C,(A,D))= 0.174715296242
P(C|B,(A,D))= 0.139904742675
P(B,(A,D))= 0.844759945818
P(B|C,(A,D))= 0.676448630334

In order to double check, I did the formal calculation for the case where A and D are known, which gave me the same results.

Python rocks! And so does the Stanford AI course!