Bayesian Networks Probability In AI

Bayes’ Theorem P(A|B) = P(B|A) P(A) / P(B)

Example Bowl 1 – 10 red and 30 white balls Randomly pick a bowl and then a ball from it What is P(Bowl 1 | white ball) ? By Bayes’ P(white|bowl 1) P(bowl 1) / P(white) P(white) = .5 P(white|bowl 1) + .5 P(white|bowl 2)

Bayesian Network G=(V,E) be a DAG Let each node be a random variable For node a the nodes connected to a are {bi} P(a|{bi}) (written in Conditional Probability Tables)

Example (From Russell and Novrig Chapt 14) Burglary Eartthquake P(B)=.001 P(E)=.002 B E P(A) T T .95 T F .94 F T .29 F F .001 Alarm JohnCalls MaryCalls A P(M) T .70 F .01 A P(J) T .9 F .04

Semantics The graph represents the full joint probability distribution P(x1, …. xn) is the probability of the set of assignments to the variables P(x1, …. xn) = Pi=1,n P(xi | parents(Xi))

Construction Nodes from the set of random variables Links Order them with causes before effects(not required, but simpler!) Links For each node determine the set of parents Link them Define Conditional Probablity Table

Query Given an observed event, what is posterior probability for the query variables. Ex: P(Burglary | JohnCalles = true , MaryCalles = true) Answer is <0.284,0.716>

Method Single Variable by enumeration Find P(X,e) X is query variable e is event : on the evidence variables Y is a set of hidden variables P(X|e) = a S P(X,e,y) The constant a is to normalize

P(Burglary | JohnCalles = true , MaryCalles = true) P(B| j,m) =a P(B,j,m) = a Se Sa P(b,j,m,e,a) For CPT entries: For burglary = true: = a Se Sa P(b)P(e)P(a|b,e)P(j|a)P(m|a) This is .00059224 For the false case - .0014919 The a is computed to give a sum = 1

Variable Elimination A query method that works by removing the unobserved variables More efficient

Summary