Summary Belief Networks zObjective: yProbabilistic knowledge base + Inference engine that computes xProb(formula | “all evidence collected so far”) zBelief (Bayes) Networks yDirected graph - arcs denote direct causal relationship yFor each root. provide unconditional (prior) probability yfor all remaining, provide a conditional probability table xProb(Child | Parent1, Parent2, Parent3) for all combos yRepresent uncertain evidence by adding a new node zInference yPoly tree case yMultiply connected networks => cluster

2 Complete Bayes Network Burglary MaryCalls JohnCalls Alarm Earthquake P(A) ATFATF P(J) ATFATF P(M) P(B).001 P(E).002 ETFTFETFTF BTTFFBTTFF

3 Cond. Independence in Bayes Nets zIf a set E d-separates paths between X & Y yThen X and Y are cond. independent given E zSet E d-separates X and Y if every undirected path between X and Y has a node Z such that, either Z Z Z Z XY E

Structural Relationships and Independence zThe basic independence assumption (simplified version): ytwo nodes X and Y are probabilistically independent conditioned on E if every undirected path from X to Y is d- separated by E xevery undirected path from X to Y is blocked by E if there is a node Z for which one of three conditions hold –Z is in E and Z has one incoming arrow on the path and one outgoing arrow –Z is in E and both arrows lead out of Z –neither Z nor any descendent of Z is in E, and both arrows lead into Z

5 Inference zGiven exact values for evidence variables zCompute posterior probability of query variable Burglary MaryCall JonCalls Alarm Earthq P(B).001 P(E).002 ATFATF P(J) ATFATF P(M) ETFTFETFTF P(A) BTTFFBTTFF Diagnostic –effects to causes Causal –causes to effects Intercausal –between causes of common effect –explaining away

6 Algorithm zIn general: NP Complete zEasy for polytrees yI.e. only one undirected path between nodes zExpress P(X|E) by y1. Recursively passing support from ancestor down x“Causal support” y2. Recursively calc contribution from descendants up x“Evidential support” zSpeed: linear in the number of nodes (in polytree)

Simplest Causal Case zSuppose know Burglary zWant to know probability of alarm yP(A|B) = 0.95 Alarm Burglary P(B).001 BTFBTF P(A).95.01

Simplest Diagnostic Case Alarm Burglary P(B).001 BTFBTF P(A) zSuppose know Alarm ringing & want to know: Burglary? zI.e. want P(B|A)

P(B|A) =P(A|B) P(B) / P(A) But we don’t know P(A)

1 =P(B|A)+P(~B|A) 1 =P(A|B)P(B)/P(A) + P(A|~B)P(~B)/P(A) 1 =[P(A|B)P(B) + P(A|~B)P(~B)] / P(A) P(A) =P(A|B)P(B) + P(A|~B)P(~B)

P(B | A) =P(A|B) P(B) / [P(A|B)P(B) + P(A|~B)P(~B)] =.95*.001 / [.95* *.999] = 0.087

General Case U1U1 UmUm X Y1Y1 YnYn Z 1j Z nj... zCompute contrib of E x + by computing effect of parents of X (recursion!) zCompute contrib of E x - by... ExEx + ExEx - zExpress P(X | E) in terms of contributions of E x + and E x -

Using Probabilities to Make Decisions zComputing the probabilities of an event is the easy part of the problem. zMaking decisions is the hard part… zThe probability of winning the lottery is Should I play? zThe probability of the professor slipping and breaking his head is Should I come to the exam?

Maximum Expected Utility zUtility Function: states -> real number zExpected Utility yEU(A|E) =  I P(Result i (A) | E,Do(A)) U(Result i (A)) zPrinciple of Maximum Expected Utility yYou’re stupid if you don’t choose the action that maximizes expected utility. (stupid is a well defined mathematical term!)

Flying To Paris zIf I fly through Chicago, it takes 11 hours. But, with prob. 0.3 I’ll miss the connection and have to wait 4 hours. zIf I fly through SF, it takes 13 hours. But with probability 0.05 I’ll miss the connection and have to wait 24 hours. zUtility: yChicago: -11 * * 0.3 = ySF: -13 * * 0.05 = -14.2

Decision Tree for Paris Travel SF Chicago miss not miss not miss