Probabilistic Reasoning; Network-based reasoning Set 7 ICS 179, Spring 2010

Propositional Reasoning Example: party problem = A = B If Alex goes, then Becky goes: If Chris goes, then Alex goes: Question: Is it possible that Chris goes to the party but Becky does not? = C = A To illustrate these concepts let me consider a simple example Consider a simple scenario involving people going to parties. Lets assume that we have three individuals, Alex, Becky and Chris. And we have some knowledge about their social relationships in regard to party-going. Show……Query. How can we do this kind of simple reasoning by a computer? We have to first formalize the information in the sentences and “represent this sentences” in some computer language . We can associate symbols with fragments in the sentence which can either be true or false (we call those “propositions”) and then describe the information In the sentence by logical rules (knowledge-representation). Subsequently we need to be able to derive answers to queries, and to do it “automatically,” namely, to engage in “automated reasoning.” Automated reasoning is the study of algorithms and computer programs that answer such queries . There are many disciplines that contributed, ideas, viewpoints and techniques to AI. Mathematical logic had a tremendous influence on AI in the early days and even today. In general to become a formal science AI relied heavily on three fundamental areas: logic, probability and computation. --------------------------------------------------------------------------------------------------------. Chavurah 5/8/2010

Probabilistic Reasoning Party example: the weather effect Alex is-likely-to-go in bad weather Chris rarely-goes in bad weather Becky is indifferent but unpredictable Questions: Given bad weather, which group of individuals is most likely to show up at the party? What is the probability that Chris goes to the party but Becky does not? P(A|W=bad)=.9 W A P(C|W=bad)=.1 C P(B|W=bad)=.5 B W A P(A|W) good .01 1 .99 bad .1 .9 Lets go back now to the party example but have a factor we did not have before. Lets say that on the day of the party the weather was bad. How would this affect our party-goers? (Remember Drew’s holiday party and the weather that day? I think it affected some of us…) Lets assume… Query… Well, in order to handle this we may use the field of probability theory and express the information in those sentences probabilistically. We can put this together into a “probabilistic network” that has a node for each proposition and directed arcs signifying Causal/probabilistic relationships. Although we express information just between several variables, we can argue that the information we have is sufficient to capture all the probabilistic relationship that the “simple” domain has. In other words, assuming that the individuals in the new story are affected only by the weather, we don’t need to specify P(a|b) because, when we know the Weather a and b are conditionally independent. The local collection of function represent a joint probability on all the variables That can be obtained by a product of all these functions. W P(W) P(A|W) P(C|W) P(B|W) B C A P(W,A,C,B) = P(B|W) · P(C|W) · P(A|W) · P(W) P(A,C,B|W=bad) = 0.9 · 0.1 · 0.5 Chavurah 5/8/2010

Mixed Probabilistic and Deterministic networks PN CN P(C|W) P(B|W) P(W) P(A|W) W B A C P(C|W) P(B|W) P(W) P(A|W) W B A C A→B C→A B A C A→B C→A B A C Query: Is it likely that Chris goes to the party if Becky does not but the weather is bad? Semantics? Algorithms? Chavurah 5/8/2010

The problem All men are mortal T All penguins are birds … Socrates is a man Men are kind p1 Birds fly p2 T looks like a penguin Turn key –> car starts P_n True propositions Uncertain propositions Q: Does T fly? P(Q)? Logic?....but how we handle exceptions Probability: astronomical

Alpha and beta are events

Burglary is independent of Earthquake

Earthquake is independent of burglary

Bayesian Networks: Representation P(D|C,B) P(B|S) P(S) P(X|C,S) P(C|S) Smoking lung Cancer Bronchitis CPD: C B D=0 D=1 0 0 0.1 0.9 0 1 0.7 0.3 1 0 0.8 0.2 1 1 0.9 0.1 X-ray Dyspnoea P(S, C, B, X, D) = P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B) Conditional Independencies Efficient Representation

Chapter 14 , Russel and Norvig Section 1 – 2 Bayesian networks Chapter 14 , Russel and Norvig Section 1 – 2

Outline Syntax Semantics

Example Topology of network encodes conditional independence assertions: Weather is independent of the other variables Toothache and Catch are conditionally independent given Cavity

Example I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar? Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects "causal" knowledge: A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call

Example contd.

Compactness A CPT for Boolean Xi with k Boolean parents has 2k rows for the combinations of parent values Each row requires one number p for Xi = true (the number for Xi = false is just 1-p) If each variable has no more than k parents, the complete network requires O(n · 2k) numbers I.e., grows linearly with n, vs. O(2n) for the full joint distribution For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)

Semantics The full joint distribution is defined as the product of the local conditional distributions: P (X1, … ,Xn) = πi = 1 P (Xi | Parents(Xi)) e.g., P(j  m  a  b  e) = P (j | a) P (m | a) P (a | b, e) P (b) P (e) n

Constructing Bayesian networks 1. Choose an ordering of variables X1, … ,Xn 2. For i = 1 to n add Xi to the network select parents from X1, … ,Xi-1 such that P (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1) This choice of parents guarantees: (chain rule) (by construction)

Example Suppose we choose the ordering M, J, A, B, E P(J | M) = P(J)?

Example Suppose we choose the ordering M, J, A, B, E No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)?

Example Suppose we choose the ordering M, J, A, B, E No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? P(B | A, J, M) = P(B)?

Example Suppose we choose the ordering M, J, A, B, E No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? P(E | B, A, J, M) = P(E | A, B)?

Example Suppose we choose the ordering M, J, A, B, E No P(A | J, M) = P(A | J)? P(A | J, M) = P(A)? No P(B | A, J, M) = P(B | A)? Yes P(B | A, J, M) = P(B)? No P(E | B, A ,J, M) = P(E | A)? No P(E | B, A, J, M) = P(E | A, B)? Yes

Example contd. Deciding conditional independence is hard in noncausal directions (Causal models and conditional independence seem hardwired for humans!) Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed