Probabilistic Reasoning; Network-based reasoning

1. Probabilistic Reasoning;Network-based reasoning COMPSCI 276 Fall 2007

2. Class Description • Instructor: Rina Dechter • Days: Monday & Wednesday • Time: 2:00 - 3:20 pm • Class page: http://www.ics.uci.edu/~dechter/ics-275b/Fall-2007/

3. Why uncertainty • Summary of exceptions • Birds fly, smoke means fire (cannot enumerate all exceptions. • Why is it difficult? • Exception combines in intricate ways • e.g., we cannot tell from formulas how exceptions to rules interact: AC BC --------- A and B - C

4. The problem True propositions Uncertain propositions Q: Does T fly? P(Q)? Logic?....but how we handle exceptions Probability: astronomical

5. Managing Uncertainty • Knowledge obtained from people is almost always loaded with uncertainty • Most rules have exceptions which one cannot afford to enumerate • Antecedent conditions are ambiguously defined or hard to satisfy precisely • First-generation expert systems combined uncertainties according to simple and uniform principle • Lead to unpredictable and counterintuitive results • Early days: logicist, new-calculist, neo-probabilist

6. Extensional vs Intensional Approaches • Extensional (e.g., Mycin, Shortliffe, 1976) certainty factors attached to rules and combine in different ways. • Intensional, semantic-based, probabilities are attached to set of worlds. AB: m P(A|B) = m

7. Certainty combination in Mycin A x If A then C (x) If B then C (y) If C then D (z) z C D y B 1.Parallel Combination: CF(C) = x+y-xy, if x,y>0 CF(C) = (x+y)/(1-min(x,y)), x,y have different sign CF( C) = x+y+xy, if x,y<0 2. Series combination… 3.Conjunction, negation Computational desire : locality, detachment, modularity

8. Burglery Example Burglery Phone call Alarm Earthquake Radio AB A more credible ------------------ B more credible IF Alarm  Burglery A more credible (after radion) But B is less credible Rule from effect to causes

9. Extensional vs Intensional Extensional Intensional

10. What’s in a rule? A and BC (m+n-mn)

11. Why networks? • Claim: the basic steps invoked while people query and update their knowledge corresponds to mental tracings of pre-established links in dependency graphs • Claim: the degree to which an explanation mirrors these tracings determines whether it is psychologically meaningful.

27. P(S) P(C|S) P(B|S) • 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 CPD: P(X|C,S) P(D|C,B) Conditional Independencies Efficient Representation Bayesian Networks: Representation Smoking lung Cancer Bronchitis 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)

28. Markov and Bayesian Networks • Pearl Chapter 3 • (Read chapter 2 for background and refresher)

29. The Qualitative Notion of Depedence • The traditional definition of independence uses equality of numerical quantities as in P(x,y)=P(x)P(y) • People can easily and confidently detect dependencies, even though they may not be able to provide precise numerical estimates of probabilities. • The notion of relevance and dependence are far more basic to human reasoning than the numerical values attached to probabilistic judgements. • Should allow assertions about dependency relationships to be expressed qualitatively, directly and explicitly. • Once asserted, these dependency relationships should remain a part of the representation scheme, impervious to variations in numerical inputs.

30. The Qualitative Notion of Depedence(continue) • Information about dependencies is essential in reasoning • If we have acquired a body of knowledge K and now wish to assess the truth of proposition A, it is important to know whether it is worthwhile to consult another proposition B, which is not in K. • How can we encode relevance information in a symbolic system? • The number of (A,B,K) combinations is astronomical. • Acquisition of new facts may destroy existing dependencies as well as create new ones (e.g.,age, hight,reading ability, or ground wet,rain,sprinkler) • The first kind of change is called “normal” . The second will be called “induced”. • Irrelevance is denoted: P(A|K,B)=P(A|K) • Dependency relationships are qualitative and can be logical

31. Dependency graphs • The nodes represent propositional variables and the arcs represent local dependencies among conceptually related propositions. • Explicitness, stability • Graph concepts are entrenched in our language (e.g., “thread of thoughts”, “lines of reasoning”, “connected ideas”) • One wonders if people can reason any other way except by tracing links and arrows and paths in some mental representation of concepts and relations. • What types of dependencies and independencies are deducible from the topological properties of a graph? • For a given probability distribution P and any three variables X,Y,Z,it is straightforward to verify whether knowing Z renders X independent of Y, but P does not dictates which variables should be regarded as neighbors. • Some useful properties of dependencies and relevancies cannot be represented graphically.

42. Why axiomatic characterization? • Allow deriving conjectures about independencies that are clearer • Axioms serve as inference rules • Can capture the principal differences between various notions of relevance or independence