Uncertainty CS570 Lecture Note by Jin Hyung Kim Computer Science Department KAIST

1. Uncertainty CS570 Lecture Note by Jin Hyung Kim Computer Science Department KAIST

2. Reasoning Under Uncertainty • Motivation • Truth value is unknown • Too complex to compute prior to make decision • characteristics of real-world applications • Source of Uncertainty • cannot be explained by deterministic model • decay of radioactive substances • Don’t understand well • disease transmit mechanism • Too complex to compute • coin tossing

3. Types of Uncertainty • Randomness • Which side will be up if I toss a coin ? • Vagueness • Am I pretty ? • Confidence • How much do you confident on your decision ? One Formalism for all vs. Separate formalism Representation + Computational Engine

4. Representing Uncertain Knowledge • Modeling • select relevant (believed) propositions and events • ignore irrelevant, independent events • quantifying relationships is difficult • Consideration of Representation • problem characteristic • degree of detail • Computational complexity • simplification by assumption • Data is obtainable ? • Computational Engine is available ?

5. Uncertain Representation Binary Logic Multi-valued Logic Probability Theory Upper/Lower Probability Possibility Theory 1 0

6. Application involving Uncertainty • Wide range of Applications • diagnose disease • language understanding • pattern recognition • managerial decision making • useful answer from uncertain, conflicting knowledge • acquiring qualitative and quantitative relationships • data fusion • multiple experts’ opinion

7. Probability Theory • Frequency Interpretation • flipping coin • generalizing future event • Subjective Interpretation • Probability of passing an exam • degree of belief

8. Degree of Belief • Elicitation of degree of belief from subject • lottery - utility functions • Elicitation of Conditional probability • conversion using Bayes Rule • Different subjective probability to same proposition • due to different background and knowledge $1 $1 0.9 Rain >< 0.1 $0 No Rain $0

9. Human Judgement and Fallibility • Systematic bias in probability judgement • Psychologist Tversky’s experiment • conservative in updating belief • A: 80% chance of $4000 • B: 100% chance of $3000 • C: 20% chance of $4000 • D: 25% chance of $3000 A or B ?, C or D ? • Risk-averse with high prob. And risk-taking with low prob.

10. Random Variable • Variable takes value from sample space • mutually exclusive, exhausitive • Joint Sample Space x1x2 ….  xn = • 0 Pr(X = x)  1, xx and • 1 =

11. More on Probability Theory • Joint Distribution • joint probability is exponential of RV n • Conditional Probability • Prior Belief + New evidence  Posterior Belief

12. Calculus of Combining Probabilities • Marginalizing • Addition Rule • Chain Rule Pr(X1, …, Xn) = Pr(X1|X2, …, Xn) Pr(X2|X3, …, X4) …. Pr(Xn-1|Xn) Pr(Xn) • factorization

13. Bayes Rule • Allows to covert Pr(A|B) to P(B|A), vice versa • Allows to elicit psychologically obtainable values • P( symptom | disease ) vs P(disease | symptom) • P(cause | effect ) vs P(effect | cause ) • P( object | attribute) vs P(attribute | object ) • Probabilistic Reasoning  Baysian School

14. Conditional Independence • Specifying distribution of n variables • require 2n numbers • Conditional Independence Assumption • Pr(A|B, C) = Pr(A|C) • A is conditionally independent of B given C • A not depend on B given C  We can ignore B in inferring A if we have C • Pr(A,B|C) = Pr(A|C)Pr(B|C) • symmetric relationship • Chain rule and conditional independence together saves storage • ex : Pr(A,B,C,D) = Pr(A|B)Pr(B|C)Pr(C|D)P(D)

15. Problems • Approximate the joint probability distribution of n variables by set of 2nd order joint probabilities Pr(ABCD) = P(A)P(B|A)P(C|A)P(D|C) or P(B)P(B|A)P(C|B)P(D|C) or ….. • From P(A) and P(B), estimate P(A, B). • Maximum Entropy Principle

16. Maintaining Consistency • Pr(a|b) = 0.7, Pr(b|a) = 0.3, Pr(b) = 0.5 • inconsisent • check by Bayes rule Pr(a,b) = Pr(b|a)Pr(a) = Pr(a|b)Pr(b) Pr(a) = Pr(a|b)Pr(b) / Pr(b|a) = 0.7 x 0.5 / 0.3 > 1.0

17. Probabilistic Networks • Bayesian Network, Causal Network, Knowledge map • Graphical model of causality and influence • directed acyclic graph, G = (V, E) • V : random variable • E : dependencies between random variables • only one edge connecting two nodes directly • direction of edge : causal influence • for X  Y  Z • evidence regarding X : causal support of Y • evidence regarding Z : diagnostic support of Y

18. H : hardware problem B : bug’s in LISP code E : editor is running L : LISP Interpreter O.K. F : Cursor is flashing P : prompt displayed W : dog is barking H W B E L Parent child successor F P

19. Burglary Earthquake Alarm MaryCalls JohnCalls

20. Independent relationship • Each node is CI of all non-successors given Parent • Pr(P|E,F,L,J,B,H) = Pr(P|L) • Topological sort of variables • variable after all children • Let X1, X2, …, Xn be a topological sort of variables of G • Pr(Xi | Xi+1, Xi+2, …, Xn) = Pr (Xi | parent(Xi))

21. H B D C S E L Pr(P,F,L,E,B,H) = Pr(P|F,L,E,B,H)Pr(F|L,E,B,H) Pr(L|E,B,H)Pr(E|B,H)Pr(B|H)Pr(H) = Pr(P|L)Pr(F|E)Pr(L|H,B)Pr(B)Pr(H) F P

22. Path-Based Characterization of Independence • A is dependent of C given B • F is independent of D given E B C A E F D

23. Direction dependent separation (D-separation) • A path P is blocked given set of node E if ZP for which 1. ZE and Z has one arrow in and one arrow out on P 2. ZE and Z has both path arrows out 3. ZE and successor(Z)E, and both path arrows lead to Z • A set of nodes E d-separates two sets of nodes X and Y if every undirected path from X to Y is blocked • If every undirected path from X to Y is d-separable by E, then X and Y are conditionally independent given E.

24. Z (1) (2) Z E (3) Z Y X Path X to Y is blocked, given evidence E

25. Quantifying Network • Supply conditional probability of each node given parent H B E L F P