1. CSE391-2005 PR 1 Reasoning - Rule-based and Probabilistic Representing relations with predicate logic Limitations of predicate logic Representing relations with probabilities

2. CSE391-2005 PR 2 Causal Rules vs. Diagnostic (Evidential) Rules Cavity(tooth1) → Toothache(tooth1) –Model of the world, causal link Toothache(tooth1) ← Cavity(tooth1) –Typical symptom, might be due to typical cause

3. CSE391-2005 PR 3 Dental Model – causal relations Cavity → Toothache Gum disease → Toothache Impacted Wisdom Teeth → Toothache Sinus Infection → Toothache Other causes?

4. CSE391-2005 PR 4 Causal Relations Cavity → Toothache Cavity → X-ray shad Gum disease → Toothache Gum disease → Red gums Impacted Wisdom Teeth → Toothache Impacted Wisdom Teeth → X-ray shad Sinus Infection → Toothache Sinus infection → X-ray shad Sinus infection → Headache

5. CSE391-2005 PR 5 Rule-based Inferencing – finding confirmatory evidence Cavity ← Toothache Cavity → X-ray shad Gum disease ← Toothache Gum disease ↔ Red gums Impacted Wisdom Teeth ← Toothache Impacted Wisdom Teeth → X-ray shad Sinus Infection ← Toothache Sinus infection → X-ray shad Sinus infection ↔ Headache

6. CSE391-2005 PR 6 First Order Logic Fails because Exhaustive listings are too labor intensive and inflexible Medical theories are incomplete Knowledge of the patient’s condition is often incomplete

7. CSE391-2005 PR 7 Artificial Intelligence Meets the Real World Search and Reasoning work well for clearly-defined problems with –Reliable axioms, complete knowledge –Accessible, Static The Real World isn’t like that –Medical Diagnosis –Refinery Control –Taxi Driving

8. CSE391-2005 PR 8 Probabilistic* reasoning works because Don’t have to predetermine order of inferencing – Bayes “reverses arrows” Accommodates uncertainty and incomplete knowledge * AKA Bayesian reasoning, bayesian networks, belief networks

9. CSE391-2005 PR 9 What are Belief Nets Used for? Dental diagnosis example – which disease is causing a toothache? cavity gum disease impacted wisdom T. sinus inf headache toothache S. X-ray Sh. Red, swollen gums T. X-ray Sh.

10. CSE391-2005 PR 10 Probabilistic Reasoning Probabilities –Joint distributions –Conditional distributions –Chain Rule –Conditional Independence Belief networks –Conditional Probability Tables –Independence Relations –Inference

11. CSE391-2005 PR 11 Probability Theory P(A) prior, unconditional probability P(cavity) = 0.05 P(A|B) conditional probability The probability of A given that all we know is B P(cavity|toothache) =.8

12. CSE391-2005 PR 12 Axioms of Probability All probabilities are between 0 and 1 P(True) = 1, P(False) = 0 P(A \/ B) = P(A) + P(B) – P(A /\ B) derive P( ¬ A) = 1 - P(A)

13. CSE391-2005 PR 13 Random Variables A term whose value isn’t necessarily known –Discrete r.v – values from a finite set –Boolean r.v. – values from {true,false} –Continuous r.v. – values from subset of real line

14. CSE391-2005 PR 14 Random Variables (2) –Discrete s2_st “state of switch 2” {ok, upside-down, short, broken, intermittent} –Boolean 11_lt “light 11 is lit” {true,false} –Continuous current(w1) “current through wire 1” real value

15. CSE391-2005 PR 15 Probabilistic Causal Rules.40 Cavity → Toothache.05 Gum disease → Toothache.10 Impacted Wisdom Teeth → Toothache.45 Sinus Infection → Toothache Empirical evidence, (approximate reality?) Subjective (based on beliefs)

16. CSE391-2005 PR 16 Joint Probability Distributions cavity¬ cavity toothache 0.04 0.06 ¬toothache 0.01 0.89 P(cavity) =.04 +.01 =.05 P(cavity  toothache) =.04 +.01 +.06 =.11

17. CSE391-2005 PR 17 Probability Distributions Joint distribution is written: P(X 1, X 2, X 3, …, X N ) N-dimensional table with 2 N entries P(X 1 =a, X 2 =b, X 3 =c, …, X N =q) is one entry of table Shorthand: P(a, b, c, …, q)

18. CSE391-2005 PR 18 Finding a marginal probability P(cavity, toothache) =.04 P(cavity, ¬ toothache) =.01 P(cavity) =.05

19. CSE391-2005 PR 19 Conditional Probability P(h) prior, unconditional probability P(Cavity= true) = P(cavity) = 0.05 P(Cavity= false) = P(¬ cavity) = 0.95 P(h|e) conditional probability The probability of h given that all we know is e P(toothache|cavity) =.8

20. CSE391-2005 PR 20 Conditional Probability P(h|e) = P(h, e) P(e) P(cavity) =.05 P(cavity, toothache) =.04 P(toothache|cavity) =.04 =.80.05 Think of each variable assignment as a possible world. Of all of the possible worlds in which “cavity” is true, in 4 out of 5 “toothache” is also true.

21. CSE391-2005 PR 21 Conditional Probability Distributions cavity¬ cavity toothache 0.80 0.064 ¬ toothache 0.20 0.936 P(toothache|cavity) = P(toothache, cavity) P(cavity)

22. CSE391-2005 PR 22 Decomposing Conjunctions via Conditional Probabilities - Chain Rule P(f 1, f 2, …, f n ) = P(f 1 ) * P(f 2 | f 1 ) * P(f 3 | f 1, f 2 ) *... P(f n | f 1, f 2,..., f n-1 )

23. CSE391-2005 PR 23 Bayes’ Theorem Bayes’ Theorem relates conditional probability distributions: P(h | e) = P(e | h) * P(h) P(e) or with additional conditioning information: P(h | e, k) = P(e | h, k) * P(h | k) P(e | k)

24. CSE391-2005 PR 24 Proof of Bayes P(h, e) = P(h | e) * P(e) (chain rule) = P(e | h) * P(h) (chain rule) P(h | e) = P(h, e) = P(e | h)  P(h) P(e) P(e) P(h, e | k) = P(h | e, k) * P(e | k) (chain rule) P(h | e, k) = P(e | k,h)  P(h | k) P(e | k)