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

CSE 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

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

CSE 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

CSE 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

CSE 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

CSE 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

CSE 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

CSE 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.

CSE PR 10 Probabilistic Reasoning Probabilities –Joint distributions –Conditional distributions –Chain Rule –Conditional Independence Belief networks –Conditional Probability Tables –Independence Relations –Inference

CSE 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

CSE 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)

CSE 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

CSE 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

CSE 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)

CSE PR 16 Joint Probability Distributions cavity¬ cavity toothache ¬toothache P(cavity) = =.05 P(cavity  toothache) = =.11

CSE 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)

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

CSE 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

CSE PR 20 Conditional Probability P(h|e) = P(h, e) P(e) P(cavity) =.05 P(cavity, toothache) =.04 P(toothache|cavity) =.04 = 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.

CSE PR 21 Conditional Probability Distributions cavity¬ cavity toothache ¬ toothache P(toothache|cavity) = P(toothache, cavity) P(cavity)

CSE 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 )

CSE 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)

CSE 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)