CSCI 121 Special Topics: Bayesian Network

Lecture #1: Reasoning Under Uncertainty

UncertaintyTraditional models of reasoning (human; computer) use “all-or-nothing” (discrete) variables and rules:

Hungry(Fido)

Toothache(Simon)

Toothache(X) → Cavity(X)

• Reality is usually more complicated:

Toothache(X) → Cavity(X) 70% of the time

Toothache(X) → Gingivitis(X) 20% of the time

Uncertainty

In general, all-or-nothing rules fail for three reasons:

1. Laziness – we don't have enough time or resources

list all such rules for a given domain.

2. Theoretical ignorance - we don't have a complete

theory of the domain.

3. Practical ignorance – even with all rules and perfect

theory, we can't make the necessary observations.

Uncertainty and Rational Decisions

• Utility – how useful is a particular outcome to the agent?

• Probability – how likely is a particular outcome?

• Utility + Probability = Decision theory

• E.g., Lottery: High utility ($$$) x extremely low

probability → bad decision!

Basic Probability• Prior probability – how likely is something, without any other knowledge? P(cavity) = 0.05

• Conditional (posterior) probability – how likely

is something, once you know something else?

• P(toothache|cavity) = 0.7

• Product Rule: • P(A|B) = P(A & B) / P(B)• P(A &B) = P(A|B) * P(B) = P(B|A)* P(A)

Basic Probability

• Probability Distribution: All possible values of a given variable, and their probabilities (sum = 1):

•cavity=0.8; gingivitis = 0.1; abcess = 0.05; ? = 0.05

• Joint probability: How likely is it that two things

occur (are observed) together?

• rainy & cloudy = 0.3; cloudy & cool = 0.4

Axioms of Probability

1) All probabilities are between 0 and 1.

2) Necessarily true propositions (A V ~A) have prob 1; necessarily false (A & ~A) have prob. 0. 3) P(A V B) = P(A) + P(B) – P(A & B)

Axioms of Probability P(A V B) = P(A) + P(B) – P(A & B)

A BA &

BE.g., in Los Angeles, maybe P(sunny) = 0.8; P(warm) = 0.7. Since P is always less than 1, can't just add 08 + 0.7 to get P(sunny V warm). Need to subtract P(sunny & warm) = 0.6 to get P(sunny V warm) = 0.9.

Bayes’ Rule

• From the Product Rule:

• P(A|B) = P(A & B) / P(B)

• P(A &B) = P(A|B) * P(B) = P(B|A) * P(A)

Rev. Thomas Bayes (1702-1761)

• We derive Bayes’ Rule by substitution:

• P(A|B) = P(A & B) / P(B) = P(B|A) * P(A) / P(B)

Bayesian (“Belief”) Nets

Burglary Earthquake

Alarm

JohnCalls MaryCallsA P(J) T .90 F .05

A P(M)T .70F .01

B E P(A)T T .95T F .94F T .29F F .001

P(B).001

P(E).002

Bayesian Nets Using recently developed techniques (Pearl 1982), we can ask, e.g., “how likely is there to be a burglary, given that John has called?”

Can also learn relationships, creating

“hidden” variables and probability tables,

based on observations.

Current “hot topic” in AI.