Chapter 13 February 19, 2004

13.1 Acting Under Uncertainty Rational Decision – Depends on the relative importance of the goals and the likelihood of their achievability First Order Logic is not appropriate 1.too much work to list antecedents/consequents 2.theoretical ignorance 3.practical ignorance

Probability – Summarizes uncertainty from laziness or ignorance, it is a “degree of belief”, not a “degree of truth”. Fuzzy logic is designed for “degree of truth”. Prior (unconditional) probability Posterior (conditional) probability

Utility Theory – Evaluates the usefulness of a state. It can be used to represent and reason with preferences about outcomes. Decision Theory – Probability Theory + Utility Theory. A rational agent seeks the maximum expected utility (MEU).

13.2 Basic Probability Notation Proposition Logic –Random variable, i.e. Cavity –Domain of values boolean discrete continuous –Connectives and or not

Atomic Event: Complete specification of a state –mutually exclusive –set of all atomic events is exhaustive –entails truth or falsehood of any proposition

Prior, Discrete Probability, P(cavity) Probability Distribution, P(weather) = Joint Probability Distribution, P(Cavity, Weather) Full Joint Probability Distrubution, P(all random variables)

Prior, Continuous Probability Density Function, P(X = x) = U[2000, 2010] (x)

Conditional P(a | b ) = P(a  b) / P(b) P(a  b) = P(a | b) * P(b) = P(b | a) * P(a) “product rule”

13.3 Axioms of Probability 0 <= P(a) <= 1 P(false) = 0, P(true) = 1 P(a or b) = P (a) + P(b) – P(a  b) de Finetti Theorem: If an agent’s beliefs violate probability theory, then the agent will not make rational decisions

13.4 Inference Using Full Joint Distributions Marginal Probability, P(cavity) Marginalization, P(Y) = ∑ P(Y, z) Conditioning P(Y) = ∑ P(Y | z ) * P (z) Normalization Constant,  P(c | t ) = P(c  t) / P(t)