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Chapter 13

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 too much work to list antecedents/consequents theoretical ignorance

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Chapter 13

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  1. Chapter 13 February 19, 2004

  2. 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 • too much work to list antecedents/consequents • theoretical ignorance • practical ignorance

  3. 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

  4. 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).

  5. 13.2 Basic Probability Notation • Proposition Logic • Random variable, i.e. Cavity • Domain of values • boolean <true, false> • discrete • continuous • Connectives • and • or • not

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

  7. Prior, Discrete • Probability, P(cavity) • Probability Distribution, P(weather) = <0.2, 0.3, 0.5> • Joint Probability Distribution, P(Cavity, Weather) • Full Joint Probability Distrubution, P(all random variables)

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

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

  10. 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

  11. 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)

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