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Navigating Uncertainty: Rational Decision Making in Probability and Utility Theory

This chapter explores rational decision-making under uncertainty, emphasizing Probability Theory and Utility Theory. Learn about Basic Probability Notation, Axioms of Probability, Inference Using Full Joint Distributions, and more. Understand how to evaluate outcomes and calculate expected utility to make informed choices.

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Navigating Uncertainty: Rational Decision Making in Probability and Utility Theory

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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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