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Chapter 16

Chapter 16. March 25, 2004. Probability Theory: What an agent should believe based on the evidence Utility Theory: What the agent wants Decision Theory: Combines the above two theories to decide what the agent should do. 16.4 Multiattribute Utility Functions. X = X 1 , … X n

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Chapter 16

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  1. Chapter 16 March 25, 2004

  2. Probability Theory: What an agent should believe based on the evidence • Utility Theory: What the agent wants • Decision Theory: Combines the above two theories to decide what the agent should do

  3. 16.4 Multiattribute Utility Functions • X = X1, … Xn • x = <x1, … xn> • By convention, higher values mean higher utilities • Strict Dominance, Figure 16.3 (no uncertainty) • Stochastic Dominance, Figure 16.4

  4. If A stochastically dominates B, then for any monotonically nondecreasing utility function U(X), the expected utility of A is at least as high as the expected utility of B. • U(x1, … xn) = f [ f1(x1), …, fn(xn) ]

  5. Definition: Two attributes X and Y are preferentially independent of a third attribute Z if the preference between outcomes <x, y, z> and <x’, y’, z> does not depend on z. • Definition: Mutual Preferential Independence (MPI)

  6. Theorem: If attributes X1, …, Xn are MPI then the agent’s preference behavior can be described as maximizing the function V(x1, …, xn) = ∑ vi (xi) where each vi is a value function referring only to the attribute Xi

  7. 16.5 Decision Networks • Augmented Bayesian Networks • Figure 16.5 • Components • chance nodes (ovals) represent random variables • decision nodes (rectangles) represent choices of decision maker • utility nodes (diamonds) represent utility func.

  8. Evaluation of Network • Set evidence variables for current state • For each possible value of decision node • Set decision node to value • Calculate posterior probability for parent nodes of utility node • Calculate resulting utility

  9. 16.6 The Value of Information • Typically, not everything is known. • Information Value Theory: Helps the agent decide what information to acquire • VPI: Value of Perfect Information

  10. Information has value to the extent that it is likely to cause a change of plan to the extent that the new plan will be significantly better than the old plan • EU( | E) =  is current best action • EU(Ej | E, Ej) = • VPIE(Ej) = Figure 16.7

  11. Theorem: j, E { VPIE(Ej) >= 0}, i.e. the value of information is non-negative • Theorem: VPIE(Ej, Ek) = VPIE(Ej) + VPIE,Ej(Ek) = VPIE(Ek) + VPIE,Ek(Ej), i.e. collecting evidence is order independent • Figure 16.8, myopic information gathering agent

  12. 16.7 Decision Theoretic Expert Systems • Create causal model • Simplify (Figure 16.9) • Assign probabilities • Assign utilities • Verify and refine model • Perform sensitivity analysis

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