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 = X 1, … X n x = 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(x 1, … x n ) = f [ f 1 (x 1 ), …, f n (x n ) ]

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

6. Theorem: If attributes X 1, …, X n are MPI then the agent’s preference behavior can be described as maximizing the function V(x 1, …, x n ) = ∑ v i (x i ) where each v i is a value function referring only to the attribute X i

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, E j ) = VPI E (Ej) = Figure 16.7

11. Theorem:  j, E { VPI E (E j ) >= 0}, i.e. the value of information is non-negative Theorem: VPI E (E j, E k ) = VPI E (E j ) + VPI E,Ej (E k ) = VPI E (E k ) + VPI E,Ek (E j ), 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