Decision Theory. What is it? Alexei A. Borissov February 1, 2007

Transcription

1 Decision Theory What is it? Alexei A. Borissov February 1, 2007

2 Agenda Background History Bayes Theorem Example of Bayes Theorem Expected Utility Expected Utility Example Further Research Conclusion

3 Background A branch of statistical theory concerned with quantifying the process of making choices between alternatives 1 Merriam-Webster Trivial (except for the mathematics) Happens everyday Ex: Making Breakfast Large field of application Political Issues -> Club of Rome Economics -> Invest or Spend Game Programmers -> Optimal Move

4 History I Marquis de Condorcet (1700s) and the 3 steps (French Constitution) 1. Discuss (brainstorm) 2. Combine (political parties) 3. Choose (debate)

5 History II (Sequential Model) Influenced by Brim et al. (1962) 1. Identification of the Problem 2. Obtaining Required Information 3. Production of Possible Solutions 4. Evaluation of Possible Solutions 5. Selecting Best Strategy

6 History III (Non-Sequential Model) Mintzberg, Raisinghani, Theoret (1976) Identification Decision Recognition -> Identify Diagnosis -> Attempt to clarify Development Search -> Attempt to find existing solutions Design -> Develop new solutions / modify existing Selection Screen -> Cut down number of solutions to evaluate Evaluation/Choice -> Choose Authorization -> Approval

7 Bayes Theorem (BT) Consists of 4 principles Coherent set of probabilistic beliefs Incoherent: (rain =.5) and (rain or snow =.6) Complete set of probabilistic beliefs Each proposition has a probability Update beliefs with new information Rain tomorrow and day after Choose option with highest utility Minimized uncertainty -> Maximized utility

8 Bayes Theorem (BT) p(c k x) -> A posteriori probability for category c k (revised beliefs in light of new evidence) p(c k ) -> A priori probability (before event occurred) p(x c k ) -> Probability of x given c k p(x) -> Probability density of x

9 BT Example (Problem) Scenario: Equal bowls & cookies (except flavour) Bowl 1: 10 chocolate & 30 plain cookies Bowl 2: 20 chocolate & 20 plain cookies Outcome: From a randomly chosen bowl and randomly chosen cookie, John gets a plain cookie. Question: What is the probability it came from Bowl 1?

10 BT Example (Solution) Event A: John picked Bowl 1 Event B: John picked a plain cookie Need: Pr(A) = 0.5 Bowls are equal -> 50/50 chance Pr(B) = (Pln_Cookie Bowl1 * Bowl_1) + (Pln_Cookie Bowl2 * Bowl_2) = [ (30/40) * (1/2) ] + [ (20/40) * (1/2) ] = 0.625

11 BT Example (Solution) Event A: John picked Bowl 1 Event B: John picked a plain cookie Need: Pr(A) = 0.5 Bowls are equal -> 50/50 chance Pr(B) = (Pln_Cookie Bowl1 * Bowl_1) + (Pln_Cookie Bowl2 * Bowl_2) = [ (30/40) * (1/2) ] + [ (20/40) * (1/2) ] = Pr(A B) = (Pr(B A) * Pr(A)) / Pr(B) = (0.75 * 0.5) / =.60 -> 60%

12 Expected Utility (EU) Obtain highest Expected Utility from: Possible action Current World State Probability affected by level of: Certainty Deterministic knowledge Risk Probabilistic knowledge Uncertainty Partial probabilistic knowledge (perhaps ignorable) Ignorance No knowledge

13 Expected Utility (EU) Maximize Expected Utility over a a -> Possible action x -> Current world state U(x,a) -> Resulting utility from doing a when x P(x a) -> Probability distribution

14 EU Example No baggage Dry clothes Umbrella No Umbrella Decision Matrix Rain No Rain

15 EU Example No baggage Dry clothes Decision Matrix Rain Umbrella 15 No Rain No Umbrella

16 EU Example No baggage Dry clothes Decision Matrix Rain No Rain Umbrella No Umbrella

17 EU Example No baggage Dry clothes Decision Matrix Rain No Rain Umbrella No Umbrella 0

18 EU Example No baggage Dry clothes Decision Matrix Rain No Rain Umbrella No Umbrella 0 18

19 EU Example Decision Matrix Rain No Rain Umbrella No Umbrella 0 18 Rain Probability =.1 Umbrella: MU =.1*15+.9*15 = 15 No umbrella: MU =.1*0+.9*18 = 16.2 Note: Max Utility not best

20 EU Example Decision Matrix Rain No Rain Umbrella No Umbrella 0 18 Rain Probability =.1 Umbrella: MU =.1*15+.9*15 = 15 No umbrella: MU =.1*0+.9*18 = 16.2 Rain Probability =.5 Umbrella: MU =.5*15+.5*15 = 15 No umbrella: MU =.5*0+.5*18 = 9 Note: Max Utility not best

21 Expected Utility Variations of EU Regret Theory 2 attributes: EU and Quantity of Regret (QR) Value received from decision vs. highest level from alternative Prospect Theory Another variation with 3 stages focused on money

22 Further Research Distributions Decision making under uncertainty Decision making under ignorance Decision instability

23 References Hansson, S. (1994). Decision Theory A Brief Introduction (2005) MacKay. Decision Theory 1 Definition of Decision Theory (January 31, 2007).