CHAPTER 3 Bayesian Decision Theory Souce Alpaypin with

CHAPTER 3: Bayesian Decision Theory Souce: Alpaypin with modifications by Christoph F. Eick; last changed on February 4, 2009. Remark: Belief Networks will be covered Early April 2011. Utility theory will be covered as port of reinforcement learning.

Probability and Inference n n Result of tossing a coin is Î {Heads, Tails} Random var X Î{1, 0} Bernoulli: P {X=1} = po Sample: X = {xt }Nt =1 Estimation: po = # {Heads}/#{Tosses} = ∑t xt / N Prediction of next toss: Heads if po > ½, Tails otherwise In theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". P(X=k)= 2

Binomial Distribution 3 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V 1. 1)

Classification n Credit scoring: Inputs are income and savings. Output is low-risk vs high-risk Input: x = [x 1, x 2]T , Output: C Î {0, 1} Prediction: 4 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V 1. 1)

Bayes’ Rule prior likelihood posterior evidence 5

Bayes’ Rule: K>2 Classes 6 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V 1. 1)

Losses and Risks n Actions: αi Loss of αi when the state is Ck : λik n Expected risk (Duda and Hart, 1973) n 7

Losses and Risks: 0/1 Loss For minimum risk, choose the most probable class Remark: This strategy is not optimal in other cases 8

Example and Homework! C 1=has cancer C 2=has not cancer 12=9 21=90 Homework: a) Determine the optimal decision making strategy Inputs: P(C 1|x), P(C 2|x) Decision Making Strategy: … b) Now assume we also have a reject option and the cost for making no decision are 3: reject, 2=3 reject, 1=3 Inputs: P(C 1|x), P(C 2|x) Decision Making Strategy: … Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V 1. 1) 9

Losses and Risks: Reject Risk for reject 10

Discriminant Functions K decision regions R 1, . . . , RK 11 Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V 1. 1)