1. 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 Utility theory will be covered as port of reinforcement learning.

2. Probability and Inference
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 the 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)=

3. Binomial Distribution
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)

4. Classification Credit scoring: Inputs are income and savings.
Output is low-risk vs high-risk
Input: x = [x1,x2]T ,Output: C Î {0,1}
Prediction:
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)

5. Bayes’ Rule
prior
likelihood
posterior
evidence

6. Bayes’ Rule: K>2 Classes
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)

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

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

9. Example and Homework! C1=has cancer C2=has not cancer 12=9 21=90 Homework: a) Determine the optimal decision making strategy Inputs: P(C1|x), P(C2|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(C1|x), P(C2|x) Decision Making Strategy: …
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)

10. Losses and Risks: Reject
Risk for
reject

11. Discriminant Functions
K decision regions R1,...,RK
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)