INTRODUCTION TO Machine Learning ETHEM ALPAYDIN © The MIT Press, Lecture Slides for

CHAPTER 3: Bayesian Decision Theory

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 3 Probability and Inference Result of tossing a coin is  {Heads,Tails} Random var X  {1,0} Bernoulli: P {X=1} = p o X (1 ‒ p o ) (1 ‒ X) Sample: X = {x t } N t =1 Estimation: p o = # {Heads}/#{Tosses} = ∑ t x t / N Prediction of next toss: Heads if p o > ½, Tails otherwise

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 = [x 1,x 2 ] T,Output: C  {0,1} Prediction:

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 5 Bayes’ Rule posterior likelihoodprior evidence

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 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: α i Loss of α i when the state is C k : λ ik Expected risk (Duda and Hart, 1973)

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 8 Losses and Risks: 0/1 Loss For minimum risk, choose the most probable class

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 9 Losses and Risks: Reject

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 10 Discriminant Functions K decision regions R 1,...,R K

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 11 K=2 Classes Dichotomizer (K=2) vs Polychotomizer (K>2) g(x) = g 1 (x) – g 2 (x) Log odds:

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 12 Utility Theory Prob of state k given exidence x: P (S k |x) Utility of α i when state is k: U ik Expected utility:

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 13 Value of Information Expected utility using x only Expected utility using x and new feature z z is useful if EU (x,z) > EU (x)

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 14 Bayesian Networks Aka graphical models, probabilistic networks Nodes are hypotheses (random vars) and the prob corresponds to our belief in the truth of the hypothesis Arcs are direct direct influences between hypotheses The structure is represented as a directed acyclic graph (DAG) The parameters are the conditional probs in the arcs (Pearl, 1988, 2000; Jensen, 1996; Lauritzen, 1996)

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 15 Causes and Bayes’ Rule Diagnostic inference: Knowing that the grass is wet, what is the probability that rain is the cause? causal diagnostic

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 16 Causal vs Diagnostic Inference Causal inference: If the sprinkler is on, what is the probability that the grass is wet? P(W|S) = P(W|R,S) P(R|S) + P(W|~R,S) P(~R|S) = P(W|R,S) P(R) + P(W|~R,S) P(~R) = = 0.92 Diagnostic inference: If the grass is wet, what is the probability that the sprinkler is on? P(S|W) = 0.35 > 0.2 P(S) P(S|R,W) = 0.21Explaining away: Knowing that it has rained decreases the probability that the sprinkler is on.

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 17 Bayesian Networks: Causes Causal inference: P(W|C) = P(W|R,S) P(R,S|C) + P(W|~R,S) P(~R,S|C) + P(W|R,~S) P(R,~S|C) + P(W|~R,~S) P(~R,~S|C) and use the fact that P(R,S|C) = P(R|C) P(S|C) Diagnostic: P(C|W ) = ?

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 18 Bayesian Nets: Local structure P (F | C) = ?

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 19 Bayesian Networks: Inference P (C,S,R,W,F) = P (C) P (S|C) P (R|C) P (W|R,S) P (F|R) P (C,F) = ∑ S ∑ R ∑ W P (C,S,R,W,F) P (F|C) = P (C,F) / P(C)Not efficient! Belief propagation (Pearl, 1988) Junction trees (Lauritzen and Spiegelhalter, 1988)

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 20 Bayesian Networks: Classification diagnostic P (C | x ) Bayes’ rule inverts the arc:

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 21 Naive Bayes’ Classifier Given C, x j are independent: p(x|C) = p(x 1 |C) p(x 2 |C)... p(x d |C)

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 22 Influence Diagrams chance node decision node utility node

Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 23 Association Rules Association rule: X  Y Support (X  Y): Confidence (X  Y): Apriori algorithm (Agrawal et al., 1996)