1. ETHEM ALPAYDIN © The MIT Press, 2010 alpaydin@boun.edu.tr http://www.cmpe.boun.edu.tr/~ethem/i2ml2e Lecture Slides for

3. Rationale Bayes’ Rule: Generative model: 3Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

4. Estimating the Parameters of a Distribution: Discrete case x t i =1 if in instance t is in state i, probability of state i is q i Dirichlet prior,  i are hyperparameters Sample likelihood Posterior Dirichlet is a conjugate prior With K=2, Dirichlet reduced to Beta 4 Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

5. 5 Estimating the Parameters of a Distribution: Continuous case p(x t )~N( ,  2 ) Gaussian prior for , p(  )~ N(  ,   2 ) Posterior is also Gaussian p(  X)~ N (  N,  N 2 ) where

6. 6Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0) 6 Estimating the Parameters of a Function: Regression r=w T x+  where p(  )~N(0,1/  ), and p(r t |x t,w,  ) ~ N(w T x t,  1/  ) Log likelihood ML solution Gaussian conjugate prior: p(w)~N(0,1/  ) Posterior: p(w|X)~N(  N  N  where

7. 7Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

8. 8 8 8 Basis/Kernel Functions For new x’, the estimate r’ is calculated as Linear kernel For any other  (x), we can write K(x’,x)=  (x’) T  (x) Dual representation

9. Kernel Functions 9Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0)

10. 10 Lecture Notes for E Alpaydın 2010 Introduction to Machine Learning 2e © The MIT Press (V1.0) 10 Gaussian Processes Assume Gaussian prior p(w)~N(0,1/  ) y=Xw, where E[y]=0 and Cov(y)=K with K ij = (x i ) T x i K is the covariance function, here linear With basis function  (x), K ij = (  (x i )) T  (x i ) r~N N (0,C N ) where C N = (1/  )I+K With new x’ added as x N+1, r N+1 ~N N+1 (0,C N+1 ) where k = [K(x’,x t ) t ] T and c=K(x’,x’)+1/ . p(r’|x’,X,r)~N(k T C N-1 r,c-k T C N-1 k)

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