BayesianCurveFitting

# BayesianCurveFitting - PATTERN RECOGNITION  ...

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Unformatted text preview: PATTERN RECOGNITION   AND MACHINE LEARNING  CHAPTER 1: INTRODUCTION  Example  Handwri/en Digit Recogni6on  Polynomial Curve Fi=ng    Sum‐of‐Squares Error Func6on  0th Order Polynomial  1st Order Polynomial  3rd Order Polynomial  9th Order Polynomial  Over‐ﬁ=ng  Root‐Mean‐Square (RMS) Error:  Polynomial Coeﬃcients    Data Set Size:   9th Order Polynomial  Data Set Size:   9th Order Polynomial  Regulariza6on  Penalize large coeﬃcient values  Regulariza6on:   Regulariza6on:   Regulariza6on:           vs.   Polynomial Coeﬃcients    Probability Theory  Marginal Probability  Joint Probability  Condi6onal Probability  Probability Theory  Sum Rule  Product Rule  The Rules of Probability  Sum Rule  Product Rule  Bayes’ Theorem  posterior ∝ likelihood × prior  Probability Theory  Apples and Oranges  4/10 6/10 Bayes’ Theorem  p(Z|X, Y) = p(Y|X, Z) p(Z|X) / p(Y|X)  p(Y|X) = sum over Z of  p(Y|X, Z) p(Z|X)   Probability Densi6es  P’(x) = p(x) Transformed Densi6es  Suppose x = g(y) Expecta6ons  Condi6onal Expecta6on  (discrete)  Approximate Expecta6on  (discrete and con6nuous)  Variances and Covariances  The Gaussian Distribu6on  Gaussian Mean and Variance  The Mul6variate Gaussian  Gaussian Parameter Es6ma6on  Likelihood func6on  Maximum (Log) Likelihood  Proper6es of          and   Curve Fi=ng Re‐visited  Maximum Likelihood  Determine            by minimizing sum‐of‐squares error,             .  Predic6ve Distribu6on  MAP: A Step towards Bayes  Determine               by minimizing regularized sum‐of‐squares error,             .  Bayesian Curve Fi=ng  Bayesian Predic6ve Distribu6on  ...
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## This note was uploaded on 02/27/2010 for the course ECE 544 taught by Professor Ray during the Spring '10 term at Rutgers.

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