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Parameter Estimation: Maximum Likelihood Estimation Chapter 3 (Duda et al.) – Sections 3.1-3.2 CS479/679 Pattern Recognition Dr. George Bebis

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Parameter Estimation Bayesian Decision Theory allows us to design an optimal classifier given that we know P( i ) and p(x/ i ): Estimating P( i ) is usually not difficult. Estimating p(x/ i ) is more difficult: – Number of samples is often too small – Dimensionality of feature space is large.

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Assumptions – A set of training samples D ={x 1, x 2,...., x n }, where the samples were drawn according to p(x| j ). – p(x| j ) has some known parametric form: Parameter estimation problem: Parameter Estimation (cont’d) Given D, find the best possible also denoted as p(x / ) where =(μ i, Σ i ) e.g., p(x / i ) ~ N(μ i, i )

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Main Methods in Parameter Estimation Maximum Likelihood (ML) Bayesian Estimation (BE)

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Main Methods in Parameter Estimation Maximum Likelihood (ML) – Assumes that the values of the parameters are fixed but unknown. – Best estimate is obtained by maximizing the probability of obtaining the samples x 1,x 2,..,x n actually observed (i.e., training data):

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Main Methods in Parameter Estimation (cont’d) Bayesian Estimation (BE) – Assumes that the parameters θ are random variables that have some known a-priori distribution p(θ . – Estimates a distribution rather than making point estimates like ML: Note: the BE solution might not be of the parametric form assumed!

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ML Estimation - Assumptions Let us assume c classes and that the training data consists of c sets (i.e., one for each class): Samples in D j have been drawn independently according to p(x/ω j ). p(x/ω j ) has known parametric form with parameters j : e.g., j =(μ j, Σ j ) for Gaussian distribution D 1, D 2,...,D c

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ML Estimation - Problem Formulation and Solution Problem: given D 1, D 2,...,D c and a model for each class, estimate If the samples in D j give no information about i ( ), we need to solve c independent problems (i.e., one for each class) The ML estimate for D={x 1,x 2,..,x n } is the value that maximizes p(D / ) (i.e., best supports the training data). 1, 2,…, c (using independence assumption)

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ML Estimation - Problem Definition and Solution (cont’d) How should we find the maximum of p(D/ ) ? where

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ML Estimation Using Log-Likelihood Consider the log-likelihood for simplicity: The solution maximizes ln p(D/ θ)

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ML Estimation Using Log-Likelihood (cont’d) ln p(D/ θ) p(D / θ) =μ=μ=μ=μ training data, unknown mean, known variance

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ML for Multivariate Gaussian Density: Case of Unknown θ=μ Consider Computing the gradient, we have

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Setting we have: The solution is given by The ML estimate is simply the “sample mean”. ML for Multivariate Gaussian Density: Case of Unknown θ=μ (cont’d)

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Special Case of ML: Maximum A-Posteriori Estimator (MAP) Assume that θ is a random variable with known p(θ). Maximize p(θ/D) or p(D/θ)p(θ) or ln p(D/ θ)p(θ): Consider:

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Special Case of ML: Maximum A-Posteriori Estimator (MAP) What happens when p(θ) is uniform? MAP is equivalent to ML

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MAP for Multivariate Gaussian Density: Case of Unknown θ=μ Assume MAP maximizes ln p(D/ μ)p(μ): maximize where (known)

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MAP for Multivariate Gaussian Density: Case of Unknown θ=μ (cont’d) If, then What happens when

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ML for Univariate Gaussian Density: Case of Unknown θ=(μ,σ 2 ) Assume p(x k /θ) θ =(θ 1,θ 2 )=(μ,σ 2 )

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ML for Univariate Gaussian Density: Case of Unknown θ=(μ,σ 2 ) (cont’d) =0 p(x k /θ)=0 The solutions are given by: =0 sample mean sample variance

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ML for Multivariate Gaussian Density: Case of Unknown θ=(μ,Σ) In the general case (i.e., multivariate Gaussian) the solutions are: sample mean sample covariance

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Biased and Unbiased Estimates An estimate is unbiased when where θ is the true value. The ML estimate is unbiased, i.e., The ML estimate and is biased:

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Biased and Unbiased Estimates (cont’d) The following are unbiased estimates of and

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Comments about ML ML estimation is usually simpler than alternative methods. It provides more accurate estimates as the number of training samples increases. If the model chosen for p(x/ θ) is correct, and independence assumptions among samples are true, ML will give very good results.

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