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Medical Applications of Finite Mixture ModelsTheory and Algorithms

Medical Applications of Finite Mixture Models: Theory and Algorithms Chapter 4 4.1 The Likelihood of Finite Mixture Models Estimation of the parameters of the mixing distribution P is predominantly done using maximum likelihood. Given a sample of iid x ∼ f (x|P), i = 1,..., n, (4.1) we are interested in finding the maximum likelihood estimates (MLEs) of P, denoted as P, that is P = arg max L(P), n k L(P)= f (x , λ ) p (4.2) i j j ∏ ∑ i=1 j=1 or alternatively finding the estimates of P which maximize the log likelihood function n k (P)= log L(P)= log f (x , λ ) p . (4.3) i j j ∑ ∑ i=1 j=1 An estimate of P can be obtained as a solution to the likelihood equation ∂ (P) S(x, P)= = 0, (4.4) ∂ P where S(x, P) is the gradient vector of the log likelihood function, where differenti- ation is with respect to the parameter vector P. Maximum likelihood estimation of P is by no means trivial, since there are mostly no closed-form solutions available. P. Schlattmann, Medical Applications of Finite Mixture Models,55 Statistics for Biology and Health, DOI: 10.1007/978-3-540-68651-4 4, Springer-Verlag Berlin Hiedelberg 2009 56 4 Theory and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Medical Applications of Finite Mixture ModelsTheory and Algorithms

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Publisher
Springer Berlin Heidelberg
Copyright
© Springer-Verlag Berlin Heidelberg 2009
ISBN
978-3-540-68650-7
Pages
1 –51
DOI
10.1007/978-3-540-68651-4_4
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Abstract

Chapter 4 4.1 The Likelihood of Finite Mixture Models Estimation of the parameters of the mixing distribution P is predominantly done using maximum likelihood. Given a sample of iid x ∼ f (x|P), i = 1,..., n, (4.1) we are interested in finding the maximum likelihood estimates (MLEs) of P, denoted as P, that is P = arg max L(P), n k L(P)= f (x , λ ) p (4.2) i j j ∏ ∑ i=1 j=1 or alternatively finding the estimates of P which maximize the log likelihood function n k (P)= log L(P)= log f (x , λ ) p . (4.3) i j j ∑ ∑ i=1 j=1 An estimate of P can be obtained as a solution to the likelihood equation ∂ (P) S(x, P)= = 0, (4.4) ∂ P where S(x, P) is the gradient vector of the log likelihood function, where differenti- ation is with respect to the parameter vector P. Maximum likelihood estimation of P is by no means trivial, since there are mostly no closed-form solutions available. P. Schlattmann, Medical Applications of Finite Mixture Models,55 Statistics for Biology and Health, DOI: 10.1007/978-3-540-68651-4 4, Springer-Verlag Berlin Hiedelberg 2009 56 4 Theory and

Published: Dec 8, 2008

Keywords: Mixture Model; Convex Hull; Bayesian Information Criterion; Expectation Maximization; Directional Derivative

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