Geoff McLachlan and Angus Ng. University of Queensland. Schlumberger Chaired Professor Univ. of Texas at Austin. + Chris Bishop
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1 EM Algorithm Geoff McLachlan and Angus Ng Department of Mathematics & Institute for Molecular Bioscience University of Queensland Adapted by Joydeep Ghosh Schlumberger Chaired Professor Univ. of Texas at Austin + Chris Bishop
2 Wiipedia An expectation-maximization (EM) algorithm is used in statistics for finding maximum lielihood estimates of parameters in probabilistic models, where the model depends on unobserved latent variables. EM alternates between performing an expectation (E) step, which computes an expectation ti of the lielihood lih by including the latent t variables as if they were observed, and a maximization (M) step, which computes the maximum lielihood estimates of the parameters by maximizing the expected lielihood found on the E step. The parameters found on the M step are then used to begin another E step, and the process is repeated.
3 Fitting Mixture Models Fitting Mixture Models K c p p = ), ( ) ( x x K c p c p p p = = 1 ) ( ) ( ) ( ) ( x K c p c p c p α θ = = 1 ) ( ) ( ) ( x x c p α θ = = 1, ) (x Weight C t Component Model parameters e.g. Mixture of two 1-Dimensional Gaussians: e.g. Mixture of two 1 Dimensional Gaussians: Fit parameters {θ, α} = {μ1, σ1, μ2, σ2, α1}
4 2 1.5 Component Models p( (x) Mixture Model p(x) x
5 Estimation of Mixture Distributions 1960s: several studies on fitting of finite mixture models by maximum lielihood (ML) Day (1969) and Wolfe (1965, 1967, 1970). Arthur Dempster, Nan Laird, and Donald Rubin. "Maximum lielihood from incomplete data via the EM algorithm". Journal of the Royal Statisticalti ti Society, Series B, 39(1):1 38, Viewed the problem as ML estimation from incomplete data - the latent (indicator) variables values are missing.
6 Complete and Incomplete Data (a) complete (b) incomplete
7 Example: Mixture of 3 Gaussians (a)
8 EM overview Observed data X ; Missing data: Z Often these are missing indicator functions complete data: Y = {X, Z} complete lielihood: l c (θ, Y) E step: (average out randomness, to get function Q) Q (Θ, Θ (t) ) = E [l c (θ, Z) X, Θ (t) ] X, Θ (t) : constants, θ : normal variable, Z: random variable a deterministic version of Gibbs sampling M step: maximize Q Θ (t + 1) = arg max Θ Q (Θ Θ (t) ) Showed Convergence of this batch-iterative process to local maximum.
9 Technique Summary: Estimate (expected value of) missing variables; Maximum Lielihood: Solve for model parameters; re-estimate t and iterate t Applications: Mixture density functions ( soft -means) Baum-Welch algorithm for HMMs Bayesian inferencing with latent variables Mixture-of-X, e.g. Mixture of experts for function approximation Mixture of probabilistic principal p components Motion segmentation Mixture of vmf distributions
10 Gaussian Mixtures Linear super-position of Gaussians Normalization and positivity require Log-lielihood
11 E-M E-step: get posterior probabilities of component membership M-step:
12 Fitting Mixtures of Exponentials (Bregman Soft Clustering, Banerjee et al, JMLR 2005)
13 Common Algorithm
14 Impact of EM Algorithm Model-based methods of clustering implemented via the EM algorithm are now being used extensively in practice. Clustering methods based on such mixture models allow estimation and hypothesis testing within the framewor of standard statistical theory.
15 Additional References The EM algorithm and Extensions, by G.J. McLachlan and T. Krishnan,(1997), Wiley A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Marov Models, by J. Bilmes Radford Neal, Geoffrey Hinton. "A view of the EM algorithm that justifies incremental, sparse, and other variants". In Michael li. Jordan (editor), Learning in Graphical Models pp Cambridge, MA: MIT Press, David Dowe's clustering, mixture modelling and unsupervised learning page. NETLAB toolbo NETLAB toolbox
16 Extensions of EM Algorithn Variants of EM algorithm for models more complex than normal models (ECM, ECME, AECM variants of EM) Incremental EM algorithm (E- and M-steps performed for blocs of observations at a time) AECM algorithm for fitting mixtures of factor analyzers models to high-dimensional data
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