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##### MISTIS - 2013
Overall Objectives
Application Domains
Software and Platforms
Partnerships and Cooperations
Dissemination
Bibliography
Overall Objectives
Application Domains
Software and Platforms
Partnerships and Cooperations
Dissemination
Bibliography

## Section: Research Program

### Mixture models

Participants : Angelika Studeny, Thomas Vincent, Christine Bakhous, Senan James Doyle, Jean-Baptiste Durand, Florence Forbes, Aina Frau Pascual, Allessandro Chiancone, Stéphane Girard, Marie-José Martinez, Darren Wraith.

Key-words: mixture of distributions, EM algorithm, missing data, conditional independence, statistical pattern recognition, clustering, unsupervised and partially supervised learning.

In a first approach, we consider statistical parametric models, $\theta$ being the parameter, possibly multi-dimensional, usually unknown and to be estimated. We consider cases where the data naturally divides into observed data $y={y}_{1},...,{y}_{n}$ and unobserved or missing data $z={z}_{1},...,{z}_{n}$. The missing data ${z}_{i}$ represents for instance the memberships of one of a set of $K$ alternative categories. The distribution of an observed ${y}_{i}$ can be written as a finite mixture of distributions,

 $\begin{array}{c}\hfill f\left({y}_{i}\mid \theta \right)=\sum _{k=1}^{K}P\left({z}_{i}=k\mid \theta \right)f\left({y}_{i}\mid {z}_{i},\theta \right)\phantom{\rule{0.277778em}{0ex}}.\end{array}$ (1)

These models are interesting in that they may point out hidden variable responsible for most of the observed variability and so that the observed variables are conditionally independent. Their estimation is often difficult due to the missing data. The Expectation-Maximization (EM) algorithm is a general and now standard approach to maximization of the likelihood in missing data problems. It provides parameter estimation but also values for missing data.

Mixture models correspond to independent ${z}_{i}$'s. They have been increasingly used in statistical pattern recognition. They enable a formal (model-based) approach to (unsupervised) clustering.