## Section: Scientific Foundations

### Mixture models

Participants : Lamiae Azizi, Christine Bakhous, Lotfi Chaari, Senan James Doyle, Jean-Baptiste Durand, Florence Forbes, Stéphane Girard, Marie-José Martinez, Darren Wraith.

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 are increasingly used in statistical pattern recognition. They enable a formal (model-based) approach to (unsupervised) clustering.