## 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,

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