## Section: Scientific Foundations

Keywords : mixture of distributions, EM algorithm, missing data, conditional independence, statistical pattern recognition, clustering, unsupervised and partially supervised learning (unsupervised learning, partially supervised learning).

### Mixture models

Participants : Caroline Bernard-Michel, Juliette Blanchet, Charles Bouveyron, Florence Forbes, Gersende Fort, Stéphane Girard, Matthieu Vignes.

In a first approach, we consider statistical parametric models,
being the parameter possibly multi-dimensional usually
unknown and to be estimated. We consider cases
where the data naturally divide 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 to one of a set of K alternative categories. The
distribution of an observed y_{i} can be written as a finite
mixture of distributions,

These models are interesting in that they may point out an 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 parameters estimation but also values
for missing data.

Mixture models correspond to independent z_{i} 's. They are more and more used
in statistical pattern recognition. They allow a formal (model-based)
approach to (unsupervised) clustering.