Section: Scientific Foundations

Mixture models

Participants : Lamiae Azizi, Senan James Doyle, Jean-Baptiste Durand, Florence Forbes, Gersende Fort, Stéphane Girard, Vasil Khalidov, Darren Wraith, Marie-José Martinez.

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 = y₁, ..., y_n and unobserved or missing data z = z₁, ..., 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,

(1)

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.