- flexible density;

- non parametric estimation of a mixing distribution; - unsupervised classification.

Maximum likelihood estimation

EM algorithm;

“Fuzzy” interpretation of EM;

ML estimation of a mixture of Gaussians.

Mixture of linear regression models

Omitted variables; Random effects; Heterogeneity; EM algorithm.

Latent class analysis

Latent variables models;

Latent class models for binary variables.

How to choose the number of components?

LR test;


Automatic selection criteria.

Principal Stratification for casual inference


ML estimation; Examples.


Frangakis, C. E., Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics, 58 21– 29.

Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer, New York. McLachlan , G.J., Peel. D. (2000). Finite Mixture Models. New York: Wiley.