The Ph.D. course on Discrete latent variable models introduces the theory of latent variables models for the analysis of cross-sectional and longitudinal data with both categorical and continuous variables arising from different fields. The lectures will be devoted to summarizing the most important aspect of this modeling framework: assumptions, advanced parameterizations, estimation, and inferential procedures also considering the computational issues. During the course, case studies and applications will be presented mainly by using different R libraries.

Schedule of the main arguments:

• Introduction to latent variable models with a focus on discrete latent variables. Maximum likelihood estimation using the expectation-maximization algorithm.
• Basic features of the latent class model with reference to the estimation methods and to the expectation-maximization algorithm. More advanced versions of the latent class model. Introduction to the R package MultiLCIRT.
• Basic features of the hidden Markov models for continuous and categorical longitudinal data. Introduction to the R package LMest.
• Recent hidden Markov model formulations and extensions with multivariate data, covariates and more complex data. Recent case studies and applications.
• Introduction to stochastic block models for network data and to the approximate maximum likelihood estimation using the variational EM algorithm. Introduction to the R package sbm.

Main reading materials:

• Bartolucci F., Farcomeni A., Pennoni F. (2013). Latent Markov models for longitudinal data. Chapman and Hall/CRC, Boca Raton.
• Bartolucci, F., Pandolfi, S., Pennoni, F. (2017). LMest: An R package for latent Markov models for longitudinal data. Journal of Statistical Software, 81, 1--38.
• Bartolucci, F., Pandolfi, S., Pennoni, F. (2022). Discrete latent variable models. Annual Review of Statistics, 9, 425--452.
• Brusa, L., Pennoni, F., (2025). Stochastic Block Model Based on Variational Inference and its Extensions: An Application to Examine Global Migration Dynamics. In: Nakai, M. (eds) Advances in Quantitative Approaches to Sociological Issues, 1--27.
• Daudin J.J., Picard F., Robin S. (2008). A mixture model for random graphs. Statistics and Computing, 18, 173--183.
• Pennoni F. (2014). Issues on the estimation of latent variable and latent class models, with applications in the social sciences. Scholars' Press, Saarbücken.