Statistical Methods and Applications

CRISTINA MOLLICA
LUCA TARDELLA

General goals
Knowledge at an intermediate and advanced level of the main issues in Bayesian statistics.
Ability to apply Bayesian statistical techniques to applicative context.

Knowledge and understanding
Knowledge and understanding of the Bayesian approach to statistical inference, of its models and of its methodologies

Applying knowledge and understanding
Ability to apply Bayesian statistical methods for inferential problems in real-data problems

Making judgements
Ability of choosing appropriate Bayesian methods and models in different inferential problems

Communication skills
Ability of communicating results of the analyses in written and oral form

Learning skills
Students acquire skills useful to approach more advanced topics in Bayesian inference, Advanced data analysis, Statistical computing and Mathematical statistics

• Understanding the main ingredients of Bayesian inference and Bayesian modelling
• Deriving posterior distributions for a variety of statistical models
• Conducting a Bayesian analysis and creating appropriate summaries of results
• Use appropriate tools for model checking
• Use appropriate tools in the presence of alternative statistical models: model comparison, model selection and model averaging.

This is an introductory course about Bayesian inference and Bayesian modelling for data analysis. We will balance between theoretical and analytical tools and practice. In particular for practical implementation of Bayesian models on real data we will make use of some software for Bayesian modelling and inference (R, BUGS, INLA[possibly]). There will be some emphasis on the specific detailed theoretical aspects of Bayesian computational tools.
List of topics:

• Introduction to Bayesian Thinking. This section covers basic definitions, Bayesian model, Bayes’ theorem. Subjective prior elicitation and asome noninformative or default prior choices (Jeffreys’ Rule). Conjugate analysis. Techniques and tools for characterizing and summarizing posterior distributions.

• Multiparameter Inference. This section covers multivariate normal and multinomial models and introduces to approximate random sampling from a multivariate distribution. Bayesian inference in the presence of missing data.

• Hierarchical Models, Model Checking and Linear Models. This section covers the Bayesian approach for hierarchical modelling and linear regression. Linear and generalized linear mixed effects models. It also considers posterior predictive checking, sensitivity analysis, and goodness-of-fit statistics.

• Bayesian computational tools. Introduction to Monte Carlo methods as approximation strategy. Monte Carlo methods for Bayesian inference. Classical asymptotic theorems and Monte Carlo methods: convergence and error control. Importance sampling techniques. Monte Carlo strategies for approximating marginal likelihood and Bayes Factor. Introduction to Markov chains on a finite state space and on general state spaces. Markov chains, stationarity, invariant measures. Limiting distributions and rate of convergence. General algorithms for Markov chain simulation with a prescribed invariant distribution: Gibbs sampling and Metropolis Hastings. Hhybrid methods: kernel composition, kernel mixtures. Approximate Bayesian Computation.

• Course lecture notes (slide available on the Moodle page of the course linked at https://corsidilaurea.uniroma1.it/it/users/lucatardellauniroma1it)
• Peter Hoff, A First Course in Bayesian Statistical Methods. Springer-Verlag Inc, 2009.
• Jean-Michel Marin and Christian P. Robert, Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer, 2007
• Ioannis Ntzoufras, Bayesian Modeling Using WinBUGS. Wiley, 2009.
• Peter Congdon. Bayesian Statistical Modelling (2nd ed.). Wiley, 2006
• Christian P. Robert and George Casella. Monte Carlo statistical methods (2nd ed.. Springer-Verlag Inc, 2004.

Traditional face-to-face teaching combined with computer sessions and labs with applications to real data and synthetic data.

In person attendance is not mandatory, but is warmly advised.

In class participation is warmly advised.

EXAM RULES

• For students attending lectures during the semester:

– 25% Homeworks (written)
– 25% Midterm test (online quiz)
– 50% Final Project (written report and oral presentation with discussion and possible questions about homeworks/test).

• For students NOT attending lectures during the semester:

– 50% Written test (online quiz with possible interview with insights on all course topics)
– 50% Final Project with discussion (written report and oral presentation)

- Bayesian model setup: what are the basic ingredients of Bayesian inference ?
- The role of the posterior distribution: how can you exploit the posterior distribution to make point estimation?
- Predictive distributions for planning experiments or carrying out predictions of future observations: what is the difference between the prior predictive distribution and the posterior predictive distribution?
- Given a Bayesian model setup how can you simulate a sample from the prior predictive distribution?
- Given a Bayesian model setup and some observed data, how can you simulate a sample from the posterior predictive distribution?
- Provide two alternative ways of addressing hypothesis testing.
- Model selections: what are the Bayesian tools for selecting the most appropriate model in the presence of alternative competing models?