Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Bayesian statistics

Bayesian statistics

Academic year 2020/2021

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Teaching tools

Course objectives

Bayesian statistics established itself as the main alternative approach to "classical" statistical inference, which is based on frequentist principles and largely relies on the often too simplistic hypothesis of independence and identity in distribution of the data. Thanks to its appealing mathematical properties and its wide interpretability, Bayesian statistics has nowadays found widespread appreciation by the most diverse scientific communities, as witnessed by the countless registered applications in virtually any applied discipline.

The course aims at providing a modern introduction to Bayesian statistical methods, covering the fundamentals of both the parametric and the nonparametric approach. The course will focus on the key probabilistic and foundational concepts, together with well established stochastic modelling tools and the most widely used computational strategies for its implementation.

A short module of the course, included in the overall courseload, will be taught by Visiting Professor David Rossell (Universitat Pompeu Fabra) on Bayesian inference in high dimensions (see International visiting professors).

Results of learning outcomes

Course delivery

The course is composed of 48 hours of lectures which (for the AY 2020/2021) will mostly be held remotely, either as live streaming or pre-recorded. Lectures can be followed via WebEx at the following link:

https://unito.webex.com/unito/j.php?MTID=m7edf3f79b8492b62859e3d8b4c89cc41

Videos of the recordings will be made available on Moodle, usually within one day (depending on WebEx performance). Teaching materials and updates will be delivered via Moodle.

Some additional activities to favour direct interaction between professors and students may be organised in presence, under appropriate conditions of social distancing and compatibly and in compliance with future existing regulations, alternatively online. 

Learning assessment methods

The exam consists in the oral verification of the material covered in class on the parametric and the nonparametric module. The final mark will be the simple average of the two evaluations given by the two lecturers on the respective modules (parametric and nonparametric).

During the Covid-19 emergency (and including the September 2020 session), the exam will be held remotely via Webex as a written test. Specific instructions will be sent to students registered to the exam via their institutional email addresses.

Program

Bayesian parametric statistics:

• The Bayesian parametric framework
  • Motivation and formal setting for the parametric approach
  • Exchengeability and de Finetti's Theorem 
• One-parameter models
  • Binomial/Poisson/Exponential models
  • Point and interval estimation
  • Exponential families and conjugate priors
  • More on priors: discrete, mixture, default, Jeffreys, non-conjugate, improper 
• (Quick review of) Monte Carlo approximation and Markov chain Monte Carlo methods for parametric inference
• The Normal model: natural conjugate priors, (independent) Jeffreys' prior, predictive distributions
• Other multi-parameters models and hierarchical modelling
• Introduction to Bayesian linear regression models

Bayesian nonparametric statistics:

• The Bayesian nonparametric framework
  • Motivation and formal setting
  • de Finetti's Theorem for general sequences
  • Generic approach to nonparametric posterior inference 
• The Dirichlet Process
  • Properties of the Dirichlet distribution
  • Definition and properties of the Dirichlet process
  • Induced clustering structure and sampling schemes
• Constructions of the Dirichlet process
  • Via finite-dimensional distributions
  • Via gamma subordinators
  • Via stick-breaking
  • Via generalised Polya urn schemes
• Beyond the Dirichlet process
  • Mixtures of Dirichlet processes and with respect to the Dirichlet process
  • Hierarchical Dirichlet process
  • Pitman-Yor processes and species sampling models
  • Normalised random measures with independent increments

Lecture notes will be made available. Additional suggested reading are:

Bayesian parametric statistics

• HOFF, P.D. (2009). A First course in Bayesian statistical methods. Springer. (Recommended)
• GELMAN, A., CARLIN, J. B., STERN, H. S., DUNSON, D. B., VETHARI, A., and RUBIN, D. B. (2014). Bayesian data analysis, Third Edition. CRC Press. (Optional)

Bayesian nonparametric statistics

• GHOSAL, S. and VAN DER VAART, A. (2017). Theory of nonparametric Bayesian inference. Cambridge University Press.
• HJORT, N., HOLMES, C., MUELLER, P. and WALKER, S.G. (eds.) (2010). Bayesian Nonparametrics. Cambridge University Press.
• GHOSH, J.K. and RAMAMOORTHI, R.V. (2003). Bayesian Nonparametrics. Springer.
  
  
  

Class schedule

• Teaching materials
• Exams and finals
• Go to Moodle
• Visit the forums