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

Bayesian statistics

Bayesian statistics

Academic year 2019/2020

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Notes
• 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 Botond Szabo (Leiden University) on Asymptotics for Bayesian posterior inference (see International visiting professors).

Results of learning outcomes

Course delivery

The course is composed of 48 hours of class lectures.

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 average of the two evaluations given by the two lecturers on the respective modules (parametric and nonparametric).

During the Covid-19 emergency, the exam will be held remotely via Webex as a written test. More 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 
• Monte Carlo approximation
• The Normal model: natural conjugate priors, (independent) Jeffreys' prior, predictive distributions
• Other multi-parameters models
• Markov chain Monte Carlo methods for parametric inference

Bayesian nonparametric statistics:

• The Bayesian nonparametric framework
  • Motivation and formal setting for the non parametric approach
  • de Finetti's Theorem for general sequences of exchangeable random variables
  • General approach to nonparametric posterior inference 
• The Dirichlet Process
  • Definition and properties of the Dirichlet distribution
  • Definition and properties of the Dirichlet process
  • Clustering structure and sampling schemes induced by the Dirichlet process
  • Exchangeable partition probability function
• Construction of the Dirichlet process
  • Construction via finite-dimensional distributions
  • Construction via gamma subordinators
  • Construction via generalised Polya urn schemes
• Nonparametric priors derived from the Dirichlet process
  • Mixtures of Dirichlet processes
  • Mixtures with respect to the Dirichlet process
  • Hierarchical Dirichlet process
• Beyond the Dirichlet process
  
  • General stick-breaking priors
  • The Pitman-Yor process 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

Note

This course will be delivered at the ESOMAS Department.

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