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Bayesian statistics


Bayesian statistics


Academic year 2020/2021

Course ID
Matteo Ruggiero
Silvia Montagna
2nd year
Teaching period
First semester
D.M. 270 TAF C - Related or integrative
Course disciplinary sector (SSD)
SECS-S/01 - statistica
Formal authority
Type of examination

Sommario del corso


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 model selection in high dimensions (see International visiting professors).


Results of learning outcomes

- Knowledge and understanding

Students will learn how to model statistical problems with Bayesian parametric and nonparametric tools, with a deep insight into their theoretical properties.

- Applying knowledge and understanding

Students will possess the ability to set up a simple Bayesian model to analyse univariate data, and to devise appropriate computational algorithms for their implementation. The knowledge will be sufficient for reading and understanding a scientific paper on topics coherent with the course contents.

- Making judgements

Students will be able to select the appropriate Bayesian model to fit different types of univariate data. They will be also able to understand when parametric models are too restrictive and a Bayesian Nonparametric approach could be more appropriate.

- Communication skills

Students will be able to use appropriate, formal statistical language to describe a Bayesian statistical model and its properties, to discuss about Bayesian modelling and when one approach is more appropriate than another, to communicate the results of their findings on model implementations, to summarize and discuss a scientific paper on a coherent topic in an oral presentation.



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

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:

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.


Suggested readings and bibliography

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 scheduleV

  • Open
    Enrollment opening date
    01/09/2020 at 00:00
    Enrollment closing date
    30/06/2021 at 00:00
    Last update: 22/09/2020 09:34
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