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Bayesian statistics

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Bayesian statistics

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Academic year 2019/2020

Course ID
MAT0070
Teachers
Matteo Ruggiero
Silvia Montagna
Botond Szabo
Year
2nd year
Teaching period
First semester
Type
D.M. 270 TAF C - Related or integrative
Credits/Recognition
6
Course disciplinary sector (SSD)
SECS-S/01 - statistica
Delivery
Formal authority
Language
English
Attendance
Optional
Type of examination
Oral
Prerequisites
STOCHASTIC MODELLING FOR STATISTICAL APPLICATIONS
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Sommario del corso

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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).

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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 suffienct 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.

 



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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
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Course delivery

The course is composed of 48 hours of class lectures.

 

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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).

 

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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.


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Notes

This course will be delivered at the ESOMAS Department.

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Class scheduleV

Enroll
  • Open
    Enrollment opening date
    01/09/2019 at 00:00
    Enrollment closing date
    30/06/2020 at 00:00
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