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Bayesian statistics

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Bayesian statistics

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Academic year 2022/2023

Course ID
MAT0070
Teaching staff
Matteo Ruggiero
Silvia Montagna
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 - statisticsm
Delivery
Blended
Language
English
Attendance
Optional
Type of examination
Written
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 MARTA CATALANO (University of Warwick, UK) on Dependent nonparametric priors via completely random measures  (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 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.

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Course delivery

The course is composed of 48 hours of lectures, comprising of two 2-hours classes per week. All classes are foreseen to be taught in presence.

Teaching materials and updates will be delivered via Moodle (see link below).

 

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Learning assessment methods

The exam consists in the written verification of the student's knowledge and mastering of the material covered in class, by exposition of the main results and constructions and through exercises. The exam will be divided into two parts, one for each half course covering the parametric and the nonparametric frameworks. The final mark will be the simple average of the two evaluations given by the two lecturers on the respective modules.

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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 
  • Specification of probability models
  • Subjective, conjugate, and non-informative prior distributions
  • Analytical and computational techniques for obtaining posterior distributions
  • Hierarchical and multilevel models
  • Model assessment, model selection, diagnostics

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

Suggested readings and bibliography

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Lecture notes will be made available. Additional suggested readings 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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Class schedule

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Last update: 14/09/2022 09:43
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