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STOCHASTIC MODELLING FOR STATISTICAL APPLICATIONS
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STOCHASTIC MODELLING FOR STATISTICAL APPLICATIONS
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Academic year 2015/2016
- Course ID
- MAT0039
- Teaching staff
- Matteo Ruggiero
Andreas Kyprianou - Year
- 1st year
- Teaching period
- Second semester
- Type
- D.M. 270 TAF B - Distinctive
- Credits/Recognition
- 6
- Course disciplinary sector (SSD)
- MAT/06 - probabilita' e statistica matematica
- Delivery
- Class Lecture
- Language
- English
- Attendance
- Optional
- Type of examination
- Oral
- Prerequisites
- PROBABILITY THEORY (MAT0034)
- Propedeutic for
- BAYESIAN NONPARAMETRIC STATISTICS (MAT0042)
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Sommario del corso
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Course objectives
The course introduces to the theory of Markov and Lévy processes, providing the necessary tools for modern temporal modelling in view of statistical applications.
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Results of learning outcomes
The student will learn the structure and properties of some classes of stochastic processes widely used in applied probability and statistical inference. This includes analysing and manipulating their main features, computing the most relevant quantites of interest, and modelling stylized observed phenomena by choosing the correct type of process based on their characteristics.
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Course delivery
The course is composed of 48 hours of class lectures.
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Learning assessment methods
The final assessment consists in an oral examination on
- the material covered during the course;
- an additional topic previously agreed with the teacher (optional).
The latter can consist in the presentation of a scientific paper whose content is coherent with the course's syllabus, or the presentation of a written essay in which the student investigates in more detail a topic of interest.
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Program
- Introduction: finite dimensional distributions; existence theorem; classes of stochastics processes based on path properties.- Markov chains: transition matrices, Chapman-Kolmogorov equations, strong Markov property, classification of states, invariant measures, reversibility, convergence to equilibrium. Examples: random walks, birth and death chains, branching processes, Wright-Fisher models.
- Elements of Markov chain Monte Carlo methods: Monte Carlo principle; Markov chain Monte Carlo; Metropolis-Hastings; Gibbs sampler; slice sampler.
- Continuous time Markov chains: transition functions and Chapman-Kolmogorov equations; transition rates and infinitesimal generators; backward and forward equations; embedded chains and holding times; uniformisation; stationarity; reversibility. Examples: Poisson process, birth and death processes, Wright-Fisher models, coalescent processes.
Additionally, a 16 hours module will be taught by visiting professor Andreas Kyprianou (University of Bath, UK) on Lévy processes and Poisson random measures.Suggested readings and bibliography
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Main references:
- NORRIS, J.R. Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics.
- BREMAUD, P. Markov Chains. Springer.
Further suggested readings:
- BILLINGSLEY, P. Probability and measure. Wiley.
- GRIMMETT, G.R. and STIRZAKER, D.R. Probability and random processes. Oxford University Press.
- KARLIN and TAYLOR. A first Course in Stochastic Processes. Academic Press.
- KARLIN and TAYLOR. A second Course in Stochastic Processes. Academic Press.
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Class schedule
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