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ANNOUNCEMENT: the lecture scheduled for Thursday 2nd is cancelled and moved to Thursday 9th, same time.
General info. The combined use of Bayesian statistics and Markov Chain Monte Carlo (MCMC) sampling methods has been one of the great successes of applied mathematics/statistics in the last 60 years. Combined with the Bayesian inference paradigm, MCMC is of current use in virtually all fields of applied science. The increasing popularity of MCMC and the need to tackle problems of growing complexity have brought higher demands on the efficiency of such algorithms, which are often undeniably costly. The answer to such demands has produced, in the last 10-15 years, both a higher level of sophistication in the design of MCMC algorithms and the introduction of alternative approaches. Overall, recent research activity has stressed the importance of the use of rigorous analytical methods in the field of statistical sampling. This principle will be the guiding point of view for this course (the title should probably have been Analytical methods for MCMC sampling, I am not adopting this title for brevity)
Course duration: 10 to 12 hours
Syllabus. The course will introduce the theory of MCMC methods, with particular emphasis on the analysis of the efficiency of such algorithms. A list of possible topics is as follows (although we would like to stress that the course will be designed depending on the interest of the audience, therefore the list below is to be intended as a guideline). This course can be intended as a theoretical complement of the course Statistics 2 (although attendance of Statistics 2 is not a prerequisite)
• Recap on Markov chains
• Recap on convergence criteria for Markov Chains: Lyapunov-functions techniques (Meyn and Tweedie approach)
• Various Algorithms: Metropolis-Hastings, in particular Random Walk Metropolis and MALA; Gibbs sampler; Hamiltonian Monte Carlo.
• Efficiency criteria: spectral gap, asymptotic variance, number of MCMC iterations, scaling with dimensions and the curse of dimensionality. In this context T. Kurtz’s general framework of diffusion (and fluid) approximations will be introduced.
• Non-reversible algorithms and the links with Freidlin-Wentzell theory
• Time allowing, one (or more) of the following further topics might be covered: i) MCMC in infinite dimensional Hilbert spaces and algorithms that do not suffer from the curse of dimensionality; ii) elements of particle methods; iii) comparison with deterministic algorithms. Only roughly half of the course will be delivered through standard lectures, the other half will be based on active student participation (reading-group style)