Reading Seminar

Markov chains on general state space

Part I (discrete time, general space)

When, where ?

• Thursday October 3, room 106, building 1R1 IMT - 13h30
• Thursday October 11, room 129, building 1R2 IMT - 13h30
• Thursday October 25, room 106, building 1R1 IMT - 13h30
• Thursday November 8, room 106, building 1R1 IMT - 13h30

Program  of the lectures

• Lecture 1 : theory of Markov chains with general state space (definition of a Markov chain, Markov transition kernels)
• Lecture 2 : (part 1) theory of Markov chains with general state space (law of a Markov chain, invariant measure). (part 2) Hastings-Metropolis sampler (the algorithm, markovian property, transition kernel)
• Lecture 3 :  (part 1) invariant measure for the Hastings-Metropolis kernel; (part 2) motivating the use of MCMC samplers in statistics : introduction du Bayesian statistics, introduction to stochastic approximation. 

Program of the talks (2018-2019)
The student will work in pairs; they will present a research paper or a chapter of a book, with possibly numerical illustrations.   A 50min talk, using either slides or blackboard - (or a mix) as the students want, with the objective to make a dynamical talk.  Each student can give his/her talk either in french or in english.

• Talk 1 : Theory of Markov chains : results on geometrical ergodicity through coupling techniques
• 13h30 - 14h30
• by Jordan Serres, Kalok Tam and Mai Trang Vu.

• Talk 2 : MCMC sampling within EM algorithm
• 14h30 - 15h30
• by Manon Digué and K. Codet

• Talk 3 : MCMC sampling within Stochastic Approximation
• 15h45 - 16h45
• by Pierre Gach and Sofiane Arradi-Alaoui

Program of the talks (2017-2018)

The student will work in pairs; they will present a research paper or a chapter of a book, with possibly numerical illustrations.   A 50min talk, using either slides or blackboard - (or a mix) as the students want, with the objective to make a dynamical talk.  The talks will be given in english.

• Talk 1 : Theory of Markov chains : results on geometrical ergodicity through coupling techniques
  

• Talk 2: MCMC methodologies : limitations of classical algorithms, and description of adaptive techniques. Some proof of ergodicity (in a simple case) and (more or less investigated, at convenience :) numerical illustrations of the behavior of the algorithm.
• paper : available on Oct 23
• suggestions for the presentation : available on Oct 23

• Talk 3 : Application of MCMC to solve optimization problems from the Computational Statistics : description of the algorithm, discussion on the convergence (proof of a simple result) and (more or less investigated, at convenience :) numerical illustrations.
• paper : available on Oct 26
• suggestions for the presentation : available on Oct 26

References

1. Inference in Hidden Markov Models, by O. Cappé, E. Moulines and T. Ryden. Springer Series in Statistics, 2005. Chapter 14.
2. Markov chains and Stochastic Stability, by S. Meyn and R.L. Tweedie. Cambridge, 2009. Chapters 2, 3, 10.