This PhD-level course runs during the spring and fall of 2022.

The intent of the course is to provide an overview of the theory and application of Monte Carlo methods, with a slight emphasis on Markov chain Monte Carlo (MCMC) methods; there may be some overlap with master's level courses at KTH on some topics.

Notes from Lecture 6: MCMC part 3

A list of the relevant book chapters and papers will also appear here as we progress in the course. In addition to attending (most of) the meetings in the course, the examination is based on (i) transcribing at least one lecture, and (ii) presenting (parts of) a book chapter of paper on a relevant topic - options for these will be presented during the course. Aside from (i) and (ii), you are expected to work on different examples and problems mentioned in class, covering both theory and implementation, but there will not be any graded homework or similar.

Main references

  • Asmussen and Glynn. Stochastic Simulation: Almgorithms and Analysis [AG]
  • Robert and Casella. Monte Carlo Statistical Methods [RC]


    • For presentations

    Below are some suggestions for papers, chapters etc. that can be used for the presentation-part of the course; own suggestions are more than welcome. The different papers/chapters etc. also contain many good references—please ask if you want helo finding a suitable topic/paper(s).

  • On the proof of Harris' ergodic theorem: Yet another look at Harris’ ergodic theorem for Markov chains, by Hairer and Mattingly.
  • Introduction to Markov semigroups, functional inequalitires: Chapters 1,3,4 and 5 in Analysis and geometry of Markov diffusion operators, by by Bakry, Gentil and Ledoux.
  • Poincaré inequality for a class of distributions: A simple proof of the Poincaré inequality for a large class of probability measures, by Bakry, Barthe, Cattiaux and Guillin.
  • Feynman-Kac semigroups and particle models: Feynman-Kac Formulae, by Del Moral.
  • The introduction of subsolutions for importance sampling: Importance Sampling, Large Deviations, and Differential Games, by Dupuis and Wang.
  • Importance sampling using Lyapunov inequalities: Efficient rare-event simulation for the maximum of heavy-tailed random walks, by Blanchet and Glynn.
  • On convergence rates for certain MCMC chains: Asymptotic Variance and Convergence Rates of Nearly-Periodic MCMC Algorithms, by Rosenthal.
  • Early example of non-reversible MCMC methods: Analysis of a nonreversible Markov chain sampler, by Diaconis, Holmes and Neal.
  • On optimal scaling in RW Metropolis methods: Optimal Scaling of Random-Walk Metropolis Algorithms on General Target Distributions, by Yang, Roberts and Rosenthal.
  • Zig-zag process: The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data, by Bierkens, Fearnhead and Roberts.
  • Intro to piecewiese deterministic MCMC methods: Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo, by Fearnhead, Bierkens, Pollock and Roberts.
  • Convergence of Langevin method for MCMC: Exponential convergence of Langevin distributions and their discrete approximations, by Roberts and Tweedie.
  • A method for unbiased estimators of equilibrium expectations for Markov chains: Exact estimation for Markov chain equilibrium expectations, by Glynn and Rhee.
  • Introduction to SMC: An introduction to Sequential Monte Carlo, by Doucet, De Freitas and Gordon.
  • SMC: Squential Monte Carlo samplers, by Del Moral, Doucet and Jasra.
  • Sequential methods for combinatorial problems: Fast Sequential Monte Carlo Methods for Counting and Optimization, by Rubenstein, Ridder and Vaisman.
  • On numerical methods for SDEs: Numerical Solution of Stochastic Differential Equations, by Kloeden and Platen.