Markov Chain Monte Carlo methods

This chain pump is not a good metaphor for how a Markov chain Monte Carlo sampler works, but it does correctly evoke the effort involved.

Despite studying within this area, I have nothing to say about MCMC broadly, but I do have some things I wish to keep notes on.

Connection to variational inference

Nifty. See (Salimans, Kingma, and Welling 2015)

Adaptive MCMC

See Adaptive MCMC.


Holden Lee, Andrej Risteski on the connection between log-concavity and convex optimisation.

Mixing rates

See ergodic theorems and mixing.

Debiasing via coupling

Pierre E. Jacob, John O’Leary, Yves Atchadé, crediting Glynn and Rhee (2014), made MCMC estimators without finite-time-bias, which is nice for parallelisation (Jacob, O’Leary, and Atchadé 2017).


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Atchadé, Yves, Gersende Fort, Eric Moulines, and Pierre Priouret. 2011. “Adaptive Markov Chain Monte Carlo: Theory and Methods.” In Bayesian Time Series Models, edited by David Barber, A. Taylan Cemgil, and Silvia Chiappa, 32–51. Cambridge: Cambridge University Press.
Bach, Francis. 2015. “On the Equivalence Between Kernel Quadrature Rules and Random Feature Expansions.” 2015.
Bales, Ben, Arya Pourzanjani, Aki Vehtari, and Linda Petzold. 2019. “Selecting the Metric in Hamiltonian Monte Carlo.” May 28, 2019.
Betancourt, Michael. 2017. “A Conceptual Introduction to Hamiltonian Monte Carlo.” January 9, 2017.
———. 2018. “The Convergence of Markov Chain Monte Carlo Methods: From the Metropolis Method to Hamiltonian Monte Carlo.” Annalen Der Physik, March.
Betancourt, Michael, Simon Byrne, Sam Livingstone, and Mark Girolami. 2017. “The Geometric Foundations of Hamiltonian Monte Carlo.” Bernoulli 23 (November): 2257–98.
Bousquet, Olivier, Ulrike von Luxburg, and Gunnar Rtsch. 2004. Advanced Lectures on Machine Learning: ML Summer Schools 2003, Canberra, Australia, February 2-14, 2003, T Bingen, Germany, August 4-16, 2003, Revised Lectures. Springer.
Calderhead, Ben. 2014. “A General Construction for Parallelizing MetropolisHastings Algorithms.” Proceedings of the National Academy of Sciences 111 (49): 17408–13.
Carpenter, Bob, Matthew D. Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt. 2015. “The Stan Math Library: Reverse-Mode Automatic Differentiation in C++.” 2015.
Caterini, Anthony L., Arnaud Doucet, and Dino Sejdinovic. 2018. “Hamiltonian Variational Auto-Encoder.” In Advances in Neural Information Processing Systems.
Chakraborty, Saptarshi, Suman K. Bhattacharya, and Kshitij Khare. 2019. “Estimating Accuracy of the MCMC Variance Estimator: A Central Limit Theorem for Batch Means Estimators.” November 3, 2019.
Cornish, Robert, Paul Vanetti, Alexandre Bouchard-Côté, George Deligiannidis, and Arnaud Doucet. 2019. “Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets.” January 28, 2019.
Cotter, S. L., G. O. Roberts, A. M. Stuart, and D. White. 2013. MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster.” Statistical Science 28 (3): 424–46.
Diaconis, Persi, and David Freedman. 1999. “Iterated Random Functions.” SIAM Review 1 (1): 45–76.
Durmus, Alain, and Eric Moulines. 2016. “High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm.” May 5, 2016.
Girolami, Mark, and Ben Calderhead. 2011. “Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2): 123–214.
Glynn, Peter W., and Chang-Han Rhee. 2014. “Exact Estimation for Markov Chain Equilibrium Expectations.” Journal of Applied Probability 51 (A): 377–89.
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Jacob, Pierre E., John O’Leary, and Yves F. Atchadé. 2017. “Unbiased Markov Chain Monte Carlo with Couplings.” August 11, 2017.
———. 2019. “Unbiased Markov Chain Monte Carlo with Couplings.” July 17, 2019.
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