# Markov Chain Monte Carlo methods

August 28, 2017 — June 8, 2022

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.

## 1 Hamiltonian Monte Carlo

## 2 Connection to variational inference

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

## 3 Adaptive MCMC

See Adaptive MCMC.

## 4 Stochastic Gradient Monte carlo

See SGD MCMC.

## 5 Tempering

e.g. Ge, Lee, and Risteski (2020);Syed et al. (2020). Saif Syed can explain this quite well. Or, as Lee and Risteski put it:

The main idea is to create a meta-Markov chain (the simulated tempering chain) which has two types of moves: change the current “temperature” of the sample, or move “within” a temperature. The main intuition behind this is that at higher temperatures, the distribution is flatter, so the chain explores the landscape faster.

## 6 Mixing rates

## 7 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 2019).

## 8 Affine invariant

J. Goodman and Weare (2010)

We propose a family of Markov chain Monte Carlo methods whose performance is unaffected by affine transformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particularly useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.

Implemented in, e.g. emcee (Foreman-Mackey et al. 2013).

## 9 Efficiency of

Want to adaptively tune the MCMC? See tuning MCMC.

## 10 Incoming

## 11 References

*Statistics and Computing*.

*Bayesian Time Series Models*.

*arXiv:1905.11916 [Stat]*.

*arXiv:1701.02434 [Stat]*.

*Annalen Der Physik*.

*arXiv:2110.07032 [Math, Stat]*.

*Bernoulli*.

*Advanced Lectures on Machine Learning: ML Summer Schools 2003, Canberra, Australia, February 2-14, 2003, T Bingen, Germany, August 4-16, 2003, Revised Lectures*.

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*. NIPS’18.

*Proceedings of the National Academy of Sciences*.

*arXiv Preprint arXiv:1509.07164*.

*Advances in Neural Information Processing Systems*.

*arXiv:1911.00915 [Math, Stat]*.

*arXiv:1901.09881 [Cs, Stat]*.

*Statistical Science*.

*SIAM Review*.

*arXiv:1605.01559 [Math, Stat]*.

*Publications of the Astronomical Society of the Pacific*.

*arXiv:1812.00793 [Cs, Math, Stat]*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Journal of Applied Probability*.

*arXiv:1206.3255*.

*Communications in Applied Mathematics and Computational Science*.

*Journal of Statistical Software*.

*arXiv:1903.12322 [Cs, Stat]*.

*SSRN Electronic Journal*.

*arXiv:1708.03625 [Stat]*.

*Neural Computation*.

*Ecology Letters*.

*Journal of the American Statistical Association*.

*Statistics and Computing*.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2*. NIPS’15.

*arXiv:1708.07114 [Math, Stat]*.

*arXiv:2004.12550 [Stat]*.

*Handbook of Uncertainty Quantification*.

*arXiv:math/0407281*.

*Handbook for Markov Chain Monte Carlo*.

*arXiv:2012.15477 [Cs, Stat]*.

*arXiv:1610.00781 [Math, Stat]*.

*SIAM/ASA Journal on Uncertainty Quantification*.

*Annual Review of Statistics and Its Application*.

*Random Structures & Algorithms*.

*Microsurveys in Discrete Probability*. DIMACS Series in Discrete Mathematics and Theoretical Computer Science.

*WIREs Computational Statistics*.

*Probability Surveys*.

*Stochastic Processes and Their Applications*.

*The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning*.

*Simulation and the Monte Carlo Method*. Wiley series in probability and statistics.

*Fast Sequential Monte Carlo Methods for Counting and Optimization*. Wiley Series in Probability and Statistics.

*Proceedings of the 32nd International Conference on Machine Learning (ICML-15)*. ICML’15.

*ECML-PKDD 2017*.

*Proceedings of the National Academy of Sciences*.

*arXiv:1905.02939 [Stat]*.

*arXiv:1412.4869 [Stat]*.

*Proceedings of the 28th International Conference on International Conference on Machine Learning*. ICML’11.

*Statistics & Probability Letters*.

*Journal of Machine Learning Research*.