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.

## Hamiltonian Monte Carlo

## Connection to variational inference

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

## Adaptive MCMC

See Adaptive MCMC.

## Stochastic Gradient Monte carlo

See SGD MCMC.

## 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 βtemperatureβ of the sample, or move βwithinβ a temperature. The main intuition behind this is that at higher temperatures, the distribution is flatter, so the chain explores the landscape faster.

## Mixing rates

## Debiasing via coupling

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

## 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).

## Efficiency of

Want to adaptively tune the MCMC? See tuning MCMC.

## References

*Statistics and Computing*18 (4): 343β73.

*Bayesian Time Series Models*, edited by David Barber, A. Taylan Cemgil, and Silvia Chiappa, 32β51. Cambridge: Cambridge University Press.

*arXiv Preprint arXiv:1502.06800*.

*arXiv:1905.11916 [Stat]*, May.

*arXiv:1701.02434 [Stat]*, January.

*Annalen Der Physik*, March.

*arXiv:2110.07032 [Math, Stat]*, October.

*Bernoulli*23 (4A): 2257β98.

*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.

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*, 8278β88. NIPSβ18. Red Hook, NY, USA: Curran Associates Inc.

*Proceedings of the National Academy of Sciences*111 (49): 17408β13.

*arXiv Preprint arXiv:1509.07164*.

*Advances in Neural Information Processing Systems*.

*arXiv:1911.00915 [Math, Stat]*, November.

*arXiv:1901.09881 [Cs, Stat]*, January.

*Statistical Science*28 (3): 424β46.

*SIAM Review*1 (1): 45β76.

*arXiv:1605.01559 [Math, Stat]*, May.

*Publications of the Astronomical Society of the Pacific*125 (925): 306.

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

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*73 (2): 123β214.

*Journal of Applied Probability*51 (A): 377β89.

*Communications in Applied Mathematics and Computational Science*5 (1): 65β80.

*arXiv:1206.3255*, June.

*Journal of Statistical Software*76 (1).

*arXiv:1903.12322 [Cs, Stat]*, March.

*SSRN Electronic Journal*.

*arXiv:1708.03625 [Stat]*, August.

*arXiv:1708.03625 [Stat]*, July.

*Neural Computation*28 (1): 45β70.

*Ecology Letters*10 (7): 551.

*Journal of the American Statistical Association*105 (492): 1617β25.

*Statistics and Computing*6 (2): 113β19.

*Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2*, 2917β25. NIPSβ15. Cambridge, MA, USA: MIT Press.

*arXiv:1708.07114 [Math, Stat]*, August.

*arXiv:2004.12550 [Stat]*, October.

*arXiv:math/0407281*, July.

*Handbook for Markov Chain Monte Carlo*, edited by Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Boca Raton: Taylor & Francis.

*arXiv:2012.15477 [Cs, Stat]*, December.

*arXiv:1610.00781 [Math, Stat]*, October.

*Random Structures & Algorithms*, 9:223β52. New York, NY, USA: John Wiley & Sons, Inc.

*Microsurveys in Discrete Probability*, edited by David Aldous and James Gary Propp, 41:181β92. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Providence, Rhode Island: American Mathematical Society.

*WIREs Computational Statistics*10 (5): e1435.

*Probability Surveys*1 (0): 20β71.

*Stochastic Processes and Their Applications*49 (2): 207β16.

*Simulation and the Monte Carlo Method*. 3 edition. Wiley series in probability and statistics. Hoboken, New Jersey: Wiley.

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

*Fast Sequential Monte Carlo Methods for Counting and Optimization*. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

*Proceedings of the 32nd International Conference on Machine Learning (ICML-15)*, 1218β26. ICMLβ15. Lille, France: JMLR.org.

*ECML-PKDD 2017*.

*Proceedings of the National Academy of Sciences*104 (6): 1760β65.

*arXiv:1905.02939 [Stat]*, November.

*arXiv:1412.4869 [Stat]*, November.

*Proceedings of the 28th International Conference on International Conference on Machine Learning*, 681β88. ICMLβ11. Madison, WI, USA: Omnipress.

*Statistics & Probability Letters*91 (Supplement C): 14β19.

*Journal of Machine Learning Research*11 (May): 1771β98.

## No comments yet. Why not leave one?