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Monte Carlo methods

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪
Incompleteness: 🚧 🚧 🚧

Finding functionals (traditionally integrals) approximately by guessing cleverly. Often, but not always, used for approximate statistical inference, especially certain Bayesian techniques.

Or don’t even guess randomly, but sample cleverly using the shiny Quasi Monte Carlo. See also sequential Monte Carlo, and and probably the most prominent use case, Bayesian statistics.

Markov chain samplers

See Markov Chain Monte Carlo.

Multi-level Monte Carlo

Hmmm. Also multi scale monte carlo, multi index monte carlo. :construction

Refs

Anderson, David F., and Desmond J. Higham. 2012. “Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics.” Multiscale Modeling & Simulation 10 (1): 146–79. https://doi.org/10.1137/110840546.

Andrieu, Christophe, and Johannes Thoms. 2008. “A Tutorial on Adaptive MCMC.” Statistics and Computing 18 (4): 343–73. https://doi.org/10.1007/s11222-008-9110-y.

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. https://doi.org/10.1017/CBO9780511984679.003.

Casella, George, and Christian P. Robert. 1996. “Rao-Blackwellisation of Sampling Schemes.” Biometrika 83 (1): 81–94. https://doi.org/10.1093/biomet/83.1.81.

Cranmer, Kyle, Johann Brehmer, and Gilles Louppe. 2019. “The Frontier of Simulation-Based Inference.” In Proceedings for the Sackler Colloquia. http://arxiv.org/abs/1911.01429.

Giles, Michael B. 2008. “Multilevel Monte Carlo Path Simulation.” Operations Research 56 (3): 607–17. https://doi.org/10.1287/opre.1070.0496.

Giles, Michael B., and Lukasz Szpruch. 2014. “Antithetic Multilevel Monte Carlo Estimation for Multi-Dimensional SDEs Without Lévy Area Simulation.” The Annals of Applied Probability 24 (4): 1585–1620. https://doi.org/10.1214/13-AAP957.

Giles, Mike, and Lukasz Szpruch. 2012. “Multilevel Monte Carlo Methods for Applications in Finance,” December. http://arxiv.org/abs/1212.1377.

Haji-Ali, Abdul-Lateef, Fabio Nobile, and Raúl Tempone. 2016. “Multi-Index Monte Carlo: When Sparsity Meets Sampling.” Numerische Mathematik 132 (4): 767–806. https://doi.org/10.1007/s00211-015-0734-5.

Higham, Desmond J. 2015. “An Introduction to Multilevel Monte Carlo for Option Valuation,” May. http://arxiv.org/abs/1505.00965.

Liu, Jun S. 1996. “Metropolized Independent Sampling with Comparisons to Rejection Sampling and Importance Sampling.” Statistics and Computing 6 (2): 113–19. https://doi.org/10.1007/BF00162521.

Propp, James, and David Wilson. 1998. “Coupling from the Past: A User’s Guide.” http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.6624&rep=rep1&type=pdf.

Rubinstein, Reuven Y, and Dirk P Kroese. 2004. The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. New York, NY: Springer New York. http://dx.doi.org/10.1007/978-1-4757-4321-0.

Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. Simulation and the Monte Carlo Method. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

Rubinstein, Reuven Y., Ad Ridder, and Radislav Vaisman. 2014. Fast Sequential Monte Carlo Methods for Counting and Optimization. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

Xia, Yuan. 2011. “Multilevel Monte Carlo Method for Jump-Diffusion SDEs,” June. http://arxiv.org/abs/1106.4730.