# Monte Carlo methods

Usefulness: đź”§
Novelty: đź’ˇ
Uncertainty: đź¤Ş đź¤Ş
Incompleteness: đźš§ đźš§ đźš§

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

Or donâ€™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.

## Multi-level Monte Carlo

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

# Refs

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

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

AtchadĂ©, Yves, Gersende Fort, Eric Moulines, and Pierre Priouret. 2011. â€śAdaptive Markov Chain Monte Carlo: Theory and Methods.â€ť In Bayesian Time Series Models, edited by David Barber, A. Taylan Cemgil, and Silvia Chiappa, 32â€“51. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511984679.003.

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

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

Giles, Michael B. 2008. â€śMultilevel Monte Carlo Path Simulation.â€ť Operations Research 56 (3): 607â€“17. https://doi.org/10.1287/opre.1070.0496.

Giles, Michael B., and Lukasz Szpruch. 2014. â€śAntithetic Multilevel Monte Carlo Estimation for Multi-Dimensional SDEs Without LĂ©vy Area Simulation.â€ť The Annals of Applied Probability 24 (4): 1585â€“1620. https://doi.org/10.1214/13-AAP957.

Giles, Mike, and Lukasz Szpruch. 2012. â€śMultilevel Monte Carlo Methods for Applications in Finance,â€ť December. http://arxiv.org/abs/1212.1377.

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

Higham, Desmond J. 2015. â€śAn Introduction to Multilevel Monte Carlo for Option Valuation,â€ť May. http://arxiv.org/abs/1505.00965.

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

Propp, James, and David Wilson. 1998. â€śCoupling from the Past: A Userâ€™s Guide.â€ť 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. â€śMultilevel Monte Carlo Method for Jump-Diffusion SDEs,â€ť June. http://arxiv.org/abs/1106.4730.