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 compressed sensing, particle filters, and matrix concentration inequalities, and probably the most important use case, Bayesian statistics.

## Markov chain samplers

## Multi-level Monte Carlo

Hmmm. Also *multi scale monte carlo*, *multi index monte carlo*. Something like adaptive importance sampling? Not sure.

## Refs

- Beta17: (2017) A Conceptual Introduction to Hamiltonian Monte Carlo.
*ArXiv:1701.02434 [Stat]*. - Cald14: (2014) A general construction for parallelizing Metropolis−Hastings algorithms.
*Proceedings of the National Academy of Sciences*, 111(49), 17408–17413. DOI - High15: (2015) An Introduction to Multilevel Monte Carlo for Option Valuation.
*ArXiv:1505.00965 [Physics, q-Fin, Stat]*. - GiSz14: (2014) Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L’{e}vy area simulation.
*The Annals of Applied Probability*, 24(4), 1585–1620. DOI - GMRB12: (2012) Church: a language for generative models.
*ArXiv:1206.3255*. - PrWi98: (1998) Coupling from the Past: a User’s Guide.
- RuRV14: (2014)
*Fast sequential Monte Carlo methods for counting and optimization*. Hoboken, New Jersey: Wiley - Neal11: (2011) MCMC using Hamiltonian dynamics. In Handbook for Markov chain Monte Carlo. Boca Raton: Taylor & Francis
- Liu96: (1996) Metropolized independent sampling with comparisons to rejection sampling and importance sampling.
*Statistics and Computing*, 6(2), 113–119. DOI - HaNT16: (2016) Multi-index Monte Carlo: when sparsity meets sampling.
*Numerische Mathematik*, 132(4), 767–806. DOI - AnHi12: (2012) Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics.
*Multiscale Modeling & Simulation*, 10(1), 146–179. DOI - Xia11: (2011) Multilevel Monte Carlo method for jump-diffusion SDEs.
*ArXiv:1106.4730 [q-Fin]*. - GiSz12: (2012) Multilevel Monte Carlo methods for applications in finance.
*ArXiv:1212.1377 [q-Fin]*. - Gile08: (2008) Multilevel Monte Carlo Path Simulation.
*Operations Research*, 56(3), 607–617. DOI - Bach15: (2015) On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions.
*ArXiv Preprint ArXiv:1502.06800*. - CaRo96: (1996) Rao-Blackwellisation of sampling schemes.
*Biometrika*, 83(1), 81–94. DOI - SiFT07: (2007) Sequential Monte Carlo without likelihoods.
*Proceedings of the National Academy of Sciences*, 104(6), 1760–1765. DOI - KoCW15: (2015) Sequential Tests for Large-Scale Learning.
*Neural Computation*, 28(1), 45–70. DOI - RuKr16: (2016)
*Simulation and the Monte Carlo Method*. Hoboken, New Jersey: Wiley - RuKr04: (2004)
*The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning*. New York, NY: Springer New York - NoFo16: (2016) Tuning of MCMC with Langevin, Hamiltonian, and other stochastic autoregressive proposals.
*ArXiv:1610.00781 [Math, Stat]*.