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 new 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. See also uncertainty quantification.
mimclib is one possible tool here.
Refs
- AnHi12
- Anderson, D. F., & Higham, D. J.(2012) Multilevel Monte Carlo for Continuous Time Markov Chains, with Applications in Biochemical Kinetics. Multiscale Modeling & Simulation, 10(1), 146–179. DOI.
- Bach15
- Bach, F. (2015) On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions. ArXiv Preprint ArXiv:1502.06800.
- Cald14
- Calderhead, B. (2014) A general construction for parallelizing Metropolis−Hastings algorithms. Proceedings of the National Academy of Sciences, 111(49), 17408–17413. DOI.
- Gile08
- Giles, M. B.(2008) Multilevel Monte Carlo Path Simulation. Operations Research, 56(3), 607–617. DOI.
- GiSz14
- Giles, M. B., & Szpruch, L. (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.
- GiSz12
- Giles, M., & Szpruch, L. (2012) Multilevel Monte Carlo methods for applications in finance. ArXiv:1212.1377 [q-Fin].
- GMRB12
- Goodman, N., Mansinghka, V., Roy, D., Bonawitz, K., & Tarlow, D. (2012) Church: a language for generative models. ArXiv:1206.3255.
- HaNT16
- Haji-Ali, A.-L., Nobile, F., & Tempone, R. (2016) Multi-index Monte Carlo: when sparsity meets sampling. Numerische Mathematik, 132(4), 767–806. DOI.
- High15
- Higham, D. J.(2015) An Introduction to Multilevel Monte Carlo for Option Valuation. ArXiv:1505.00965 [Physics, q-Fin, Stat].
- KoCW15
- Korattikara, A., Chen, Y., & Welling, M. (2015) Sequential Tests for Large-Scale Learning. Neural Computation, 28(1), 45–70. DOI.
- Liu96
- Liu, J. S.(1996) Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statistics and Computing, 6(2), 113–119. DOI.
- NoFo16
- Norton, R. A., & Fox, C. (2016) Tuning of MCMC with Langevin, Hamiltonian, and other stochastic autoregressive proposals. ArXiv:1610.00781 [Math, Stat].
- PrWi98
- Propp, J., & Wilson, D. (1998) Coupling from the Past: a User’s Guide.
- RuKr04
- Rubinstein, R. Y., & Kroese, D. P.(2004) The Cross-Entropy Method a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning. . New York, NY: Springer New York
- RuKr08
- Rubinstein, R. Y., & Kroese, D. P.(2008) Simulation and the monte carlo method. (2nd ed.). Hoboken, N.J: John Wiley & Sons
- RuRV14
- Rubinstein, R. Y., Ridder, A., & Vaisman, R. (2014) Fast sequential Monte Carlo methods for counting and optimization. . Hoboken, New Jersey: Wiley
- SiFT07
- Sisson, S. A., Fan, Y., & Tanaka, M. M.(2007) Sequential Monte Carlo without likelihoods. Proceedings of the National Academy of Sciences, 104(6), 1760–1765. DOI.
- Xia11
- Xia, Y. (2011) Multilevel Monte Carlo method for jump-diffusion SDEs. ArXiv:1106.4730 [q-Fin].