The Living Thing / Notebooks :

Monte Carlo methods

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

See Markov Chain Monte Carlo.

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].