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This short note is a self-contained and basic introduction to the Metropolis-Hastings algorithm, this ubiquitous tool used for producing dependent simulations from an arbitrary distribution.
“The Monte Carlo method.”
Metropolis, N. and Ulam, S. (1949) · 1949
Earlier work this paper cites.
“Equations of state calculations by fast computing machines.”
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E. (1953) · 1953
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Monte Carlo Methods
Hammersley, J. and Handscomb, D. (1964) · 1964
Earlier work this paper cites.
“Monte Carlo calculations of the radial distribution functions for a proton electron plasma.”
Barker, A. (1965) · 1965
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“Monte Carlo sampling methods using Markov chains and their application.”
Hastings, W. (1970) · 1970
Earlier work this paper cites.
“Optimum Monte Carlo sampling using Markov chains.”
Peskun, P. (1973) · 1973
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“Guidelines for chosing the transition matrix in Monte Carlo methods using Markov chains.”
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Simulation and the Monte Carlo Method
Rubinstein, R. (1981) · 1981
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“A spectral method for confidence interval generation and run length control in simulations.”
Heidelberger, P. and Welch, P. (1983) · 1983
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“Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images.”
Geman, S. and Geman, D. (1984) · 1984
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“Hybrid Monte Carlo.”
Duane, S., Kennedy, A. D., Pendleton, B. J., , and Roweth, D. (1987) · 1987
Earlier work this paper cites.
“Stan Ulam, John Von Neumann, and the Monte Carlo Method.”
Eckhardt, R. (1987) · 1987
Earlier work this paper cites.
“The calculation of posterior distributions by data augmentation.”
Tanner, M. and Wong, W. (1987) · 1987
Earlier work this paper cites.
“Sampling based approaches to calculating marginal densities.”
Gelfand, A. and Smith, A. (1990) · 1990
Earlier work this paper cites.
“Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments (with discussion).”
Geweke, J. (1992) · 1992
Earlier work this paper cites.
“Computable bounds for convergence rates of Markov chains.”
Meyn, S. and Tweedie, R. (1994) · 1994
Earlier work this paper cites.
“Markov chains for exploring posterior distributions (with discussion).”
Tierney, L. (1994) · 1994
Cited alongside, same era.
“Exponential convergence for Langevin diffusions and their discrete approximations.”
Roberts, G. and Tweedie, R. (1995) · 1995
Cited alongside, same era.
“Rates of convergence of the Hastings and Metropolis algorithms.”
Mengersen, K. and Tweedie, R. (1996) · 1996
Cited alongside, same era.
“Weak convergence and optimal scaling of random walk Metropolis algorithms.”
Roberts, G., Gelman, A., and Gilks, W. (1997) · 1997
Cited alongside, same era.
“Adaptive Proposal Distribution for Random Walk Metropolis Algorithm.”
Haario, H., Saksman, E., and Tamminen, J. (1999) · 1999
Cited alongside, same era.
“Estimation of population growth or decline in genetically monitored populations.”
Beaumont, M. (2003) · 2003
“Examples of Adaptive MCMC.”
Roberts, G. and Rosenthal, J. (2009) · 2009
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“A history of Markov Chain Monte Carlo-Subjective recollections from incomplete data.”
— (2010) · 2010
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“Efficient parallelisation of Metropolis–Hastings algorithms using a prefetching approach.”
Strid, I. (2010) · 2010
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“Particle Markov chain Monte Carlo (with discussion).”
Andrieu, C., Doucet, A., and Holenstein, R. (2011) · 2011
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“Riemann manifold Langevin and Hamiltonian Monte Carlo methods.”
Girolami, M. and Calderhead, B. (2011) · 2011
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“The particle marginal Metropolis–Hastings (PMMH) particle MCMC algorithm.”
Wilkinson, D. (2011) · 2011
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Cited alongside, same era.
“A mixture representation of π \pi with applications in Markov chain Monte Carlo and perfect sampling.”
Hobert, J. and Robert, C. (2004) · 2004
Cited alongside, same era.
Monte Carlo Statistical Methods
Robert, C. and Casella, G. (2004) · 2004
Cited alongside, same era.
“Parallel Markov chain Monte Carlo Simulation by Pre-Fetching.”
Brockwell, A. (2006) · 2006
Cited alongside, same era.
“Sequential Monte Carlo samplers.”
Del Moral, P., Doucet, A., and Jasra, A. (2006) · 2006
Cited alongside, same era.
“An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants.”
Møller, J., Pettitt, A. N., Reeves, R., and Berthelsen, K. K. (2006) · 2006
Cited alongside, same era.
“MCMC for doubly-intractable distributions.”
Murray, I., Ghahramani, Z., , and MacKay, D. (2006a) · 2006
Cited alongside, same era.
“SMC2: an efficient algorithm for sequential analysis of state space models.”
Chopin, N., Jacob, P. E., and Papaspiliopoulos, O. (2013) · 2013
Later among the works it cites.
“MCMC using Hamiltonian dynamics.”
Neal, R. (2013) · 2013
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“Bayes and big data: The consensus Monte Carlo algorithm.”
Scott, S., Blocker, A., Bonassi, F., Chipman, H., George, E., and McCulloch, R. (2013) · 2013
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“Parallizing MCMC via Weierstrass Sampler.”
Wang, X. and Dunson, D. (2013) · 2013
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“The Geometric Foundations of Hamiltonian Monte Carlo.”
Betancourt, M. J., Byrne, S., Livingstone, S., and Girolami, M. (2014) · 2014
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“STAN: A C++ Library for Probability and Sampling, Version 2.5.0, http://mc-stan.org/.”
Stan Development Team (2014) · 2014
Later among the works it cites.
“Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms.”
Andrieu, C. and Vihola, M. (2015) · 2015
Closest in time.
“Accelerating Metropolis-Hastings algorithms by Delayed Acceptance.”
Banterle, M., Grazian, C., Lee, A., and Robert, C. P. (2015) · 2015
Closest in time.
“Testing hypotheses as a mixture estimation model.”
Kamary, K., Mengersen, K., Robert, C., and Rousseau, J. (2014) · 2044
Closest in time.