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| Metropolis–Hastings | |
|---|---|
| Name | Metropolis–Hastings |
| Paradigms | Markov chain Monte Carlo |
| Inventors | Nicholas Metropolis, W. K. Hastings |
| First appeared | 1953 (Metropolis), 1970 (Hastings) |
| Related | Gibbs sampling, Markov chain Monte Carlo, Monte Carlo method |
Metropolis–Hastings Metropolis–Hastings is a Markov chain Monte Carlo technique used to draw samples from complex probability distributions by constructing a Markov chain with a desired stationary distribution, and it plays a central role alongside Gibbs sampling, Hamiltonian Monte Carlo, Sequential Monte Carlo and Importance sampling. It originated in computational contexts linked to Los Alamos National Laboratory, Manhattan Project alumni and later formalized within statistical communities such as Institute of Mathematical Statistics and Royal Statistical Society. The method connects to historical work by Nicholas Metropolis and formal extension by W. K. Hastings, and it has influenced methodologies used at institutions like Princeton University, University of Cambridge, Stanford University and Massachusetts Institute of Technology.
Metropolis–Hastings constructs a Markov chain whose stationary distribution equals a target distribution often specified up to a normalizing constant, a problem encountered in Bayesian inference at University of Oxford, Harvard University, Yale University, Columbia University and University of California, Berkeley. The algorithm generalizes the Metropolis algorithm and is applied in fields from computational physics in the tradition of Enrico Fermi and John von Neumann to modern statistics influenced by figures like Bradley Efron, David R. Cox, Andrew Gelman and Geoffrey Hinton. Connections exist to stochastic simulation projects at Los Alamos National Laboratory, numerical work at Argonne National Laboratory and machine learning research at Google and DeepMind.
The Metropolis–Hastings procedure iteratively proposes a candidate state via a proposal distribution and accepts or rejects it using an acceptance ratio that balances the target density and proposal probabilities, a mechanism analyzed by researchers at Bell Labs, IBM Research and Microsoft Research. Algorithmic components such as proposal kernels can be symmetric as in the original Metropolis work or asymmetric as in Hastings' extension, echoing sampling approaches used in Metropolis (software) style simulations and in later developments at Los Alamos National Laboratory and Princeton University. Practical proposal choices range from random-walk proposals inspired by studies at Cambridge University and Imperial College London to informed proposals using gradients as in techniques from Stanford University and Massachusetts Institute of Technology.
The chain produced by the algorithm is ergodic under broad conditions provided by Markov chain theory developed by scholars linked to Kolmogorov, Andrey Markov, Andrey Kolmogorov and later formalized by academics at Princeton University and University of Chicago, guaranteeing convergence to the target distribution. Detailed balance and reversibility conditions tie to mathematical treatments found in texts from Cambridge University Press and research by statisticians such as Persi Diaconis, J. Michael Steele, Christian Robert and George Casella. Central limit theorems and spectral gap analyses, topics advanced by groups at University of California, San Diego and University of Oxford, provide quantitative convergence rates and variance estimates used in evaluations at Los Alamos National Laboratory and Columbia University.
Extensions include Metropolis-adjusted Langevin algorithms developed in collaborations at University of Toronto and University College London, adaptive schemes influenced by work at University of Warwick and Imperial College London, and blockupdating strategies used in hierarchical modeling at Harvard University and Stanford University. Other variants encompass independence samplers studied at Princeton University, reversible jump methods introduced by researchers associated with University of Cambridge and University of Exeter for trans-dimensional problems, and hybrid approaches that combine ideas from Hamiltonian Monte Carlo research at Princeton University and University of California, Berkeley. Tempering techniques such as simulated tempering and parallel tempering have roots in statistical physics research at Los Alamos National Laboratory and computational chemistry labs like Columbia University and University of California, Santa Barbara.
Implementations are available in software ecosystems supported by institutions like R Project for Statistical Computing, Python Software Foundation through packages developed at Stan Development Team and PyMC Community and in platforms maintained by GitHub and Bioconductor. Diagnostics for convergence, mixing and effective sample size draw on methods popularized by authors affiliated with University of Chicago, Duke University, Johns Hopkins University and Princeton University, including trace plots, autocorrelation analysis and Gelman–Rubin diagnostics attributed to work at Columbia University and Harvard University. Performance tuning often references benchmark problems from UCI Machine Learning Repository, optimization ideas from AT&T Bell Laboratories and numerical linear algebra libraries originating in work at Argonne National Laboratory and Lawrence Berkeley National Laboratory.
Metropolis–Hastings supports Bayesian computation in applied projects at National Institutes of Health, Centers for Disease Control and Prevention, World Health Organization and financial modeling in firms such as Goldman Sachs and Morgan Stanley, as well as population genetics studies at University of California, Davis and phylogenetics analyses using tools from Natural History Museum, London and Smithsonian Institution. It underpins inference in cosmology pipelines developed at European Space Agency and NASA teams, image reconstruction methods from Max Planck Society groups, and machine learning models deployed by research labs at Google, Facebook and DeepMind. Cross-disciplinary applications appear in ecology research at National Oceanic and Atmospheric Administration, neuroscience projects at Massachusetts General Hospital and econometrics studies at London School of Economics and University of Chicago.
Category:Markov chain Monte Carlo algorithms