---
title: "Markov chain Monte Carlo (Metropolis-Hastings)"
description: "Markov chain Monte Carlo (Metropolis-Hastings) explained: variables, a worked example, related methods and assumptions. Draws from a posterior distribution by a random walk that accepts proposals with probability alpha."
url: https://scholar4ai.com/equations/markov-chain-monte-carlo-metropolis-hastings
markdown_url: https://scholar4ai.com/equations/markov-chain-monte-carlo-metropolis-hastings.md
license: CC BY-NC 4.0 (https://creativecommons.org/licenses/by-nc/4.0/)
share_text: "Markov chain Monte Carlo (Metropolis-Hastings) explained: variables, a worked example, related methods and assumptions. Draws from a posterior distribution by a random walk that accepts proposals with probability alpha. (as of 2026-11-05)"
data_url: https://scholar4ai.com/data/rankings.json
updated: 2026-11-05T06:02:37+09:00
window: 2026-10-07 to 2026-11-05
source: OpenAlex (CC0)
---

# Markov chain Monte Carlo (Metropolis-Hastings)

$$
\alpha = \min\left(1,\ \frac{\pi(\theta')\, q(\theta \mid \theta')}{\pi(\theta)\, q(\theta' \mid \theta)}\right)
$$

Draws from a posterior distribution by a random walk that accepts proposals with probability alpha.

- area: Statistics and Bayesian methods
- 3 papers in the last 7 days
- Names searched for: Markov chain Monte Carlo, Metropolis-Hastings, Gibbs sampling, Gibbs sampler, MCMC

Back to the ranking: https://scholar4ai.com/rankings/equations.md

## Variables and notation

pi is the target density, q the proposal density, theta current state, theta' proposed state and alpha the acceptance probability.

## Worked example

For a symmetric proposal and target-density ratio pi(theta')/pi(theta)=0.4, accept with probability 0.4. A uniform draw of 0.3 accepts; 0.7 rejects.

## Related methods and when to use them

Metropolis–Hastings draws a dependent chain; independent sampling does not reuse the current state. Proposal asymmetry requires the q correction; convergence and effective sample size need diagnostics.

## Sources and further reading

- [Hastings (1970), Monte Carlo sampling methods using Markov chains and their applications](https://doi.org/10.1093/biomet/57.1.97)

Examples use illustrative inputs, not observed research results. Logarithms are natural unless stated otherwise. Guide updated 2026-11-05. Prepared by Scholar4AI. [Editorial and verification policy](/editorial-policy).

## Key findings

- Markov chain Monte Carlo (Metropolis-Hastings) was mentioned in 22 papers in the last 30 days (2026-10-07 to 2026-11-05), ranking #76 of the 248 equations and models mentioned.
- By journal category: Statistics (17), Operations Research (2), AI (1), Health Economics and Policy (1) and Higher Education and Research (1).
- Most of them appeared in Statistics and Computing (5), Computational Statistics (1) and Statistical Papers (1).
- Often mentioned together with Gaussian process regression (6), Logistic regression (logit) (5) and Maximum likelihood estimation (5).
- Datasets named in the same papers: S&P 500 (1), American Community Survey (1) and MovieLens (1).
- From Nov 2023 to Oct 2026, 1,087 archived papers mentioned Markov chain Monte Carlo (Metropolis-Hastings) in their title, abstract or keywords (the search of earlier months' full texts is still under way), most of them in Jul 2026 (49).
- From 2017 to 2026 (so far), Markov chain Monte Carlo (Metropolis-Hastings) was mentioned in 1,857 papers in the tracked journals, most of them in statistics (934), AI (339) and economics (265).
- 331 of them were published in 2025, against 104 in 2017.
- Since 2023, the institutions with the most of these papers were Centre National de la Recherche Scientifique (30), University of Warwick (21) and University College London (19).
- The journals publishing most of them were Statistics and Computing (172), Computational Statistics (56) and Computational Statistics & Data Analysis (53).
- The most-cited of them is “A review of ensemble learning and data augmentation models for class imbalanced problems: Combination, implementation and evaluation” (Expert Systems with Applications, 2023), with 548 citations.

## Mentions over 36 months

*Papers mentioning it by publication month, Nov 2023 to Oct 2026 (titles, abstracts and keywords; the full texts of earlier months are still being searched)*

| Trend | Nov 23–Oct 24 | Nov 24–Oct 25 | Nov 25–Oct 26 | 36 months |
|---|---:|---:|---:|---:|
|  | 364 | 319 | 404 | 1,087 |

## Papers per year

*Papers per publication year and category*

| Category | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 | 2025 | 2026 (so far) |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| [Statistics](https://scholar4ai.com/categories/statistics.md) | 64 | 54 | 60 | 57 | 70 | 87 | 89 | 124 | 150 | 179 |
| [AI](https://scholar4ai.com/categories/ai.md) | 22 | 21 | 17 | 25 | 22 | 19 | 26 | 83 | 61 | 43 |
| [Economics](https://scholar4ai.com/categories/economics.md) | 10 | 15 | 7 | 13 | 14 | 11 | 18 | 64 | 46 | 67 |
| [Operations Research](https://scholar4ai.com/categories/operations-research.md) | 7 | 10 | 11 | 12 | 6 | 5 | 17 | 18 | 37 | 32 |
| [Finance](https://scholar4ai.com/categories/finance.md) | 0 | 2 | 3 | 3 | 4 | 5 | 6 | 25 | 15 | 18 |
| [Management](https://scholar4ai.com/categories/management.md) | 1 | 0 | 0 | 1 | 2 | 0 | 1 | 10 | 6 | 2 |
| [Health Economics and Policy](https://scholar4ai.com/categories/health-economics-and-policy.md) | 0 | 3 | 1 | 0 | 2 | 0 | 2 | 4 | 4 | 6 |
| [Higher Education and Research](https://scholar4ai.com/categories/higher-education-and-research.md) | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 3 | 6 | 1 |
| [Information Systems](https://scholar4ai.com/categories/information-systems.md) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 4 | 1 | 0 |
| [Tourism and Hospitality](https://scholar4ai.com/categories/tourism-and-hospitality.md) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 3 |
| [Marketing](https://scholar4ai.com/categories/marketing.md) | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 3 | 0 |
| [Accounting](https://scholar4ai.com/categories/accounting.md) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| All categories | 104 | 108 | 99 | 113 | 121 | 129 | 162 | 339 | 331 | 351 |

## Where the papers come from

*Journals with the most papers, since 2023*

| # | Journal | Papers |
|---:|---|---:|
| 1 | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | 172 |
| 2 | [Computational Statistics](https://scholar4ai.com/journals/computational-statistics.md) | 56 |
| 3 | [Computational Statistics & Data Analysis](https://scholar4ai.com/journals/computational-statistics-data-analysis.md) | 53 |
| 4 | [Spatial Statistics](https://scholar4ai.com/journals/spatial-statistics.md) | 36 |
| 5 | [Journal of Econometrics](https://scholar4ai.com/journals/journal-of-econometrics.md) | 26 |
| 6 | [Journal of Multivariate Analysis](https://scholar4ai.com/journals/journal-of-multivariate-analysis.md) | 21 |
| 7 | [IEEE Transactions on Pattern Analysis and Machine Intelligence](https://scholar4ai.com/journals/ieee-transactions-on-pattern-analysis-and-machine-intelligence.md) | 21 |
| 8 | [Artificial Intelligence Review](https://scholar4ai.com/journals/artificial-intelligence-review.md) | 19 |
| 9 | [Machine Learning](https://scholar4ai.com/journals/machine-learning.md) | 18 |
| 10 | [Statistical Papers](https://scholar4ai.com/journals/statistical-papers.md) | 17 |

*Institutions with the most papers, since 2023*

| # | Institution | Papers |
|---:|---|---:|
| 1 | [Centre National de la Recherche Scientifique](https://scholar4ai.com/institutions/centre-national-de-la-recherche-scientifique.md) | 30 |
| 2 | [University of Warwick](https://scholar4ai.com/institutions/university-of-warwick.md) | 21 |
| 3 | [University College London](https://scholar4ai.com/institutions/university-college-london.md) | 19 |
| 4 | [Lancaster University](https://scholar4ai.com/institutions/lancaster-university.md) | 18 |
| 5 | [University of Cambridge](https://scholar4ai.com/institutions/university-of-cambridge.md) | 18 |
| 6 | [University of Edinburgh](https://scholar4ai.com/institutions/university-of-edinburgh.md) | 17 |
| 7 | [UNSW Sydney](https://scholar4ai.com/institutions/unsw-sydney.md) | 15 |
| 8 | [Monash University](https://scholar4ai.com/institutions/monash-university.md) | 14 |
| 9 | [Australian National University](https://scholar4ai.com/institutions/australian-national-university.md) | 13 |
| 10 | [Université Paris-Saclay](https://scholar4ai.com/institutions/universite-paris-saclay.md) | 12 |

*Countries with the most papers, since 2023*

| # | Country | Papers |
|---:|---|---:|
| 1 | United States | 252 |
| 2 | United Kingdom | 211 |
| 3 | China | 164 |
| 4 | Italy | 118 |
| 5 | Germany | 101 |
| 6 | Australia | 77 |
| 7 | France | 72 |
| 8 | Canada | 56 |
| 9 | Spain | 55 |
| 10 | Netherlands | 51 |

*Keywords with the most papers, since 2023*

| # | Keyword | Papers |
|---:|---|---:|
| 1 | Markov chain Monte Carlo | 158 |
| 2 | Gibbs sampling | 96 |
| 3 | Bayesian inference | 76 |
| 4 | Bayesian estimation | 58 |
| 5 | [uncertainty quantification](https://scholar4ai.com/keywords/uncertainty-quantification.md) | 45 |
| 6 | [computational efficiency](https://scholar4ai.com/keywords/computational-efficiency.md) | 43 |
| 7 | [maximum likelihood estimation](https://scholar4ai.com/keywords/maximum-likelihood-estimation.md) | 33 |
| 8 | Metropolis-Hastings algorithm | 31 |
| 9 | [parameter estimation](https://scholar4ai.com/keywords/parameter-estimation.md) | 31 |
| 10 | [COVID-19 pandemic](https://scholar4ai.com/keywords/covid-19-pandemic.md) | 29 |

## Most-cited papers

*Most-cited papers since 2017*

| # | Citations | Per month | Title | Journal | Published | Authors |
|---:|---:|---:|---|---|---|---|
| 1 | 548 | 15.7 | [A review of ensemble learning and data augmentation models for class imbalanced problems: Combination, implementation and evaluation](https://doi.org/10.1016/j.eswa.2023.122778) | [Expert Systems with Applications](https://scholar4ai.com/journals/expert-systems-with-applications.md) | 2023-12-09 | Azal Ahmad Khan, Omkar Chaudhari, Rohitash Chandra |
| 2 | 401 | 21.3 | [A comprehensive survey of loss functions and metrics in deep learning](https://doi.org/10.1007/s10462-025-11198-7) | [Artificial Intelligence Review](https://scholar4ai.com/journals/artificial-intelligence-review.md) | 2025-04-11 | Juan R. Terven, Diana Margarita Córdova Esparza, Julio-Alejandro Romero-González et al. |
| 3 | 263 | 3.8 | [Bayesian Temporal Factorization for Multidimensional Time Series Prediction](https://doi.org/10.1109/tpami.2021.3066551) | [IEEE Transactions on Pattern Analysis and Machine Intelligence](https://scholar4ai.com/journals/ieee-transactions-on-pattern-analysis-and-machine-intelligence.md) | 2021-01-01 | Xinyu Chen, Lijun Sun |
| 4 | 242 | 3.3 | [Database-independent molecular formula annotation using Gibbs sampling through ZODIAC](https://doi.org/10.1038/s42256-020-00234-6) | [Nature Machine Intelligence](https://scholar4ai.com/journals/nature-machine-intelligence.md) | 2020-10-13 | Marcus Ludwig, Louis‐Félix Nothias, Kai Dührkop et al. |
| 5 | 176 | 5.6 | [A review of predictive uncertainty estimation with machine learning](https://doi.org/10.1007/s10462-023-10698-8) | [Artificial Intelligence Review](https://scholar4ai.com/journals/artificial-intelligence-review.md) | 2024-03-18 | Hristos Tyralis, Georgia Papacharalampous |
| 6 | 164 | 4.8 | [A survey of multi-agent deep reinforcement learning with communication](https://doi.org/10.1007/s10458-023-09633-6) | [Autonomous Agents and Multi-Agent Systems](https://scholar4ai.com/journals/autonomous-agents-and-multi-agent-systems.md) | 2024-01-06 | Changxi Zhu, Mehdi M. Dastani, Shihan Wang |
| 7 | 159 | 1.4 | [Bayesian estimation of agent-based models](https://doi.org/10.1016/j.jedc.2017.01.014) | [Journal of Economic Dynamics and Control](https://scholar4ai.com/journals/journal-of-economic-dynamics-and-control.md) | 2017-02-05 | Jakob Grazzini, Matteo Guido Richiardi, Efthymios G. Tsionas |
| 8 | 150 | 4.9 | [Automated machine learning: past, present and future](https://doi.org/10.1007/s10462-024-10726-1) | [Artificial Intelligence Review](https://scholar4ai.com/journals/artificial-intelligence-review.md) | 2024-04-18 | Mitra Baratchi, Can Wang, Steffen Limmer et al. |
| 9 | 132 | 4.0 | [A survey on neural topic models: methods, applications, and challenges](https://doi.org/10.1007/s10462-023-10661-7) | [Artificial Intelligence Review](https://scholar4ai.com/journals/artificial-intelligence-review.md) | 2024-01-25 | Xiaobao Wu, Thong Nguyen, Anh Tuan Luu |
| 10 | 123 | 1.3 | [Community-based influence maximization for viral marketing](https://doi.org/10.1007/s10489-018-1387-8) | [Applied Intelligence](https://scholar4ai.com/journals/applied-intelligence.md) | 2019-01-01 | Huimin Huang, Hong Shen, Zaiqiao Meng et al. |

Archive counts by publication year: a paper counts when its title, abstract or keywords name the equation, or when OpenAlex's full-text search finds it there (searched for the last 30 days' papers and, by a pass over the past years, for older ones). Names added to the list later are looked for in older papers as the archive is read again.

*Papers mentioning Markov chain Monte Carlo (Metropolis-Hastings), newest first.*

| Date | Title | Journal | Authors | Affiliations |
|---|---|---|---|---|
| 2026-11-02 | [Simultaneous covariate selection and change-point detection in dependent binary time series: a sparse Bayesian approach](https://doi.org/10.1007/s00180-026-01819-2) | [Computational Statistics](https://scholar4ai.com/journals/computational-statistics.md) | Masoud Majidizadeh | Shahid Beheshti University |
| 2026-10-31 | [Gaussian variational Bayes approximate inference for structural equation models](https://doi.org/10.1007/s00362-026-01909-9) | [Statistical Papers](https://scholar4ai.com/journals/statistical-papers.md) | Ying Wu, Hongyu Zhu, Ping Ren et al. | Yunnan University; Luoyang Normal University |
| 2026-10-30 | [Chance constrained optimization of hub transport systems](https://doi.org/10.1016/j.cor.2026.107709) | [Computers & Operations Research](https://scholar4ai.com/journals/computers-operations-research.md) | Max Bourgeat, Öykü Naz Attila, Rémy Chevrier et al. | Centre interuniversitaire de recherche sur les réseaux d'entreprise, la logistique et le transport; Polytechnique Montréal; Société Nationale des Chemins de Fer Français (France); … |
| 2026-10-28 | [Efficient optimization for sparse large-scale problems: A Markov chain Monte Carlo-driven population sampling strategy](https://doi.org/10.1016/j.eswa.2026.134575) | [Expert Systems with Applications](https://scholar4ai.com/journals/expert-systems-with-applications.md) | Tao Zhang, Sheng Qi, Benlu Gan et al. | National University of Defense Technology; China Tobacco; China Tobacco Hunan Industrial Co., Ltd. (China); … |
| 2026-10-23 | [Generalized and scalable deep Gaussian process emulation](https://doi.org/10.1016/j.csda.2026.108485) | [Computational Statistics & Data Analysis](https://scholar4ai.com/journals/computational-statistics-data-analysis.md) | Deyu Ming, Daniel Williamson | University College London; University of Exeter |
| 2026-10-23 | [Global–local shrinkage priors for modeling random effects in multivariate spatial small area estimation](https://doi.org/10.1016/j.spasta.2026.101062) | [Spatial Statistics](https://scholar4ai.com/journals/spatial-statistics.md) | Shushi Nishina, Takahiro Onizuka, Shintaro Hashimoto | Hiroshima University; Chiba University |
| 2026-10-23 | [An infinite BART model](https://doi.org/10.1007/s11222-026-11006-4) | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | Marco Battiston, Yu Luo | Lancaster University; King's College London |
| 2026-10-22 | [Long term survival model with frailt terms based on the power variance function family](https://doi.org/10.1007/s10260-026-00889-2) | [Statistical Methods & Applications](https://scholar4ai.com/journals/statistical-methods-applications.md) | Michelle M. Sacramento, Gladys D. C. Barriga, Vicente G. Cancho et al. | Universidade de São Paulo; Universidade Estadual Paulista (Unesp) |
| 2026-10-22 | [High-dimensional normal approximations for sums of Langevin Markov chains](https://doi.org/10.1016/j.spa.2026.105124) | [Stochastic Processes and their Applications](https://scholar4ai.com/journals/stochastic-processes-and-their-applications.md) | Tian Shen, Zhonggen Su, Xiaolin Wang | Shanghai University of International Business and Economics; Zhejiang University; Chinese University of Hong Kong |
| 2026-10-21 | [Unamortized black-box variational inference for joint response and response time models](https://doi.org/10.1007/s41237-026-00321-9) | [Behaviormetrika](https://scholar4ai.com/journals/behaviormetrika.md) | Rudolf Debelak, Dylan Molenaar | École Polytechnique Fédérale de Lausanne; University of Amsterdam |
| 2026-10-20 | [Sequential Monte Carlo sampling with integrator snippets](https://doi.org/10.1007/s42081-026-00381-x) | [Japanese Journal of Statistics and Data Science](https://scholar4ai.com/journals/japanese-journal-of-statistics-and-data-science.md) | Christophe Andrieu, Mauro Camara Escudero, Chang Zhang | University of Bristol |
| 2026-10-20 | [Fast simulation of Gaussian random fields with flexible correlation models in Euclidean spaces](https://doi.org/10.1007/s11222-026-11002-8) | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | Moreno Bevilacqua, Xavier Emery, Francisco Cuevas‐Pacheco | Ca' Foscari University of Venice; Adolfo Ibáñez University; Advanced Mining Technology Center; … |
| 2026-10-17 | [Annealed leap-point sampler for multimodal target distributions](https://doi.org/10.1007/s11222-026-10988-5) | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | Nicholas G. Tawn, Matthew T. Moores, Hugo Queniat et al. | University of Warwick; Lappeenranta-Lahti University of Technology |
| 2026-10-15 | [Infinite hidden Markov models for cylindrical data](https://doi.org/10.1016/j.spl.2026.110995) | [Statistics & Probability Letters](https://scholar4ai.com/journals/statistics-probability-letters.md) | Federico P. Cortese, Luca Rossini | University of Milan; National Research Council; Fondazione Eni Enrico Mattei |
| 2026-10-14 | [Semi-parametric Bayesian inference to shape-restricted partially linear additive mixed-effects negative binomial models for longitudinal data](https://doi.org/10.1007/s42952-026-00394-y) | [Journal of the Korean Statistical Society](https://scholar4ai.com/journals/journal-of-the-korean-statistical-society.md) | Miaojie Xia, Ya Liu, Jiang Du | Beijing University of Technology |
| 2026-10-14 | [Increasing domain asymptotics for covariate-based nonparametric Bayesian intensity estimation with Gaussian and Besov–Laplace priors](https://doi.org/10.1016/j.jmva.2026.105715) | [Journal of Multivariate Analysis](https://scholar4ai.com/journals/journal-of-multivariate-analysis.md) | Patric Dolmeta, Matteo Giordano | University of Turin |
| 2026-10-13 | [Posterior sampling of word embeddings](https://doi.org/10.1007/s11222-026-10996-5) | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | Väinö Yrjänäinen, Isac Boström, Johan Jonasson et al. | Uppsala University; Chalmers University of Technology |
| 2026-10-13 | [Machine Learning Parameters Identification for a New Unit Power-Logistic Quadratic Lindley Model](https://doi.org/10.1007/s42519-026-00644-6) | [Journal of Statistical Theory and Practice](https://scholar4ai.com/journals/journal-of-statistical-theory-and-practice.md) | Ahmed Elhassanein | University of Bisha |
| 2026-10-08 | [Wealth and weight revisited: A Bayesian model averaging approach to the Obesity Kuznets Curve](https://doi.org/10.1007/s10198-026-01998-1) | [The European Journal of Health Economics](https://scholar4ai.com/journals/the-european-journal-of-health-economics.md) | Katerine Restrepo Gómez, H. Du, C. Mendez | Nagoya University |
| 2026-10-08 | [Are large language models able to predict highly cited papers? Evidence from statistical publications](https://doi.org/10.1007/s11192-026-05839-z) | [Scientometrics](https://scholar4ai.com/journals/scientometrics.md) | Zhanshuo Ye, Yiming Hou, Rui Pan et al. | Central University of Finance and Economics; Peking University |
| 2026-10-08 | [SAT-sampling for statistical significance testing in sparse contingency tables](https://doi.org/10.1007/s11222-026-10985-8) | [Statistics and Computing](https://scholar4ai.com/journals/statistics-and-computing.md) | Patrick Scharpfenecker, Tobias Windisch | Kempten University of Applied Sciences |
| 2026-10-07 | [Bayesian Transfer Learning of Spatial Lag Stochastic Frontier Analysis](https://doi.org/10.1007/s11067-026-09793-4) | [Networks and Spatial Economics](https://scholar4ai.com/journals/networks-and-spatial-economics.md) | Yitao Zhu, Yunquan Song | China University of Petroleum, East China |

## Cite this page

Scholar4AI. (2026). Markov chain Monte Carlo (Metropolis-Hastings). Journal and Conference Research Rankings. Retrieved November 5, 2026, from https://scholar4ai.com/equations/markov-chain-monte-carlo-metropolis-hastings

```bibtex
@misc{scholar4ai-equations-markov-chain-monte-carlo-metropolis-hastings,
  author = {{Scholar4AI}},
  title = {Markov chain Monte Carlo (Metropolis-Hastings)},
  year = {2026},
  howpublished = {\url{https://scholar4ai.com/equations/markov-chain-monte-carlo-metropolis-hastings}},
  urldate = {2026-11-05},
  note = {Journal and Conference Research Rankings; data from OpenAlex (CC0); licensed CC BY-NC 4.0}
}
```

License: CC BY-NC 4.0 (https://creativecommons.org/licenses/by-nc/4.0/).