Scholar4AI Updated 2026-11-05 06:02 UTC+09:00

Markov chain Monte Carlo (Metropolis-Hastings)

Share: X LinkedIn Bluesky Threads Facebook Reddit Hacker News Hatena LINE Email
Copy:

area: Statistics and Bayesian methods · 3 papers in the last 7 days

← Most-used equations and models · All statistics and bayesian methods equations

α=min(1, π(θ′)q(θ∣θ′)π(θ)q(θ′∣θ))

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

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

Names searched for: Markov chain Monte Carlo, Metropolis-Hastings, Gibbs sampling, Gibbs sampler, MCMC

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.

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#

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.

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)
TrendNov 23–Oct 24Nov 24–Oct 25Nov 25–Oct 2636 months
3643194041,087

Papers per year#

2017: 104 papers2018: 108 papers2019: 99 papers2020: 113 papers2021: 121 papers2022: 129 papers2023: 162 papers2024: 339 papers2025: 331 papers2026 (so far): 351 papers
20172026 (so far)
Papers per publication year and category
Category2017201820192020202120222023202420252026 (so far)
Statistics64546057708789124150179
AI22211725221926836143
Economics1015713141118644667
Operations Research71011126517183732
Finance0233456251518
Management10012011062
Health Economics and Policy0310202446
Higher Education and Research0000101361
Information Systems0101010410
Tourism and Hospitality0101010113
Marketing0000002330
Accounting0100000010
All categories10410899113121129162339331351

Where the papers come from#

Journals with the most papers, since 2023
#JournalPapers
1Statistics and Computing172
2Computational Statistics56
3Computational Statistics & Data Analysis53
4Spatial Statistics36
5Journal of Econometrics26
6Journal of Multivariate Analysis21
7IEEE Transactions on Pattern Analysis and Machine Intelligence21
8Artificial Intelligence Review19
9Machine Learning18
10Statistical Papers17
Institutions with the most papers, since 2023
#InstitutionPapers
1Centre National de la Recherche Scientifique30
2University of Warwick21
3University College London19
4Lancaster University18
5University of Cambridge18
6University of Edinburgh17
7UNSW Sydney15
8Monash University14
9Australian National University13
10Université Paris-Saclay12
Countries with the most papers, since 2023
#CountryPapers
1United States252
2United Kingdom211
3China164
4Italy118
5Germany101
6Australia77
7France72
8Canada56
9Spain55
10Netherlands51
Keywords with the most papers, since 2023
#KeywordPapers
1Markov chain Monte Carlo158
2Gibbs sampling96
3Bayesian inference76
4Bayesian estimation58
5uncertainty quantification45
6computational efficiency43
7maximum likelihood estimation33
8Metropolis-Hastings algorithm31
9parameter estimation31
10COVID-19 pandemic29

Most-cited papers#

Most-cited papers since 2017
#CitationsPer monthTitleJournalPublishedAuthors
154815.7A review of ensemble learning and data augmentation models for class imbalanced problems: Combination, implementation and evaluationExpert Systems with Applications2023-12-09Azal Ahmad Khan, Omkar Chaudhari, Rohitash Chandra
240121.3A comprehensive survey of loss functions and metrics in deep learningArtificial Intelligence Review2025-04-11Juan R. Terven, Diana Margarita Córdova Esparza, Julio-Alejandro Romero-González et al.
32633.8Bayesian Temporal Factorization for Multidimensional Time Series PredictionIEEE Transactions on Pattern Analysis and Machine Intelligence2021-01-01Xinyu Chen, Lijun Sun
42423.3Database-independent molecular formula annotation using Gibbs sampling through ZODIACNature Machine Intelligence2020-10-13Marcus Ludwig, Louis‐Félix Nothias, Kai Dührkop et al.
51765.6A review of predictive uncertainty estimation with machine learningArtificial Intelligence Review2024-03-18Hristos Tyralis, Georgia Papacharalampous
61644.8A survey of multi-agent deep reinforcement learning with communicationAutonomous Agents and Multi-Agent Systems2024-01-06Changxi Zhu, Mehdi M. Dastani, Shihan Wang
71591.4Bayesian estimation of agent-based modelsJournal of Economic Dynamics and Control2017-02-05Jakob Grazzini, Matteo Guido Richiardi, Efthymios G. Tsionas
81504.9Automated machine learning: past, present and futureArtificial Intelligence Review2024-04-18Mitra Baratchi, Can Wang, Steffen Limmer et al.
91324.0A survey on neural topic models: methods, applications, and challengesArtificial Intelligence Review2024-01-25Xiaobao Wu, Thong Nguyen, Anh Tuan Luu
101231.3Community-based influence maximization for viral marketingApplied Intelligence2019-01-01Huimin 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.
DateTitleJournalAuthorsAffiliations
2026-11-02Simultaneous covariate selection and change-point detection in dependent binary time series: a sparse Bayesian approachComputational StatisticsMasoud MajidizadehShahid Beheshti University
2026-10-31Gaussian variational Bayes approximate inference for structural equation modelsStatistical PapersYing Wu, Hongyu Zhu, Ping Ren et al.Yunnan University; Luoyang Normal University
2026-10-30Chance constrained optimization of hub transport systemsComputers & Operations ResearchMax 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); …
2026-10-28Efficient optimization for sparse large-scale problems: A Markov chain Monte Carlo-driven population sampling strategyExpert Systems with ApplicationsTao Zhang, Sheng Qi, Benlu Gan et al.National University of Defense Technology; China Tobacco; China Tobacco Hunan Industrial Co., Ltd. (China); …
2026-10-23Generalized and scalable deep Gaussian process emulationComputational Statistics & Data AnalysisDeyu Ming, Daniel WilliamsonUniversity College London; University of Exeter
2026-10-23Global–local shrinkage priors for modeling random effects in multivariate spatial small area estimationSpatial StatisticsShushi Nishina, Takahiro Onizuka, Shintaro HashimotoHiroshima University; Chiba University
2026-10-23An infinite BART modelStatistics and ComputingMarco Battiston, Yu LuoLancaster University; King's College London
2026-10-22Long term survival model with frailt terms based on the power variance function familyStatistical Methods & ApplicationsMichelle M. Sacramento, Gladys D. C. Barriga, Vicente G. Cancho et al.Universidade de São Paulo; Universidade Estadual Paulista (Unesp)
2026-10-22High-dimensional normal approximations for sums of Langevin Markov chainsStochastic Processes and their ApplicationsTian Shen, Zhonggen Su, Xiaolin WangShanghai University of International Business and Economics; Zhejiang University; Chinese University of Hong Kong
2026-10-21Unamortized black-box variational inference for joint response and response time modelsBehaviormetrikaRudolf Debelak, Dylan MolenaarÉcole Polytechnique Fédérale de Lausanne; University of Amsterdam
2026-10-20Sequential Monte Carlo sampling with integrator snippetsJapanese Journal of Statistics and Data ScienceChristophe Andrieu, Mauro Camara Escudero, Chang ZhangUniversity of Bristol
2026-10-20Fast simulation of Gaussian random fields with flexible correlation models in Euclidean spacesStatistics and ComputingMoreno Bevilacqua, Xavier Emery, Francisco Cuevas‐PachecoCa' Foscari University of Venice; Adolfo Ibáñez University; Advanced Mining Technology Center; …
2026-10-17Annealed leap-point sampler for multimodal target distributionsStatistics and ComputingNicholas G. Tawn, Matthew T. Moores, Hugo Queniat et al.University of Warwick; Lappeenranta-Lahti University of Technology
2026-10-15Infinite hidden Markov models for cylindrical dataStatistics & Probability LettersFederico P. Cortese, Luca RossiniUniversity of Milan; National Research Council; Fondazione Eni Enrico Mattei
2026-10-14Semi-parametric Bayesian inference to shape-restricted partially linear additive mixed-effects negative binomial models for longitudinal dataJournal of the Korean Statistical SocietyMiaojie Xia, Ya Liu, Jiang DuBeijing University of Technology
2026-10-14Increasing domain asymptotics for covariate-based nonparametric Bayesian intensity estimation with Gaussian and Besov–Laplace priorsJournal of Multivariate AnalysisPatric Dolmeta, Matteo GiordanoUniversity of Turin
2026-10-13Posterior sampling of word embeddingsStatistics and ComputingVäinö Yrjänäinen, Isac Boström, Johan Jonasson et al.Uppsala University; Chalmers University of Technology
2026-10-13Machine Learning Parameters Identification for a New Unit Power-Logistic Quadratic Lindley ModelJournal of Statistical Theory and PracticeAhmed ElhassaneinUniversity of Bisha
2026-10-08Wealth and weight revisited: A Bayesian model averaging approach to the Obesity Kuznets CurveThe European Journal of Health EconomicsKaterine Restrepo Gómez, H. Du, C. MendezNagoya University
2026-10-08Are large language models able to predict highly cited papers? Evidence from statistical publicationsScientometricsZhanshuo Ye, Yiming Hou, Rui Pan et al.Central University of Finance and Economics; Peking University
2026-10-08SAT-sampling for statistical significance testing in sparse contingency tablesStatistics and ComputingPatrick Scharpfenecker, Tobias WindischKempten University of Applied Sciences
2026-10-07Bayesian Transfer Learning of Spatial Lag Stochastic Frontier AnalysisNetworks and Spatial EconomicsYitao Zhu, Yunquan SongChina 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

@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}
}

Free to reuse for non-commercial purposes with attribution (CC BY-NC 4.0).