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Wang G. On the theoretical properties of the exchange algorithm. BERNOULLI 2022. [DOI: 10.3150/21-bej1405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Guanyang Wang
- Department of Statistics, Rutgers University, Piscataway, NJ 08854, USA
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Morina G, Łatuszyński K, Nayar P, Wendland A. From the Bernoulli factory to a dice enterprise via perfect sampling of Markov chains. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1679] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | | | - Piotr Nayar
- Institute of Mathematics, University of Warsaw
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Andrieu C, Livingstone S. Peskun–Tierney ordering for Markovian Monte Carlo: Beyond the reversible scenario. Ann Stat 2021. [DOI: 10.1214/20-aos2008] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Jin R, Tan A. Fast Markov Chain Monte Carlo for High-Dimensional Bayesian Regression Models With Shrinkage Priors. J Comput Graph Stat 2021. [DOI: 10.1080/10618600.2020.1864383] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Rui Jin
- Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA
| | - Aixin Tan
- Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA
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Abstract
Abstract
Different Markov chains can be used for approximate sampling of a distribution given by an unnormalized density function with respect to the Lebesgue measure. The hit-and-run, (hybrid) slice sampler, and random walk Metropolis algorithm are popular tools to simulate such Markov chains. We develop a general approach to compare the efficiency of these sampling procedures by the use of a partial ordering of their Markov operators, the covariance ordering. In particular, we show that the hit-and-run and the simple slice sampler are more efficient than a hybrid slice sampler based on hit-and-run, which, itself, is more efficient than a (lazy) random walk Metropolis algorithm.
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Abstract
We propose adaptive incremental mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis–Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a stochastic process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions. Finally, the methodology is successfully applied to two real data examples, including the Bayesian inference of a semiparametric regression model for the Boston Housing dataset. Supplementary materials for this article are available online.
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Affiliation(s)
- Florian Maire
- Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Quebec, Canada
| | - Nial Friel
- School of Mathematics and Statistics, University College Dublin, Dublin, Ireland.,The Insight Centre for Data Analytics, Dublin, Ireland
| | - Antonietta Mira
- Data Science Lab, ICS, Università della Svizzera Italiana, Lugano, Switzerland.,DISAT, Università dell'Insubria, Varese, Italy
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Gonçalves FB, Łatuszyński K, Roberts GO. Barker’s algorithm for Bayesian inference with intractable likelihoods. BRAZ J PROBAB STAT 2017. [DOI: 10.1214/17-bjps374] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Duncan AB, Lelièvre T, Pavliotis GA. Variance Reduction Using Nonreversible Langevin Samplers. JOURNAL OF STATISTICAL PHYSICS 2016; 163:457-491. [PMID: 27453589 PMCID: PMC4939425 DOI: 10.1007/s10955-016-1491-2] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/16/2015] [Accepted: 02/29/2016] [Indexed: 06/06/2023]
Abstract
A standard approach to computing expectations with respect to a given target measure is to introduce an overdamped Langevin equation which is reversible with respect to the target distribution, and to approximate the expectation by a time-averaging estimator. As has been noted in recent papers [30, 37, 61, 72], introducing an appropriately chosen nonreversible component to the dynamics is beneficial, both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution. In this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility. Our theoretical findings are supported by numerical simulations.
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Affiliation(s)
- A. B. Duncan
- />Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ England
| | - T. Lelièvre
- />CERMICS, Ecole des ponts, Universit Paris-Est, 6-8 avenue Blaise Pascal, 77455 Marne la Valle Cedex 2, France
| | - G. A. Pavliotis
- />Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ England
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Lee A, atuszy ski K. Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation. Biometrika 2014. [DOI: 10.1093/biomet/asu027] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022] Open
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Maire F, Douc R, Olsson J. Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods. Ann Stat 2014. [DOI: 10.1214/14-aos1209] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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13
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Goldberg JF, Hebblewhite M, Bardsley J. Consequences of a refuge for the predator-prey dynamics of a wolf-elk system in Banff National Park, Alberta, Canada. PLoS One 2014; 9:e91417. [PMID: 24670632 PMCID: PMC3966774 DOI: 10.1371/journal.pone.0091417] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2013] [Accepted: 02/12/2014] [Indexed: 11/18/2022] Open
Abstract
Refugia can affect predator-prey dynamics via movements between refuge and non-refuge areas. We examine the influence of a refuge on population dynamics in a large mammal predator-prey system. Wolves (Canis lupus) have recolonized much of their former range in North America, and as a result, ungulate prey have exploited refugia to reduce predation risk with unknown impacts on wolf-prey dynamics. We examined the influence of a refuge on elk (Cervus elaphus) and wolf population dynamics in Banff National Park. Elk occupy the Banff townsite with little predation, whereas elk in the adjoining Bow Valley experience higher wolf predation. The Banff refuge may influence Bow Valley predator-prey dynamics through source-sink movements. To test this hypothesis, we used 26 years of wolf and elk population counts and the Delayed Rejection Adaptive Metropolis Markov chain Monte Carlo method to fit five predator-prey models: 1) with no source-sink movements, 2) with elk density-dependent dispersal from the refuge to the non-refuge, 3) with elk predation risk avoidance movements from the non-refuge to the refuge, 4) with differential movement rates between refuge and non-refuge, and 5) with short-term, source-sink wolf movements. Model 1 provided the best fit of the data, as measured by Akaike Information Criterion (AIC). In the top model, Banff and Bow Valley elk had median growth rates of 0.08 and 0.03 (95% credibility intervals [CIs]: 0.027-0.186 and 0.001-0.143), respectively, Banff had a median carrying capacity of 630 elk (95% CI: 471.9-2676.9), Bow Valley elk had a median wolf encounter rate of 0.02 (95% CI: 0.013-0.030), and wolves had a median death rate of 0.23 (95% CI: 0.146-0.335) and a median conversion efficiency of 0.07 (95% CI: 0.031-0.124). We found little evidence for potential source-sink movements influencing the predator-prey dynamics of this system. This result suggests that the refuge was isolated from the non-refuge.
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Affiliation(s)
- Joshua F. Goldberg
- Wildlife Biology Program, Department of Ecosystem and Conservation Science, College of Forestry and Conservation, University of Montana, Missoula, Montana, United States of America
| | - Mark Hebblewhite
- Wildlife Biology Program, Department of Ecosystem and Conservation Science, College of Forestry and Conservation, University of Montana, Missoula, Montana, United States of America
| | - John Bardsley
- Department of Mathematics and Statistics, University of Montana, Missoula, Montana, United States of America
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Roberts GO, Rosenthal JS. Minimising MCMC variance via diffusion limits, with an application to simulated tempering. ANN APPL PROBAB 2014. [DOI: 10.1214/12-aap918] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Bédard M, Mireuta M. On the empirical efficiency of local MCMC algorithms with pools of proposals. CAN J STAT 2013. [DOI: 10.1002/cjs.11196] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Mylène Bédard
- Département de mathématiques et de statistique; Université de Montréal; Montréal Québec Canada
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Woodard DB, Rosenthal JS. Convergence rate of Markov chain methods for genomic motif discovery. Ann Stat 2013. [DOI: 10.1214/12-aos1075] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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22
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Rigat F, Mira A. Parallel hierarchical sampling: A general-purpose interacting Markov chains Monte Carlo algorithm. Comput Stat Data Anal 2012. [DOI: 10.1016/j.csda.2011.11.020] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Höhna S, Drummond AJ. Guided Tree Topology Proposals for Bayesian Phylogenetic Inference. Syst Biol 2011; 61:1-11. [DOI: 10.1093/sysbio/syr074] [Citation(s) in RCA: 94] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sebastian Höhna
- Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Alexei J. Drummond
- Department of Computer Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
- Allan Wilson Centre for Molecular Ecology and Evolution, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
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Rosenthal J. Optimal Proposal Distributions and Adaptive MCMC. CHAPMAN & HALL/CRC HANDBOOKS OF MODERN STATISTICAL METHODS 2011. [DOI: 10.1201/b10905-5] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/03/2022]
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Mira A. Stationarity preserving and efficiency increasing probability mass transfers made possible. Comput Stat 2006. [DOI: 10.1007/s00180-006-0009-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Levine RA. A note on Markov chain Monte Carlo sweep strategies. J STAT COMPUT SIM 2005. [DOI: 10.1080/0094965042000223671] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Wu SS, Wells MT. An Extension of the Metropolis Algorithm. COMMUN STAT-THEOR M 2005. [DOI: 10.1081/sta-200052116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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Levine RA, Yu Z, Hanley WG, Nitao JJ. Implementing componentwise Hastings algorithms. Comput Stat Data Anal 2005. [DOI: 10.1016/j.csda.2004.02.002] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Bressanini D, Morosi G, Tarasco S, Mira A. Delayed rejection variational Monte Carlo. J Chem Phys 2004; 121:3446-51. [PMID: 15303908 DOI: 10.1063/1.1775789] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
An acceleration algorithm to address the problem of multiple time scales in variational Monte Carlo simulations is presented. After a first attempted move has been rejected, the delayed rejection algorithm attempts a second move with a smaller time step, so that even moves of the core electrons can be accepted. Results on Be and Ne atoms as test cases are presented. Correlation time and both average accepted displacement and acceptance ratio as a function of the distance from the nucleus evidence the efficiency of the proposed algorithm in dealing with the multiple time scales problem.
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Affiliation(s)
- Dario Bressanini
- Dipartimento di Scienze Chimiche ed Ambientali, Universita' dell'Insubria, Sede di Como, via Lucini 3, 22100 Como, Italy.
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