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ON THE ACCEPT-REJECT MECHANISM FOR metropolis-hastings algorithms

ANNALS OF APPLIED PROBABILITY 2023年第6B期33卷 5279-5333页

作者： Glatt-holtz, Nathan Krometis, Justin Mondaini, Cecilia Tulane Univ Dept Math New Orleans LA 70118 USA Virginia Tech Natl Secur Inst Blacksburg VA USA Virginia Tech Dept Math Blacksburg VA USA Drexel Univ Dept Math Philadelphia PA USA

This work develops a powerful and versatile framework for determin-ing acceptance ratios in metropolis-hastings-type Markov kernels widely used in statistical sampling problems. Our approach allows us to derive new classes of kernels which unify random walk or diffusion-type sampling meth-ods with more complicated "extended phase space" algorithms based around ideas from Hamiltonian dynamics. Our starting point is an abstract result de-veloped in the generality of measurable state spaces that addresses proposal kernels that possess a certain involution structure. Note that, while this under-lying proposal structure suggests a scope which includes Hamiltonian-type kernels, we demonstrate that our abstract result is, in an appropriate sense, equivalent to an earlier general state space setting developed in (Ann. Appl. Probab. 8 (1998) 1-9) where the connection to Hamiltonian methods was more *** the basis of our abstract results we develop several new classes of extended phase space, HMC-like algorithms. First we tackle the classical finite-dimensional setting of a continuously distributed target measure. We then consider an infinite-dimensional framework for targets which are ab-solutely continuous with respect to a Gaussian measure with a trace-class covariance. Each of these algorithm classes can be viewed as "surrogate -trajectory" methods, providing a versatile methodology to bypass expensive gradient computations through skillful reduced order modeling and/or data driven approaches as we begin to explore in a forthcoming companion work (Glatt-Holtz et al. (2023)). On the other hand, along with the connection of our main abstract result to the framework in (Ann. Appl. Probab. 8 (1998) 1- 9), these algorithm classes provide a unifying picture connecting together a number of popular existing algorithms which arise as special cases of our gen-eral frameworks under suitable parameter choices. In particular we show that, in the finite-dimensional setting, we ca

关键词： Markov chain Monte Carlo (MCMC) algorithms metropolis-hastings algorithms sampling on abstract state spaces Hamiltonian Monte Carlo surrogate trajectory methods

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An Efficient Sampling Algorithm for Non-smooth Composite Potentials

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JOURNAL OF MACHINE LEARNING RESEARCH 2022年第1期23卷 1-50页

作者： Mou, Wenlong Flammarion, Nicolas Wainwright, Martin J. Bartlett, Peter L. Univ Calif Berkeley Dept EECS Berkeley CA 94720 USA Ecole Polytech Fed Lausanne Sch Comp & Commun Sci CH-1015 Lausanne Switzerland Univ Calif Berkeley Dept Stat Berkeley CA 94720 USA

We consider the problem of sampling from a density of the form p(x) ? exp(-f (x) - g(x)), where f : Rd-+ R is a smooth function and g : R-d-+ R is a convex and Lipschitz function. We propose a new algorithm based on the metropolis-hastings framework. Under certain isoperimetric inequalities on the target density, we prove that the algorithm mixes to within total variation (TV) distance e of the target density in at most O(d log(d/e)) iterations. This guarantee extends previous results on sampling from distributions with smooth log densities (g = 0) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions g. Simulation results on posterior sampling problems that arise from the Bayesian Lasso show empirical advantage over previous proposal distributions.

关键词： Markov Chain Monte Carlo mixing time metropolis-hastings algorithms Langevin diffusion non-smooth functions Bayesian inference

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Extending computations for disparity testing when data sources are uncertain

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HEALTH SERVICES AND OUTCOMES RESEARCH METHODOLOGY 2023年第2期23卷 207-226页

作者： McDonald, Gary C. Oakley, Rachel H. Oakland Univ Dept Math & Stat Rochester MI 48063 USA

The topic of this article is one-sided hypothesis testing on the means of two populations when there is uncertainty as to which population a datum is drawn. Along with each datum, a probability is given as to which of the populations the datum emanated. Such situations arise, for example, in the use of Bayesian imputation methods to assess racial and ethnic disparities with certain survey, health, and financial data. By use of a Bayesian framework and Markov Chain Monte Carlo sampling from the joint posterior distribution of the population means, the probability of a disparity hypothesis is estimated. This approach extends sample size limitations of previous methods given in the literature from a few dozen to well into the thousands. Four methods are developed and compared. Three methods are implemented in R codes and one method in WinBUGS. All the codes are provided in the appendices.

关键词： Bayesian improved surname and geocoding (BISG) Laplace method Posterior distribution metropolis-hastings algorithms Random walk chain Independence chain Gibbs sampling WinBUGS

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A Bayesian Mixture Model Approach to Disparity Testing

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Applied Mathematics 2024年第3期15卷 214-234页

作者： Gary C. McDonald Department of Mathematics and Statistics Oakland University Rochester USA

The topic of this article is one-sided hypothesis testing for disparity, i.e., the mean of one group is larger than that of another when there is uncertainty as to which group a datum is drawn. For each datum, the uncertainty is captured with a given discrete probability distribution over the groups. Such situations arise, for example, in the use of Bayesian imputation methods to assess race and ethnicity disparities with certain insurance, health, and financial data. A widely used method to implement this assessment is the Bayesian Improved Surname Geocoding (BISG) method which assigns a discrete probability over six race/ethnicity groups to an individual given the individual’s surname and address location. Using a Bayesian framework and Markov Chain Monte Carlo sampling from the joint posterior distribution of the group means, the probability of a disparity hypothesis is estimated. Four methods are developed and compared with an illustrative data set. Three of these methods are implemented in an R-code and one method in WinBUGS. These methods are programed for any number of groups between two and six inclusive. All the codes are provided in the appendices.

关键词： Bayesian Improved Surname and Geocoding (BISG) Mixture Likelihood Function Posterior Distribution metropolis-hastings algorithms Random Walk Chain Independence Chain Gibbs Sampling WinBUGS

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Parameter estimation of gravitational waves with a quantum metropolis algorithm

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CLASSICAL AND QUANTUM GRAVITY 2023年第4期40卷 045001-045001页

作者： Escrig, Gabriel Campos, Roberto Casares, Pablo A. M. Martin-Delgado, M. A. Univ Complutense Madrid Dept Fis Teor Madrid Spain Quasar Sci Resources SL Madrid Spain Univ Politecn Madrid CCS Ctr Computat Simulat Madrid Spain

After the first detection of a gravitational wave in 2015, the number of successes achieved by this innovative way of looking through the Universe has not stopped growing. However, the current techniques for analyzing this type of events present a serious bottleneck due to the high computational power they require. In this article we explore how recent techniques based on quantum algorithms could surpass this obstacle. For this purpose, we propose a quantization of the classical algorithms used in the literature for the inference of gravitational wave parameters based on the well-known quantum walks technique applied to a metropolis-hastings algorithm. Finally, we develop a quantum environment on classical hardware, implementing a metric to compare quantum versus classical algorithms in a fair way. We further test all these developments in the real inference of several sets of parameters of all the events of the first detection period GWTC-1 and we find a polynomial advantage in the quantum algorithms, thus setting a first starting point for future algorithms.

关键词： gravitational waves quantum walks metropolis-hastings algorithms

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An efficient sampling algorithm for non-smooth composite potentials

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2022年第1期23卷 10611-10660页

作者： Wenlong Mou Nicolas Flammarion Martin J. Wainwright Peter L. Bartlett Department of EECS University of California Berkeley Berkeley CA School of Computer and Communication Sciences EPFL Lausanne Switzerland Department of EECS and Department of Statistics University of California Berkeley Berkeley CA

We consider the problem of sampling from a density of the form p(x) ∝ exp(-f(x) - g(x)), where f : ℝd → ℝ is a smooth function and g : ℝd → ℝ is a convex and Lipschitz function. We propose a new algorithm based on the metropolis-hastings framework. Under certain isoperimetric inequalities on the target density, we prove that the algorithm mixes to within total variation (TV) distance ε of the target density in at most O(d log(d/ε)) iterations. This guarantee extends previous results on sampling from distributions with smooth log densities (g = 0) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions g. Simulation results on posterior sampling problems that arise from the Bayesian Lasso show empirical advantage over previous proposal distributions.

关键词： Markov Chain Monte Carlo mixing time metropolis-hastings algorithms Langevin diffusion non-smooth functions Bayesian inference

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Weak convergence and optimal tuning of the reversible jump algorithm

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MATHEMATICS AND COMPUTERS IN SIMULATION 2019年 161卷 32-51页

作者： Gagnon, Philippe Bedard, Mylene Desgagne, Alain Univ Oxford Dept Stat 24-29 St Giles Oxford OX1 3LB England Univ Montreal Dept Math & Stat CP 6128Succursale Ctr Ville Montreal PQ H3C 3J7 Canada Univ Quebec Montreal Dept Math CP 8888Succursale Ctr Ville Montreal PQ H3C 3P8 Canada

The reversible jump algorithm is a useful Markov chain Monte Carlo method introduced by Green (1995) that allows switches between subspaces of differing dimensionality, and therefore, model selection. Although this method is now increasingly used in key areas (e.g. biology and finance), it remains a challenge to implement it. In this paper, we focus on a simple sampling context in order to obtain theoretical results that lead to an optimal tuning procedure for the considered reversible jump algorithm, and consequently, to easy implementation. The key result is the weak convergence of the sequence of stochastic processes engendered by the algorithm. It represents the main contribution of this paper as it is, to our knowledge, the first weak convergence result for the reversible jump algorithm. The sampler updating the parameters according to a random walk, this result allows to retrieve the well-known 0.234 rule for finding the optimal scaling. It also leads to an answer to the question: "with what probability should a parameter update be proposed comparatively to a model switch at each iteration?" Crown Copyright (C) 2018 Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS). All rights reserved.

关键词： Markov chain Monte Carlo methods metropolis-hastings algorithms Model selection Optimal scaling Random walk metropolis algorithms

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A computable bound of the essential spectral radius of finite range metropolis-hastings kernels

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STATISTICS & PROBABILITY LETTERS 2016年 117卷 72-79页

作者： Herve, Loic Ledoux, James INSA Rennes IRMAR F-35042 Rennes France CNRS UMR 6625 F-35708 Rennes France Univ Bretagne Loire Rennes France

Let pi be a positive continuous target density on R. Let P be the metropolis-hastings operator on the Lebesgue space L-2 (pi) corresponding to a proposal Markov kernel Q on R. When using the quasi-compactness method to estimate the spectral gap of P, a mandatory first step is to obtain an accurate bound of the essential spectral radius re (P) of P. In this paper a computable bound of r(ess)(P) is obtained under the following assumption on the proposal kernel: Q has a bounded continuous density q(x,y) on R-2 satisfying the following finite range assumption : vertical bar u vertical bar > s double right arrow (x, x + u) = 0 (for some s > 0). This result is illustrated with Random Walk metropolis-hastings kernels. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Markov chain operator metropolis-hastings algorithms Spectral gap

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An Empirical Study of an Adaptive Langevin Algorithm for Bounded Target Densities

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Journal of Data Science 2013年第3期11卷 501-536页

作者： Christopher H. Mehl OMNITEC Solutions Inc

Markov chain Monte Carlo simulation techniques enable the application of Bayesian methods to a variety of models where the posterior density of interest is too difficult to explore analytically. In practice, however, multivariate posterior densities often have characteristics which make implementation of MCMC methods more difficult. A number of techniques have been explored to help speed the convergence of a Markov chain. This paper presents a new algorithm which employs some of these techniques for cases where the target density is bounded. The algorithm is tested on several known distributions to empirically examine convergence properties. It is then applied to a wildlife disease model to demonstrate real-world applicability. [ABSTRACT FROM AUTHOR]

关键词： Adaptive MCMC controlled MCMC Langevin algorithms metropolis-hastings algorithms stochastic approximation tempered MCMC

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Probit and Logit Model Selection

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2011年第1期40卷 159-175页

作者： Chen, Guo Tsurumi, Hiroki Rutgers State Univ Dept Econ New Brunswick NJ 08901 USA

Monte Carlo experiments are conducted to compare the Bayesian and sample theory model selection criteria in choosing the univariate probit and logit models. We use five criteria: the deviance information criterion (DIC), predictive deviance information criterion (PDIC), Akaike information criterion (AIC), weighted, and unweighted sums of squared errors. The first two criteria are Bayesian while the others are sample theory criteria. The results show that if data are balanced none of the model selection criteria considered in this article can distinguish the probit and logit models. If data are unbalanced and the sample size is large the DIC and AIC choose the correct models better than the other criteria. We show that if unbalanced binary data are generated by a leptokurtic distribution the logit model is preferred over the probit model. The probit model is preferred if unbalanced data are generated by a platykurtic distribution. We apply the model selection criteria to the probit and logit models that link the ups and downs of the returns on S&P500 to the crude oil price.

关键词： Deviance information criterion Exponential power distribution Logit metropolis-hastings algorithms Probit

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