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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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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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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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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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Markov chain Monte Carlo for dynamic generalised linear models

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BIOMETRIKA 1998年第1期85卷 215-227页

作者： Gamerman, D Univ Fed Rio de Janeiro Inst Matemat BR-21945970 Rio De Janeiro Brazil

This paper presents a new methodological approach for carrying out Bayesian inference about dynamic models for exponential-family observations. The approach is;simulation-based and involves the use of Markov chain Monte Carlo techniques. A metropolis-hastings algorithm is combined with the Gibbs sampler in repeated use of an adjusted version of normal dynamic linear models. Different alternative schemes based on sampling from the system disturbances and state parameters separately and in a block are derived and compared. The approach is fully Bayesian in obtaining posterior samples with state parameters and unknown hyperparameters. Illustrations with real datasets with sparse counts and missing values are presented. Extensions to accommodate more general evolution forms and distributions for observations and disturbances are outlined.

关键词： adjusted time series Bayesian metropolis-hastings algorithms reparameterisation sampling schemes system disturbances

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Phylogenetic inference for binary data on dendograms using Markov chain Monte Carlo

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 1997年第1期6卷 122-131页

作者： Mau, B Newton, MA UNIV WISCONSIN DEPT STATMADISONWI 53706 UNIV WISCONSIN DEPT BIOSTATMADISONWI 53706

Using a stochastic model for the evolution of discrete characters among a group of organisms, we derive a Markov chain that simulates a Bayesian posterior distribution on the space of dendograms. A transformation of the tree into a canonical cophenetic matrix form, with distinct entries along its superdiagonal, suggests a simple proposal distribution for selecting candidate trees ''close'' to the current tree in the chain. We apply the consequent metropolis algorithm to published restriction site data on nine species of plants. The Markov chain mixes well from random starting trees, generating reproducible estimates and confidence sets for the path of evolution.

关键词： labeled history metropolis-hastings algorithms phylogeny reconstruction

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The ODE method for stability of skip-free Markov chains with applications to MCMC

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ANNALS OF APPLIED PROBABILITY 2008年第2期18卷 664-707页

作者： Fort, Gersende Meyn, Sean Moulines, Eric Priouret, Pierre Telecom Paris CNRS Lab Traitement & Commun & Informat F-75634 Paris 13 France Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA Univ Illinois Coordinated Sci Lab Urbana IL 61801 USA Univ Paris 06 Lab Probabil & Modeles Aleatoires F-75252 Paris 05 France

In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in "most" of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.

关键词： Markov chain fluid limit subgeometric ergodicity state-dependent drift criteria Markov chain Monte Carlo metropolis-hastings algorithms

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An adaptive version for the metropolis Adjusted Langevin algorithm with a truncated drift

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2006年第2期8卷 235-254页

作者： Atchade, Yves F. Univ Ottawa Dept Math & Stat Ottawa ON K1N 6N5 Canada

This paper extends some adaptive schemes that have been developed for the Random Walk metropolis algorithm to more general versions of the metropolis-hastings (MH) algorithm, particularly to the metropolis Adjusted Langevin algorithm of Roberts and Tweedie (1996). Our simulations show that the adaptation drastically improves the performance of such MH algorithms. We study the convergence of the algorithm. Our proves are based on a new approach to the analysis of stochastic approximation algorithms based on mixingales theory.

关键词： adaptive Markov Chain Monte Carlo Langevin algorithms metropolis-hastings algorithms stochastic approximation algorithms

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On the Flexibility of metropolis-hastings Acceptance Probabilities in Auxiliary Variable Proposal Generation

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SCANDINAVIAN JOURNAL OF STATISTICS 2011年第2期38卷 342-358页

作者： Storvik, Geir Univ Oslo Dept Math N-0316 Oslo Norway

Use of auxiliary variables for generating proposal variables within a metropolis-hastings setting has been suggested in many different settings. This has in particular been of interest for simulation from complex distributions such as multimodal distributions or in transdimensional approaches. For many of these approaches, the acceptance probabilities that are used turn up somewhat magic and different proofs for their validity have been given in each case. In this article, we will present a general framework for construction of acceptance probabilities in auxiliary variable proposal generation. In addition to showing the similarities between many of the proposed algorithms in the literature, the framework also demonstrates that there is a great flexibility in how to construct acceptance probabilities. With this flexibility, alternative acceptance probabilities are suggested. Some numerical experiments are also reported.

关键词： acceptance probabilities auxiliary variables Markov chain Monte Carlo metropolis-hastings algorithms

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