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Cauchy Markov random field priors for Bayesian inversion

STATISTICS AND COMPUTING 2022年第2期32卷 1-26页

作者： Suuronen, Jarkko Chada, Neil K. Roininen, Lassi Lappeenranta Lahti Univ Technol Sch Engn Sci POB 20 Lappeenranta 53851 Finland King Abdullah Univ Sci & Technol Kingdom Saudi Arabia Appl Math & Computat Sci Program Thuwal 23955 Saudi Arabia

The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed. In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo methods are usually required. In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison. We firstly propose a one-dimensional second-order Cauchy difference prior, and construct new first- and second-order two-dimensional isotropic Cauchy difference priors. Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Matern type Gaussian presentation. The comparison also includes Cauchy sheets. Our numerical computations are based on both maximum a posteriori and conditional mean estimation. We exploit state-of-the-art MCMC methodologies such as metropolis-within-Gibbs, Repelling-Attracting metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo. We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems. Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors.

关键词： Cauchy priors Markov models inverse problems Bayesian approach metropolis algorithms HMC-NUTS

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Efficacy of the metropolis Algorithm for the Minimum-Weight Codeword Problem Using Codeword and Generator Search Spaces

Efficacy of the Metropolis Algorithm for the Minimum-Weight ...

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IEEE Transactions on Evolutionary Computation

作者： Ajitha Shenoy, K.B. Biswas, Somenath Kurur, Piyush P. Department of Information and Communication Technology Manipal Institute of Technology Manipal Academy of Higher Education Manipal India School of Mathematics and Computer Science Indian Institute of Technology Goa Ponda India Department of Computer Science and Engineering Indian Institute of Technology Palakkad Palakkad India

This article studies the efficacy of the metropolis algorithm for the minimum-weight codeword problem. The input is a linear code C given by its generator matrix and our task is to compute a nonzero codeword in the code C of least weight. In particular, we study the metropolis algorithm on two possible search spaces for the problem: 1) the codeword space and 2) the generator space. The former is the space of all codewords of the input code and is the most natural one to use and hence has been used in previous work on this problem. The latter is the space of all generator matrices of the input code and is studied for the first time in this article. In this article, we show that for an appropriately chosen temperature parameter the metropolis algorithm mixes rapidly when either of the search spaces mentioned above are used. Experimentally, we demonstrate that the metropolis algorithm performs favorably when compared to previous attempts. When using the generator space, the metropolis algorithm is able to outperform the previous algorithms in most of the cases. We have also provided both theoretical and experimental justification to show why the generator space is a worthwhile search space to use for this problem. © 1997-2012 IEEE.

关键词： C (programming language) Approximation algorithms Combinatorial optimization Gas generators Heuristic algorithms Markov chains Markov processes Matrix algebra Optimization Space time codes Codeword Linear codes metropolis algorithms Mixing time Randomized local search Search problem

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On the Markov commutator

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BULLETIN DES SCIENCES MATHEMATIQUES 2019年 154卷 1-35页

作者： Miclo, Laurent Univ Toulouse UMR 5219 Inst Math Toulouse Toulouse France CNRS Paris France Univ Toulouse UMR 5314 Toulouse Sch Econ Toulouse France

The Markov commutator associated to a finite Markov kernel P is the convex semigroup consisting of all Markov kernels commuting with P. Its interest comes from its relation with the hypergroup property and with the notion of Markovian duality by intertwining. In particular, it is shown that the discrete analogue of the Achour-Trimeche's theorem, asserting the preservation of non-negativity by the wave equations associated to certain metropolis birth and death transition kernels, cannot be extended to all convex potentials. But it remains true for symmetric and monotone potentials which are sufficiently convex. (C) 2019 Elsevier Masson SAS. All rights reserved.

关键词： Symmetry group of a Markov kernel Hypergroup property Duality by intertwining Birth and death chains metropolis algorithms One-dimensional discrete wave equations

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A metropolis algorithm combined with Hooke-Jeeves local search method applied to global optimization

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APPLIED MATHEMATICS AND COMPUTATION 2010年第2期217卷 843-853页

作者： Rios-Coelho, A. C. Sacco, W. F. Henderson, N. Univ Fed Oeste do Para BR-68040070 Santarem PA Brazil Univ Estado Rio de Janeiro Inst Politecn BR-28630050 Nova Friburgo RJ Brazil

A hybridization of a recently introduced metropolis algorithm named the Particle Collision Algorithm (PCA) and the Hooke-Jeeves local search method is applied to a testbed of global optimization functions and to real-world chemical equilibrium nonlinear systems. The results obtained by this method, called HJPCA, are compared against those achieved by two state-of-the-art global optimization methods, C-GRASP and GLOBAL. HJPCA performs better than both algorithms, thus demonstrating its potential for other applications. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Global optimization Metaheuristics Hybrid methods metropolis algorithms Hooke-Jeeves method

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A metropolis algorithm applied to a nuclear power plant auxiliary feedwater system surveillance tests policy optimization

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PROGRESS IN NUCLEAR ENERGY 2008年第1期50卷 15-21页

作者： Sacco, Wagner F. Lapa, Celso M. F. Pereira, Claudio M. N. A. Alves Filho, Hennes Univ Estado Rio de Janeiro Inst Politecn Dept Modelagem Computac TOG BR-28601970 Nova Friburgo RJ Brazil Ilha Fundao DIRE IEN Comissao Nacl Energia Nucl BR-21945970 Rio De Janeiro Brazil

The Particle Collision Algorithm (PCA) is a metropolis-based algorithm loosely inspired by the physics of nuclear particle collision reactions, particularly scattering and absorption. This metaheuristic is applied to a Nuclear Power Plant Auxiliary Feedwater System surveillance tests policy optimization, and its performance is compared to previous results. The optimization problem consists in maximizing the system's average availability for a given period of time, considering realistic features such as (i) aging effects on standby components during the tests;(ii) revealing failures in the tests imply on corrective maintenance, increasing outage times;(iii) components have distinct test parameters (outage time, aging factors, etc.);(iv) tests are not necessarily periodic. The PCA outperforms the canonical genetic algorithm and two of its variants, ratifying the previous performance in another optimization problem. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： metaheuristics metropolis algorithms surveillance tests policy optimization

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Necessary conditions for geometric and polynomial ergodicity of random-walk-type Markov chains

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BERNOULLI 2003年第4期9卷 559-578页

作者： Jarner, SF Tweedie, RL Danish Labour Market Supplementary Fixed Income Quantitat Res DK-3400 Hillerod Denmark

We give necessary conditions for geometric and polynomial convergence rates of random-walk-type Markov chains to stationarity in terms of existence of exponential and polynomial moments of the invariant distribution and the Markov transition kernel. These results complement the use of Foster-Lyapunov drift conditions for establishing geometric and polynomial ergodicity. For polynomially ergodic Markov chains, the results allow us to derive exact rates of convergence and exact relations between the moments of the invariant distribution and the Markov transition kernel. In an application to Markov chain Monte Carlo we derive tight rates of convergence for symmetric random walk metropolis algorithms and Langevin algorithms with polynomial target densities.

关键词： geometric and polynomial moments Markov chains metropolis algorithms

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Perfect sampling from independent metropolis-Hastings chains

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2002年第2期104卷 297-314页

作者： Corcoran, JN Tweedie, RL Univ Colorado Dept Appl Math Boulder CO 80309 USA

"Perfect sampling" enables exact draws from the invariant measure of a Markov chain. We show that the independent metropolis-Hastings chain has certain stochastic monotonicity properties that enable a perfect sampling algorithm to be implemented, at least when the candidate is overdispersed with respect to the target distribution. We prove that the algorithm has an optimal geometric convergence rate, and applications show that it may converge extreme rapidly. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： metropolis algorithms Hastings algorithms Markov chains invariant measures backwards coupling coupling from the past exact sampling perfect sampling

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A note on acceptance rate criteria for CLTs for metropolis-Hastings algorithms

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JOURNAL OF APPLIED PROBABILITY 1999年第4期36卷 1210-1217页

作者： Roberts, GO Univ Cambridge Cambridge CB2 1TN England

This paper considers positive recurrent Markov chains where the probability of remaining in the current state is arbitrarily close to 1, Specifically, conditions are given which ensure the non-existence of central limit theorems for ergodic averages of functionals of the chain. The results are motivated by applications for metropolis-Hastings algorithms which are constructed in terms of a rejection probability (where a rejection involves remaining at the current state). Two examples for commonly used algorithms are given, for the independence sampler and the metropolis-adjusted Langevin algorithm. The examples are rather specialized, although, in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used.

关键词： Hastings algorithms metropolis algorithms Markov chain Monte Carlo central limit theorems

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Rates of convergence of the Hastings and metropolis algorithms

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ANNALS OF STATISTICS 1996年第1期24卷 101-121页

作者： Mengersen, KL Tweedie, RL COLORADO STATE UNIV DEPT STATFT COLLINSCO 80523

We apply recent results in Markov chain theory to Hastings and metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi. In the independence case (in R(k)) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi;in the symmetric case (in R only) we show geometric convergence essentially occurs if and only if pi has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective.

关键词： posterior distributions Hastings algorithms metropolis algorithms Gibbs sampling Markov chain Monte Carlo irreducible Markov processes geometric ergodicity stochastic monotonicity log-concave distributions

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Exponential convergence of Langevin distributions and their discrete approximations

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Bernoulli 1996年第4期2.0卷 341-363页

作者： Roberts, Gareth O. Tweedie, Richard L. Statistical Laboratory Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane Cambridge CB2 1SB United Kingdom Department of Statistics Colorado State University Fort Collins 80523 CO United States

In this paper we consider a continuous-time method of approximating a given distribution π using the Langevin diffusion dLt=dWt+1/2∇log π(Lt)dt. We find conditions under this diffusion converges exponentially quickl... 详细信息

In this paper we consider a continuous-time method of approximating a given distribution π using the Langevin diffusion dL_t=dW_t+1/2∇log π(L_t)dt. We find conditions under this diffusion converges exponentially quickly to π or does not: In one dimension, these are essentially that for distributions with exponential tails of the form π(x) ∝ exp (-γ|x|^β) 0<β<∞, exponential onvergence occurs if and only if β≥1. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretizations need not do so. We then consider a 'metropolis-adjusted' version of the algorithm, and find conditions under which this also converges at an exponential rate: Perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms. © 1996 Chapman & Hall.

关键词： Diffusions Discrete approximations Geometric ergodicity Hastings algorithms Irreducible Markov processes Langevin models Markov chain Monte Carlo metropolis algorithms Posterior distributions

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