The Poisson equation for reversible Markov chains: Analysis and application to Markov chain samplers

2012 ◽  
Author(s):  
Randy Cogill ◽  
Erik Vargo
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Poisson Equation ◽  
The Poisson Equation ◽  
Reversible Markov Chains

scholarly journals Event-Chain Monte Carlo: Foundations, Applications, and Prospects

2021 ◽  
Vol 9 ◽  
Author(s):  
Werner Krauth
Keyword(s):  
Aqueous Solution ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Real World ◽  
Continuous Time ◽  
Statistical Physics ◽  
Reversible Markov Chains ◽  
Sampling Problem ◽  
Review Reports

This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply ECMC to the sampling problem in molecular simulation, i.e., to real-world models of peptides, proteins, and polymers in aqueous solution.

scholarly journals The Prelimit Generator Comparison Approach of Stein’s Method

2021 ◽  
Author(s):  
Anton Braverman
Keyword(s):  
Markov Chain ◽  
Poisson Equation ◽  
Diffusion Processes ◽  
Technical Difficulty ◽  
Stein’S Method ◽  
Stein's Method ◽  
The Poisson Equation ◽  
Comparison Approach ◽  
Derivatives Of ◽  
Generator Comparison

This paper uses the generator comparison approach of Stein’s method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the [Formula: see text] model as a simple working example to illustrate our approach.

Invariant measures of time-reversible Markov chains

1979 ◽  
Vol 16 (01) ◽  
pp. 226-229 ◽  
Author(s):  
P. Suomela
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
Explicit Formula ◽  
Invariant Measures ◽  
Transition Probabilities ◽  
Time Reversibility ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.

Reversible Markov Chains

2019 ◽  
pp. 405-427
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Gaussian Approximation ◽  
Building Blocks ◽  
Partial Sums ◽  
Reversible Markov Chain ◽  
Maximal Inequalities ◽  
Functional Clt ◽  
Reversible Markov Chains ◽  
Functional Central Limit

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.

Invariant measures of time-reversible Markov chains

1979 ◽  
Vol 16 (1) ◽  
pp. 226-229 ◽  
Author(s):  
P. Suomela
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
Explicit Formula ◽  
Invariant Measures ◽  
Transition Probabilities ◽  
Time Reversibility ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.

Monotonicity of Positive Dependence with Time for Stationary Reversible Markov Chains

1995 ◽  
Vol 9 (2) ◽  
pp. 227-237 ◽  
Author(s):  
Taizhong Hu ◽  
Harry Joe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Real Line ◽  
Continuous Time ◽  
Marginal Distribution ◽  
Random Vectors ◽  
Positive Dependence ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

Let (X1, X2) and (Y1, Y2) be bivariate random vectors with a common marginal distribution (X1, X2) is said to be more positively dependent than (Y1, Y2) if E[h(X1)h(X2)] ≥ E[h(Y1)h(Y2)] for all functions h for which the expectations exist. The purpose of this paper is to study the monotonicity of positive dependence with time for a stationary reversible Markov chain [X1]; that is, (Xs, Xl+s) is less positively dependent as t increases. Both discrete and continuous time and both a denumerable set and a subset of the real line for the state space are considered. Some examples are given to show that the assertions established for reversible Markov chains are not true for nonreversible chains.

Geometric Approaches to the Estimation of the Spectral Gap of Reversible Markov Chains

1993 ◽  
Vol 2 (3) ◽  
pp. 301-323 ◽  
Author(s):  
Salvatore Ingrassia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Spectral Gap ◽  
Systematic Treatment ◽  
Reversible Markov Chain ◽  
Underlying Graph ◽  
Reversible Markov Chains

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.

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