 # coupling of markov chains introductions

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• ### Coupling of Markov Chains - Texas A&M University

2020-8-10 · In other words, a coupling consists of two copies of the Markov chain M running simultaneously. These two copies are not literal copies; the two chains are not necessarily in same state, nor do they necessarily make the same move. Instead, we mean that each copy behaves exactly like the original Markov chain in terms of its transition probabilities.

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• ### Coupling of Markov Chains - Texas A&M University

2020-8-10 · In other words, a coupling consists of two copies of the Markov chain M running simultaneously. These two copies are not literal copies; the two chains are not necessarily in same state, nor do they necessarily make the same move. Instead, we mean that each copy behaves exactly like the original Markov chain in terms of its transition probabilities.

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• ### Markov Chains and Coupling - Duke University

2015-9-9 · 2 Coupling Coupling is a powerful technique that will help us bound the convergence rates of a Markov chain. De nition 1. Let Xand Y be random variables with probability distributions and on . A distribution !on is a coupling if 8x2; X y2Omega w(x;y) = (x) 8x2; X x2Omega w(x;y) = (y) 2.1 Coupling Lemma Lemma 1. Consider a pair of distributions and over .

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• ### Markov Chains, Eigenvalues, and Coupling

2014-12-12 · Markov Chains, Eigenvalues, and Coupling (December, 1993. Revised, April 1994.) by Je rey S. Rosenthal Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1 Phone: (416) 978-4594. Internet: [email protected] Technical Report No. 9320 (A later version of this paper, called Convergence rates

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• ### Coupling and Mixing Times in a Markov Chains

2013-5-21 · The derivation of the expected time to coupling in a Markov chain and its relation to the expected time to mixing (as introduced by the author in “Mixing times with applications to perturbed Markov chains” Linear Algebra Appl. (417, 108-123 (2006)) are explored. The two-state cases and three-state cases are examined in detail. 1. Introduction

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• ### The Coupling/Minorization/Drift Approach to Markov

2020-11-18 · Abstract: This review paper provides an introduction of Markov chains and their convergence rates { an important and interesting mathematical topic which also has important applications for very widely used Markov chain Monte Carlo (MCMC) algorithm. We rst discuss eigenvalue anal-ysis for Markov chains on nite state spaces. Then, using the coupling

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• ### On coupling of Markov chains | SpringerLink

Griffeath, D.: Coupling methods for Markov processes. Thesis, Cornell University (1975) 6. Griffeath, D.: Uniform coupling of nonhomogeneous Markov chains. [To appear in J. Appl. Probability (1975)] 7. Griffeath, D.: Partial coupling and loss of memory for Markov chains. [To appear in Ann. Probability (1976)] 8. Hodges, J.L., LeCam, L.:

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• ### Everything about Markov Chains - cl.cam.ac.uk

2019-9-3 · Coupling itself is not mysterious. Xand Y can be correlated or not. As long as they have the correct distributions we are good. An important theorem here is: Theorem 3.2. For all couplings (X;Y) of and , we have: k k TV P[X6= Y] (3.1) Furthermore, there always is a coupling …

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• ### Introduction to Markov Chains and Ri†e Shu†ing

2012-10-6 · A Markov chains is a type of stochastic process that was ﬂrst studied by Markov in 1906. A process consists of a sequence of states; in a Markov chain, each state is independent of the previous one. Markov chains are of great interest, because they can model many diﬁerent problems.

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• ### Probability Theory: The Coupling Method - Universiteit

2020-3-6 · This is the standard Markov Chain Convergence Theorem (MCCT) (see e.g. H¨aggstro¨m , Chapter 5, or Kraaikamp , Section 2.2). A coupling proof of (1.1) goes as follows. Let X′ = (X′ n)n∈N0 be an independent copy of the same Markov chain, but starting from π. Since πPn = πfor all n, X′ is stationary. Run Xand X′ together, and let

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• ### Notes 23 : Markov chains: asymptotic behavior

2018-4-18 · A coupling of Markov chains with transition probability pis a Markov chain f(X n;Y n)gon S Ssuch that both fX ngand fY ngare Markov chains with transition probability p. For our purposes, the following special type of coupling will sufﬁce. DEF 23.20 (Markovian coupling) A Markovian coupling …

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• ### 1 Introduction - cs.purdue.edu

2021-5-27 · 1.1 Basics of Markov Chains We begin by recalling some basic de nitions of Markov chains. De nition 1 (Markov Chain). A Markov chain is a discrete-time stochastic process fx n: n 0gwith each random variable taking values in a countable state space X, that satis es the Markov property: P(x n= x njx n 1 = x n 1;:::;x 0 = x 0) = P(x n= x njx n 1 ...

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• ### Path Coupling: a Technique for Proving Rapid Mixing in ...

2017-2-10 · Path Coupling: a Technique for Proving Rapid Mixing in Markov Chains Russ Bubley Martin Dyer School of Computer Studies University of Leeds Leeds LS2 9JT United Kingdom Abstract The main technique used in algorithm design for approxi- mating #P-hard counting problems is the Markov chain Monte Carlo method.

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• ### COUPLING AND ERGODICITY OF ADAPTIVE MARKOV

2009-11-16 · under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant. Keywords: Markov chains; computational methods 2000 Mathematics Subject Classiﬁcation: Primary 60J10 Secondary 60J22; 65C40 1 ...

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• ### Exact sampling with coupled Markov chains and

Guichong Li, Sampling Graphical Networks via Conditional Independence Coupling of Markov Chains, Advances in Artificial Intelligence, 10.1007/978-3-319-34111-8_36, (298-303), (2016). Crossref.

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• ### Markov Chains and Computer Science

2001-1-28 · Markov Chain Formalisation Long run behavior Cache modeling Synthesis Formal deﬁnition Let fX ng n2N a random sequence of variables in a discrete state-space X fX ng n2N is a Markov chain with initial law ˇ(0) iff X0 ˘ˇ(0) and for all n 2N and for all (j;i;in 1; ;i0) 2Xn+2 P(X n+1 = jjX = i;Xn 1 = in 1; ;X0 = i0) = P(Xn+1 = jjXn = i): fX ng n2N is a homogeneous Markov chain iff

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• ### Lecture Notes Markov Chains

2021-7-28 · Markov Chains are widely used as stochastic models to study a broad spectrum of system performance and dependability characteristics. This monograph is devoted to compositional specification and analysis of Markov chains. Based on principles known from process algebra, the author systematically develops an algebra of interactive Markov chains. By

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• ### Markov Chains - University of Cambridge

2013-4-23 · Chapter 11 is on Markov Chains. This book it is particulary interesting about absorbing chains and mean passage times. There are many nice exercises, some notes on the history of probability, and on pages 464-466 there is information about A. A. Markov …

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• ### Lecture Notes Markov Chains -

2021-7-12 · Markov chains are universal and the resulting transition matrix can be estimated more accu rately. ... including coupling, strong stationary times, and spectral methods. ... probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are ...

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• ### Coupling - University of Wisconsin–Madison

2020-11-20 · For instance, in the classical application of coupling to the convergence of Markov chains (Theorem 1.22), one simultaneously constructs two copies of a Markov chain—one of which is at stationarity—and shows that they can be made to coincide after a random amount of time called the coupling …

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• ### Couplings of Markov chains by randomized stopping

Summary. We consider a 0-recurrent ergodic Markov chain on (E,N), generated by a kernel P. Again we consider couplings of two chains (~X,), (uX,) starting with the initial distributions v respectively ~t and evolving with P. The coupling consists of two randomized stopping times: T, S, with ~(~xT) = ~(~,Xs). Under additional regularity assumptions we characterize the existence of 'short ...

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• ### Coupling of Markov chains and cellular automata

2016-11-1 · This study aims to predict land cover changes using coupling of Markov chains and cellular automata. One of the most rapid land cover changes is occurs at upper Ci Leungsi catchment area that located near Bekasi City and Jakarta Metropolitan Area. Markov chains has a good ability to predict the probability of change statistically while cellular ...

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• ### COUPLING AND ERGODICITY OF ADAPTIVE MARKOV

2009-11-16 · under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant. Keywords: Markov chains; computational methods 2000 Mathematics Subject Classiﬁcation: Primary 60J10 Secondary 60J22; 65C40 1 ...

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• ### On Mixing of Markov Chains: Coupling, Spectral ...

For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the ...

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• ### Lectures on Coupling (Exercises) - Warwick

2006-10-24 · Exercise 1.5 on Doeblin coupling (III): I Obtain better bounds by analyzing the derived Markov chain which represents the Doeblin coupling as a new Markov chain X,Xe, keeping track of the states of two coupled copies of the random walk. You will need to write an Rfunction which takes the transition matrix of the original chain and returns the

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• ### Optimal markovian couplings and applications |

This paper is devoted to studying a new topic: optimal Markovian couplings, mainly for time-continuous Markov processes. The study emphasizes the analysis of the coupling operators rather than the processes. Some constructions of optimal Markovian couplings for Markov chains and diffusions are presented, which are often unexpected. Then, the results are applied to study theL 2-convergence for ...

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• ### James Norris Markov Chains - mercury.uvaldetx.gov

2021-7-15 · Two excellent introductions are James Norris's 'Markov Chains' and Pierre Bremaud's 'Markov Chains: Gibbs fields, Monte Carlo simulation, and queues'. Both books assume a motivated student who is somewhat mathematically mature, though Bremaud reviews basic …

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• ### Markov Chains and Mixing Times: With a Chapter on

Compre online Markov Chains and Mixing Times: With a Chapter on Coupling from the Past, de Levin, David A., Peres, Yuval, Wilmer, Elizabeth L. na Amazon. Frete GRÁTIS em milhares de produtos com o Amazon Prime. Encontre diversos livros escritos por Levin, David A., Peres, Yuval, Wilmer, Elizabeth L. com ótimos preços.

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• ### Markov Chains and Mixing Times: With a Chapter on

Noté /5. Retrouvez Markov Chains and Mixing Times: With a Chapter on Coupling from the Past et des millions de livres en stock sur Amazon.fr. Achetez neuf ou d'occasion

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• ### Introduction to Markov Chains and Ri†e Shu†ing

2012-10-6 · A Markov chains is a type of stochastic process that was ﬂrst studied by Markov in 1906. A process consists of a sequence of states; in a Markov chain, each state is independent of the previous one. Markov chains are of great interest, because they can model many diﬁerent problems.

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• ### [1710.10026] A note on faithful coupling of Markov

2017-10-27 · Title:A note on faithful coupling of Markov chains. A note on faithful coupling of Markov chains. Authors: Debojyoti Dey, Pranjal Dutta, Somenath Biswas. (Submitted on 27 Oct 2017) Abstract: One often needs to turn a coupling of a Markov chain into a sticky coupling where once at some , then from then on, at each subsequent time step , we shall ...

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• ### On asymptotics for Vaserstein coupling of Markov

2013-9-1 · Construction of coupling and main results. 2.1. Coupling. Vaserstein’s coupling construction was proposed in . It provides a coupling for two Markov chains which have a countable state space. This chapter contains two main lemmas which allow to construct a Vaserstein-type coupling for two homogeneous Markov chains.

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• ### pr.probability - 'Permutation Coupling' for Markov

2018-3-11 · Say I now want to define a coupling on a collection of Markov chains with this kernel, { X k ( i) } i = 1 N, such that. The chains remain at different locations throughout, i.e. for i ≠ j, k ⩾ 0, X k ( i) ≠ X k ( j). By the Birkhoff-von Neumann theorem, this is possible: P can be written as a convex combination of permutation matrices ...

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• ### Chuck Norris' Coupling of Markov Chains: An Invariant ...

2015-12-12 · Browse other questions tagged stochastic-processes markov-chains or ask your own question. The Overflow Blog Prosus’s Acquisition of Stack Overflow: Our Exciting Next Chapter

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• ### On asymptotics for Vaserstein coupling of a Markov chain

The Vaserstein’s coupling construction was proposed in . It provides a coupling for two Markov chains which have a countable state space. This chapter contains two main lemmas which allow to construct a Vaserstein-type coupling for two homogeneous Markov 2

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• ### On Mixing of Markov Chains: Coupling, Spectral ...

For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the ...

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• ### Mixing times of Markov chains

2020-12-8 · 1.2 Total variation distance and coupling Recall the convergence to equilibrium theorem for Markov chains. Theorem 1.3. Suppose that Xis an irreducible and aperiodic Markov chain on a nite state space with invariant distribution ˇ. Then for all x;ywe have Pt(x;y) !ˇ(y) as t!1:

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• ### Handout 8 1 Markov Chains and Random Walks on Graphs

2011-5-23 · 1 Markov Chains and Random Walks on Graphs Recall from last time that a random walk on a graph gave us an RL algorithm for the problem of undirected graph connectivity. In this class, we also saw an RP algorithm for solving 2-SAT (see [2, Chapter 7] for details). We now develop some of the theory behind Markov chains and random

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• ### Monotonicity in coupling of Markov chains - Stack

2020-10-19 · When we define a coupling of Markov chains, there is partial order on the state space but I don't understand where exactly we use monotonicity defined in the process of coupling? For example, consider a simple random walk on a line segment {0,1,cdots,n}.

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• ### A note on faithful coupling of Markov chains -

2017-10-27 · 10/27/17 - One often needs to turn a coupling (X_i, Y_i)_i≥ 0 of a Markov chain into a sticky coupling where once X_T = Y_T at some T, then...

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• ### MarkovChains - Yale University

2014-7-1 · If you want to read more about coupling, a good place to start might be Chapter 11 of MitzenmacherUpfal, this chapter of the unpublished but nonetheless famous Aldous-Fill manuscript (which is a good place to learn about Markov chains and Markov chain Monte Carlo methods in general 1, or even an entire book: Lindvall, Torgny, Lectures on the ...

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• ### On asymptotics for Vaserstein coupling of Markov

2013-9-1 · Construction of coupling and main results. 2.1. Coupling. Vaserstein’s coupling construction was proposed in . It provides a coupling for two Markov chains which have a countable state space. This chapter contains two main lemmas which allow to construct a Vaserstein-type coupling for two homogeneous Markov chains.

Get Price
• ### On asymptotics for Vaserstein coupling of a Markov chain

The Vaserstein’s coupling construction was proposed in . It provides a coupling for two Markov chains which have a countable state space. This chapter contains two main lemmas which allow to construct a Vaserstein-type coupling for two homogeneous Markov 2

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• ### Probability Theory: The Coupling Method - Universiteit

2020-3-6 · Lindvall  explains how coupling was invented in the late 1930’s by Wolfgang Doeblin, and provides some historical context. Standard references for coupling are Lindvall  and Thorisson . 1.1 Markov chains Let X= (Xn)n∈N0 be a Markov chain …

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• ### Partial Coupling and Loss of Memory for Markov Chains

Coupling methods are used to obtain a structure theorem for the atomic decomposition of the tail \sigma-algebra of an arbitrary nonhomogeneous Markov chain. Various related results are also derived by coupling.

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• ### pr.probability - 'Permutation Coupling' for Markov

2018-3-11 · Say I now want to define a coupling on a collection of Markov chains with this kernel, { X k ( i) } i = 1 N, such that. The chains remain at different locations throughout, i.e. for i ≠ j, k ⩾ 0, X k ( i) ≠ X k ( j). By the Birkhoff-von Neumann theorem, this is possible: P can be written as a convex combination of permutation matrices ...

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• ### Ricci curvature of Markov chains on metric spaces

2010-9-5 · of discrete Markov chains, our techniques reduce, respectively, to Bakry–Émery theory or to a metric version of the coupling method. As far as I know, it had not been observed that these can actually be viewed as the same phenomenon. From the discrete Markov chain point of view, the techniques presented here are just a ver-

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• ### Stochastic Complementation, Uncoupling Markov

2012-2-17 · A concept called stochastic complementation is an idea which occurs naturally, although not always explicitly, in the theory and application of finite Markov chains. This paper brings this idea to the forefront with an explicit definition and a development of some of its properties.

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