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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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  • SOME MULTI-STEP COUPLING CONSTRUCTIONS FOR

    2012-10-12 · SOME MULTI-STEP COUPLING CONSTRUCTIONS FOR MARKOV CHAINS 3 2. Flows Let a transition kernel P(x;dy) be given on a su ciently regular state space E. It is assumed that a Markov chain with transition probabilities Pcan be constructed as a ow. This usually proceeds as follows: X s F F F F F F F F F F F F s+1 s+2 s+4 s+5 s+6 s+7 s+8 s+9 s+10 s+11 s+12 X s+12 s+3

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  • Markov Chains, Eigenvalues, and Coupling

    2014-12-12 · p= q= 1. If p= q= 0 the Markov chain is decomposable, meaning that the state space X can be partitioned into two non-empty subsets X 1 and X 2 such that P(x;y) = 0 whenever xand y are not in the same subset. If p= q= 1 the Markov chain is periodic, meaning that the state space Xcan be partitioned into non-empty subsets X 1;:::;X d (with d 2) such that for x 2X

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  • On coupling of Markov chains | SpringerLink

    Griffeath, D.: A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 95–106 (1975) Google Scholar 5. 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.

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  • A maximal coupling for Markov chains | SpringerLink

    2020-10-27 · Coupling of W t, Z t Markov chains: At each time a fair coin is tossed to determine which of the two chains W t, Z t moves. The chosen chain then moves according to the transition probabilities given by (1), (2). The two chains move together once they have met for the first time. I am trying to find

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  • Finding joint probability from coupling of Markov

    2020-10-19 · It refers to the coupling of two probability distributions. This is different than coupling two (or more) sample paths of a Markov chain. Imagine running sample paths of a Markov chain starting from all points in the state space, and using the same random inputs. If all of the paths come together, they will stay together or 'coupled' for all future ...

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  • Monotonicity in coupling of Markov chains - Stack

    2013-9-1 · Otherwise we obviously have X n 1 = d X n 2 for all n, and the coupling is trivial, namely we set X ˜ n 1 = X ˜ n 2 = X n 1. Introduce a Markov chain ( η n 1, η n 2, ξ n, ζ n) n ∈ Z + on the state space ( E 3 × { 0, 1 }, E 3 ⊗ 2 { 0, 1 }). If q 0 = 0 then we set η 0 1 ≔ X 0 1, η 0 2 ≔ X 0 2, ξ 0 ≔ 0, ζ 0 ≔ 1.

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  • On asymptotics for Vaserstein coupling of Markov

    2021-3-12 · 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 ...

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  • SOME MULTI-STEP COUPLING CONSTRUCTIONS FOR

    2012-10-12 · SOME MULTI-STEP COUPLING CONSTRUCTIONS FOR MARKOV CHAINS 3 2. Flows Let a transition kernel P(x;dy) be given on a su ciently regular state space E. It is assumed that a Markov chain with transition probabilities Pcan be constructed as a ow. This usually proceeds as follows: X s F F F F F F F F F F F F s+1 s+2 s+4 s+5 s+6 s+7 s+8 s+9 s+10 s+11 ...

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  • 1. Markov chains - Yale University

    2013-1-10 · Section 7. Introduces the idea of coupling. Section 8. Uses coupling to prove the Basic Limit Theorem. Section 9. A Strong Law of Large Numbers for Markov chains. Markov chains are a relatively simple but very interesting and useful class of random processes. A Markov chain describes a system whose state changes over time. The changes

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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.

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  • [2103.07459v1] On Mixing of Markov Chains: Coupling ...

    2021-3-13 · 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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  • Finding joint probability from coupling of Markov

    2020-10-27 · Finding joint probability from coupling of Markov chains. 0. Let W t, Z t be Ehrenfest urn Markov chains i.e. the transition probabilities for W t, Z t are given by. (1) P ( j, j + 1) = n − j n (2) P ( j, j − 1) = j n. i.e. we can consider two urns with n balls distributed among them. A ball is selected uniformly at random and transferred ...

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

    2017-8-22 · ing the coupling method to certain Markov chains, which we call path coupling. The power of the path coupling method is that it requires only comparisons between ad- jacent states, rather than arbitrary staters, and this results in much simpler analyses. While we stress that the methodol- ogy is rather more general than our main theorems here, in

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  • Mixing of Markov Chains - MIT - Massachusetts Institute

    2019-11-18 · Markov Chain Monte Carlo (MCMC) methods address the problem of sampling from a given distribution by rst constructing a Markov chain whose stationary distribution is the given distribution, and then sampling from this Markov chain. Since there are broad classes of Markov chains for which the distribution over states converges to 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 [22]. 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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  • 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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  • Speed of d-convergence for Markov approximations of

    2010-4-17 · Markov chains are aperiodic [Friedman and Ornstein (1970)]. The use of the distance dis de nitory. Indeed, every process can be approximated in the vague topology by the so-called canonical k-step Markov approximations, de ned so to have the same transitions from kto k+ 1 states as the original process (De nition 2 below).

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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.

    Get Price
  • 2.1 Markov Chains - gatech.edu

    2006-9-7 · Markov Chains, Coupling, Stationary Distribution Eric Vigoda 2.1 Markov Chains In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (X t) on Ω is a Markov

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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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  • 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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  • 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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  • (PDF) Path Coupling: A Technique for Proving Rapid

    Path Coupling: a Technique for Proving Rapid Mixing in Markov Chains Russ Bubley Martin Dyer School of Computer Studies zyxw zyxwvutsr University of Leeds Leeds LS2 9JT United Kingdom Abstract zyxwvut 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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  • [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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  • 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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  • Markov Chains - University of Cambridge

    2013-4-23 · Markov Chains Course information, a blog, discussion and resources for a course of 12 lectures on Markov Chains to second year mathematicians at Cambridge in autumn 2012. Until recently my home page linked to content for the 2011 course.

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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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  • MARKOV CHAINS AND COUPLING FROM THE PAST

    2017-10-31 · Markov Chains 3 4. Convergence and Coupling 8 5. Glauber dynamics and the hardcore con guration 13 6. Coupling from the Past 16 6.1. The Ising Model and CFTP 16 6.2. Design choices of CFTP 18 6.3. Monotone CFTP Formalized and Bounded 20 6.4. Bounding the Mixing Time of a Random Walk 24 6.5. The hardcore model: why we can use CFTP even though it ...

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  • 2.1 Markov Chains - gatech.edu

    2006-9-7 · Markov Chains, Coupling, Stationary Distribution Eric Vigoda 2.1 Markov Chains In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (X t) on Ω is a Markov

    Get Price
  • 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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  • 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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  • 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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  • Mixing times of Markov chains - University of British

    2020-1-7 · Mixing times of Markov chains Jonathan Hermon January 7, 2020 Jonathan Hermon Mixing times of Markov chains January 7, 20201/31. General information ... powerful technique called coupling. This technique is indispensible part of the toolkit of anyone working in discrete-probability.

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  • Damage spreading and coupling in Markov chains -

    In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire ...

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  • Markov Chains - University of Cambridge

    2013-4-23 · Markov Chains Course information, a blog, discussion and resources for a course of 12 lectures on Markov Chains to second year mathematicians at Cambridge in autumn 2012. Until recently my home page linked to content for the 2011 course.

    Get Price
  • General state space Markov chains and MCMC algorithms

    2007-2-1 · constructing (and running) such Markov chains is often surprisingly straightfor-ward. Remark. In the practical use of MCMC, rather than start a fresh Markov chain for each new sample, often an entire tail of the Markov chain run fX ngis used to create an estimate such as (N B) 1 …

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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 ...

    Get Price
  • 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 …

    Get Price
  • Rapidly Mixing Markov Chains: A Comparison of

    2005-12-9 · Coupling which is an entirely different approach to bounding the mixing time, gives an illustrative example of Coupling in action, and also discusses Path Coupling, which is a useful design tool in constructing ... all Markov chains we use for sampling to be ergodic, so next we turn to conditions on the chain which will ensure ergodicity. 2 ...

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  • Convergence Rate of Markov Chains - Will Perkins

    2016-4-22 · The rst technique we will consider is Coupling. Recall: A coupling of two Markov Chains X n and Y n is a way to de ned them on the same probability space - i.e. specify a joint distribution so the marginal distributions are correct but somehow the dependence of the chains tells us something. We can specify a coupling of P and Q both ...

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  • Diagonal Couplings of Quantum Markov Chains

    Quantum Markov Chains Burkhard Kümmerer and Kay Schwieger Abstract In this article we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by analyzing couplings.

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  • A Romantic View of Markov Chains - Gregory

    2019-10-28 · A Romantic View of Markov Chains A Markov chain is ergodic if and only if it has at most one recurrent class and is aperiodic. A sketch of a proof of this theorem hinges on an intuitive probabilistic idea called 'coupling' that is worth understanding.

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  • Exact Sampling with Markov Chains

    2019-4-10 · algorithms that use Markov chains to return samples distributed exactly according to r. The algorithms determine on their own how long to run the Markov chain. Two of the algorithms may be used with any Markov chain, but are useful only if the state space is not too large.

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  • Damage spreading and coupling in Markov chains -

    In this paper, we relate the coupling of Markov chains, at the basis of perfect sampling methods, with damage spreading, which captures the chaotic nature of stochastic dynamics. For two-dimensional spin glasses and hard spheres we point out that the obstacle to the application of perfect-sampling schemes is posed by damage spreading rather than by the survey problem of the entire ...

    Get Price
  • 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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  • CONVERGENCE OF MARKOV CHAIN MONTE CARLO

    2005-2-28 · state spaces, we are able to use the method developed in this chapter to improve the results of Chapter 3. The methods described above require the analytic calculation of param-eters of the Markov chains relating the distance between two coupled re-alisations of the chain to the distance at the previous iteration. In most

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  • 2.1 Markov Chains - cc.gatech.edu

    2006-2-23 · Markov Chains, Coupling, Stationary Distribution Eric Vigoda 2.1 Markov Chains In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (X t) on Ω is a Markov

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  • Estimating Convergence of Markov chains with L-Lag

    coupled Markov chains. A comparison with our proposed method will be given in Section 2.4. We propose to use L-lag couplings of Markov chains to estimate the distance between ˇ tand ˇ for a fixed time t, building on 1-lag couplings used to obtain unbiased estimators in [23, 29]. The discussion of [29] mentions that upper bounds on the TV ...

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  • Counting and Sampling Fall 2017 Lecture 4: Coupling

    2017-12-7 · this lecture we will discuss coupling as an important tool in studying Markov chains. First, we will discuss the Heat-Bath chain as a general rule in designing Markov chains with a given stationary distribution, then we introduce coupling and see its applications. 4.1 …

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  • Lectures on Coupling (Exercises) - Warwick

    2006-10-24 · An example of Doeblin’s coupling for Markov chains: consider the utterly simple chain with transition matrix and equilibrium distribution P = 0.9 0.1 0.1 0.9 ; π = 1 2 1 2. Use Rto compare convergence to equilibrium against coupling time distribution. LECTURE ANSWER

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  • COUPLING AND ERGODICITY OF ADAPTIVE MARKOV

    2009-11-16 · We then use a bivariate coupling construction to prove the validity of adaptive MCMC in uniform settings (Section 5) and nonuniform (Section 6) settings. We make connections to drift conditions (Section 7) and ... The goal of MCMC is to approximately sample from π(·) through the use of Markov chains,particularlywhenπ ...

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  • Diagonal Couplings of Quantum Markov Chains

    Quantum Markov Chains Burkhard Kümmerer and Kay Schwieger Abstract In this article we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by analyzing couplings.

    Get Price
  • Speed of -convergence for Markov approximations of

    Speed of -convergence for Markov approximations of chains with complete connections. A coupling approach. Stochastic Processes and their Applications, 1999. Roberto Fernandez. PDF. Download Free PDF. ... Speed of -convergence for Markov approximations of chains with complete connections. A coupling approach

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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-

    Get Price
  • Perfect Sampling: The Basics

    2004-5-18 · Coupling Markov chains Coupling from the past (Propp, Wilson 1996) Fill, Machida, Murdoch, Rosenthal (1999) Read-Once CFTP (Wilson 2000) High noise CFTP (Häggström, Steif 2000) Modified Acceptance/Rejection Popping Algorithms (Propp, …

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  • Markov approximations of chains of infinite order

    2005-7-12 · chain and its canonical Markov approximations of different orders is at worst propor-tional to the continuity rate of the chain. The result generalizes previous bounds ob-tained by X. Bressaud and ourselves, while relying on a similar coupling argument. Keywords: Chains of infinite order, chains with complete connections, Ornstein dis-

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