J. S. Rosenthal, \Minorization conditions and convergence rates for Markov chain Monte Carlo," Journal of the American Statistical Association, vol. 90, no. 430, pp. 558-566, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Moreover, our results are framed in terms of Foster Lyapunov drift conditions, which have already proved very successful in the analysis of homogeneous Markov chains, especially those related to MCMC (Markov chain Monte Carlo) algorithms [Roberts and Tweedie (1996) Roberts and Tweedie (1996) [Rosenthal (1995)] Stramer and Tweedie (1999a) Stramer and Tweedie (1999b) To the best of our knowledge, such drift conditions have not been used before in the formulation of simulated annealing problems. Consider the problem of nding the set of global maxima of some density whose domain is a set X , ....

....contracting property of the transition kernel. We now study the asymptotic behaviour of each of these terms. 2. 1 Bounding the estimation error In this section, we bound the estimation error P by a standard coupling construction [Lindvall (1992) and using a method pioneered by [Rosenthal (1995)] for time homogeneous Markov chains. In our time inhomogeneous setup, most of his arguments can be salvaged, but some subtle changes are required (see Subsections 2.1.1 and 2.1.2) The main result here is Proposition 2, where a bound on the estimation error is obtained that consists of a term due ....

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Rosenthal, J.S. (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association, 90, 558-566.

....condition, which together imply that the underlying Markov chain is geometrically ergodic. In this paper, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2) The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an analytic upper bound on the distance to stationarity. A rigorous answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are ....

....15 16) It is dicult to discuss drift and minorization before describing some basic ideas and notation from Markov chain theory (see Subsection 2. 1) but we can describe how they are used to give rigorous answers to (Q1) and (Q2) Once drift and minorization have been established, the results of Rosenthal (1995) or Roberts and Tweedie (1999) can be employed to calculate a bound on exactly how many iterations are necessary to get within a prespeci ed (total variation) distance of the target distribution. In other words, we can nd an n 0 such that 1 2 Z j f n 0 (x) x) j dx 0:01; say. The value n ....

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Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90: 558-566.

....Markov process SHORT TITLE: Stochastic monotonicity and rate bounds 1 Introduction 1 1 Introduction There has been considerable recent work on the problem of determining, in some computable way, the rate of convergence of Markov chains and processes. Recently, Meyn and Tweedie [11] Rosenthal [14], and Roberts and Tweedie [13] have given bounds on total variation distance from stationarity of positive recurrent Markov chains, in terms of Lyapunov Foster type drift conditions. Such results are extended to continuous time models by Roberts and Rosenthal [12] Alternative approaches to this ....

....small set was introduced. Let be the stochastic majorant of and , that is P (Y y) Gamma1; y] Gamma1; y] 10) Analogous to (5) we define = log (V ) log Gamma1 log(b= 1 Gamma ) V ) log Gamma1 ; 11) the inequality follows since (V ) b= 1 Gamma ) as in [6] or [14] for example. Theorem 2.2 Suppose that X is a stochastically monotone Markov chain satisfying (2) and (3) and started from x. Suppose also that C takes the form [ Gamma1; c] for some c 2 X. Then we have kP n (x; Delta) Gamma ( Delta)k r(n; b; V; d) 12) with r(n; b; V; d) given ....

J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90:558--566, 1995.

.... Moreover, our results are framed in terms of Foster Lyapunov drift conditions, which have already proved very successful in the analysis of homogeneous Markov chains, especially those related to MCMC (Markov chain Monte Carlo) algorithms [Roberts and Tweedie (1996) Roberts and Tweedie (1996) [Rosenthal (1995)] Stramer and Tweedie (1999a) Stramer and Tweedie (1999b) Consider the problem of finding the set of global maxima of some density whose domain is a set Theta, usually taken to be R k . This problem arises commonly in Bayesian statistics, in the context of maximum a posteriori (MAP) ....

....of each of these terms. A. Bounding the estimation error In this section, we bound the estimation error fl fl flP i X (i) 2 Delta j Gamma P i X (i) 2 Delta fi fi fi X (m) fl m jfl fl fl (13) by a standard coupling construction [Lindvall (1992) and using a method pioneered by [Rosenthal (1995)] for time homogeneous Markov chains. In our time inhomogeneous setup, most of his arguments can be salvaged, but some subtle changes are required (see Subsections II.1. and II.2. The main result here is Proposition 2, where a bound on the estimation error is obtained that consists of a term due ....

[Article contains additional citation context not shown here]

Rosenthal, J.S. (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo, J. of the Am. Stat. As., 90, 558-566.

....et al. 16] Ingrassia [20] and Belsley [5] proved results speci cally for Markov chain Monte Carlo. On general state spaces, not many results have been found yet. For partial results, see Amit and Grenander [2] Amit [1] Hwang et al. 19] Lawler and Sokal [24] Meyn and Tweedie [25] Rosenthal [33, 34, 35, 36], Baxter and Rosenthal [4] and Roberts and Rosenthal [30, 31] In particular, Diaconis and Stroock [13] and Sinclair [38] used geomet1 CHAPTER 1. INTRODUCTION 2 ric arguments with paths to bound the second largest eigenvalue of a selfadjoint discrete time Markov chain. On the other hand, Lawler ....

J.S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995b) 558-566.

....condition, which together imply that the underlying Markov chain is geometrically ergodic. In this paper, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2) The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an exact upper bound on the distance to stationarity. A de nitive answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are ....

.... will see, allows the user to formally answer (Q1) and (Q2) k The method described herein for developing exact answers to (Q1) and (Q2) requires that one establish a drift condition and an associated minorization condition for the underlying Markov chain, which together imply geometric ergodicity (Rosenthal 1995, Roberts and Tweedie 1999) It is dicult to discuss drift and minorization before describing some basic ideas and notation from Markov chain theory (see Section 2.1) but we can describe how they are used to answer (Q1) and (Q2) Once drift and minorization have been established, the results of ....

[Article contains additional citation context not shown here]

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90: 558-566.

....[4] For results on discrete time 1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3. Internet: yuen math.toronto.edu 1 general state spaces, see Amit and Grenander [2] Amit [1] Hwang et al. 17] Lawler and Sokal [22] Meyn and Tweedie [23] Rosenthal [29, 30, 31, 32], Baxter and Rosenthal [3] and Roberts and Rosenthal [26, 27, 28] In particular, there are results related to discrete approximations to a Langevin di usion (See e.g. Roberts and Rosenthal [28] which is a continuous time Markov process. Lawler and Sokal [22] took an idea from the literature of ....

J.S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995b) 558-566.

....say, iterations of the algorithm so that the Markov chain will forget the starting value. Actually, h n is still a consistent estimator of E h if there is no burn in; i.e. B = 0. However, if burn in is used then the preferred method is to determine B before any simulation starts (Rosenthal [34], Roberts and 12 Tweedie [33] Unfortunately, this can be a dicult task for a realistic model (see e.g. Jones and Hobert [22] The alternative to a priori calculation of B is the so called convergence diagnostics which try to determine when to stop burn in and start sampling based on the ....

J. S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association, 90:558-566, 1995.

....a probability measure Q( a small set C for which (C) 0, and an 0 such that for any x 2 C P (x; A) Q(A) 8 A 2 B (X ) 2) This implies that is strongly aperiodic. See Meyn and Tweedie (1993, Chapter 5) for a thorough discussion of minorization conditions, including existence issues. Rosenthal (1995) provides a general technique for establishing (2) in the context of MCMC (see also Mykland, Tierney and Yu, 1995) The random variables T and N t are de ned in terms of the split chain that is now developed. If (2) holds and x 2 C then we can write P (x; as a two component mixture P (x; ....

....start sampling the Markov chain. We are, of course, assuming that perfect sampling from is not an viable option. Otherwise, getting close to would be a mute point. It is possible to nd n by constructing exact upper bounds on kP n ( k through drift and minorization conditions (Rosenthal, 1995; Roberts and Tweedie, 1999) The phrase dicult theoretical analysis is used by Fill, Machida, Murdoch and Rosenthal (2000) to describe this method. While this approach has been successfully applied to a few relatively simple MCMC algorithms (Roberts and Rosenthal, 1999; Jones and Hobert, 2000) ....

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Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90: 558-566.

....See Jones and Hobert (2000a) for an introduction to these ideas. One can prove that MA is geometrically ergodic and, simultaneously, get an upper bound on M( 0 ; 0 )t n by establishing drift and minorization conditions. There are several different ways to do this (Meyn and Tweedie, 1994; Rosenthal, 1995; Roberts and Tweedie, 1999) The version I describe here is based on the work of Rosenthal (1995) A drift condition holds if for some function V : R R R , some 0 1, and some L 1 E[V ( j( 0 ; 0 ) V ( 0 ; 0 ) L : 6) As the notation suggests, this ....

....ergodic and, simultaneously, get an upper bound on M( 0 ; 0 )t n by establishing drift and minorization conditions. There are several different ways to do this (Meyn and Tweedie, 1994; Rosenthal, 1995; Roberts and Tweedie, 1999) The version I describe here is based on the work of Rosenthal (1995). A drift condition holds if for some function V : R R R , some 0 1, and some L 1 E[V ( j( 0 ; 0 ) V ( 0 ; 0 ) L : 6) As the notation suggests, this expectation is with respect to k( j 0 ; 0 ) the Markov transition density. An associated ....

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Rosenthal, J. S. (1995), "Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo," Journal of the American Statistical Association, 90, 558--566.

....sufficiently good approximation is reached PETROLEUM GEOSTATISTICS 9 and much research has concentrated on this topic. One approach used is to look for theoretical bounds on the number of iterations necessary to reach within a given distance from the specified distribution (Meyn and Tweedie, 1994; Rosenthal, 1994) but so far no good bounds seem to exist for models of the complexity necessary in reservoir characterization. A technique frequently used also for complex models is output analysis; see Ripley (1981) and references therein. One approach is to plot important univariate characteristics of the ....

Rosenthal, J.S. (1994). "Minorization conditions and convergence rates for Markov chain Monte Carlo", Technical Report 9321, Department of Statistics, University of Toronto.

.... to control convergence to equilibrium: as Nummelin wrote, the elementary techniques and constructions based on the notion of regeneration, and common in the study of discrete chains, can now be applied in the general case [14, page ix] More recently small sets have been used by Rosenthal [17] to establish rates of convergence for Markov chain Monte Carlo (see also the extended notion of pseudo small sets described by Roberts and Rosenthal [16] and also (under the rubric of gamma coupling) to produce effective Coupling from the Past (CFTP) constructions in the work of Murdoch and ....

J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of American Statistical Association, 90:558--566, 1995.

.... satisfactory (Cowles, Roberts Rosenthal, 1997) There has been much work on developing rigorous, a priori, quantitative bounds on the convergence time; see for example Sinclair Jerrum (1989) Diaconis Stroock (1991) Frieze, Kannon Polson (1994) Ingrassia (1994) Meyn Tweedie (1994) Rosenthal (1995), Mengersen Tweedie (1996) Polson (1996) and Frigessi, Martinelli Stander (1997) Application of many of these results is difficult in practice except to the simplest of problems, and the constant of proportionality is not always available. In this paper, we show how coupling methodology can ....

Rosenthal, J.S. (1995). Minorization condition and convergence rates for Markov chain Monte Carlo. J. Am. Statist. Assoc. 90, 558--66.

....Much of the success in bounding convergence in total variation distance arises from its coupling characterization TV ( inf Pr(X 6= Y ) where the infimum is taken over random variables X and Y whose distributions are and respectively. For examples of this, see Aldous and Diaconis (1987) Rosenthal (1995), Luby, Randall and Sinclair (1995) and Gibbs (2000) In this paper, we introduce the use of the Wasserstein metric to assessing convergence of Markov chain Monte Carlo algorithms. If , are two probability measures on the same space X , the Wasserstein metric is W ( inf E[d(X; Y ) 2.1) ....

Rosenthal, J. S. (1995). Minorization condition and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association 90: 558-- 566.

.... norm i.e. the determination of a rate function r = fr(n)g 1 and of the dependence on the initial distribution of the form kP n (x; k TV B r (x; n) lim sup n r(n)B r (x; n) 1: 1) Explicit expressions for the bounds have been recently obtained by Meyn and Tweedie (1994) Rosenthal (1995), Mengersen and Tweedie (1996) and Roberts and Tweedie (1999) under conditions implying the V uniform ergodicity (i.e. geometrical rates r(n) n , 1 ) The purpose of this paper is to construct such bounds under typically weaker conditions implying polynomial, subgeometrical, ....

....and Roberts and Tweedie (1999) under conditions implying the V uniform ergodicity (i.e. geometrical rates r(n) n , 1 ) The purpose of this paper is to construct such bounds under typically weaker conditions implying polynomial, subgeometrical, ergodicity. As illustrated by Rosenthal (1995) and Roberts and Tweedie (1999) computational bounds for the total variation distance can be obtained by using the so called Lindvall s inequality, which relates kP n (x; k TV to the tail probability of a coupling time T . The construction of this coupling time involves to de ne a ....

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Rosenthal, J. (1995). Minorization conditions and convergence rates for Markov Chain Monte Carlo.

....to add another small stone to the edifice of geometric ergodicity. 2 Main results There has been considerable recent work on assessing when a Markov chain is geometrically ergodic [5, Chapters 15,16] and on the related problem of determining rates of convergence for geometrically ergodic chains [6, 12, 3, 11]. One particular use of such work is in verifying whether the algorithms used in Markov chain Monte Carlo (MCMC) have a geometric rate of convergence, and if so to bound the number of iterations needed for convergence (see, amongst others, 4, 10, 9] for references) 2 Main results 3 Most of this ....

J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90:558--566, 1995.

.... no general guidelines are available for deciding on an appropriate number of iterations for a general state space Markov model, although much work has been done on this problem recently in many different contexts and from many different directions, both in the probabilistic literature (see e.g. [21, 28, 7]) and in the computer science literature (see e.g. 13, 12] In this paper we consider instead methods of perfect simulation, based on a backward coupling technique that enables exact draws from . These are closely linked to the couplingfrom the past (CFTP) algorithm introduced recently by ....

J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90:558--566, 1995.

....until the observations are satisfactorily close to stationarity. Recent references are Schervish and Carlin (1992) Chan (1993) Frigessi, Hwang, Sheu and Di Stefano (1993) Tierney (1994) Meyn and Tweedie (1994) Ingrassia (1994) Roberts and Polson (1994) Athreya, Doss and Sethuraman (1996) Rosenthal (1995), Mengersen and Tweedie (1996) Roberts and Tweedie (1996) Johnson (1996) Roberts and Sahu (1997) Kira and Ji (1997) Robert (1998) and Diaconis and Salo Coste (1998) The initial observations from this burn in period are usually discarded. At this point the transition distribution, Q, used ....

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558-566.

....the success in bounding convergence in total variation distance arises from its coupling characterisation d TV ( inf P (X 6= Y ) where the infimum is taken over random variables X and Y whose distributions are and respectively. For examples of this, see Aldous and Diaconis [2] Rosenthal [21], Luby, Randall and Sinclair [15] and Gibbs [10] To the author s knowledge, this paper is the first application of convergence in the Wasserstein metric to Markov chain Monte Carlo algorithms. The Wasserstein metric was chosen for this application because of its coupling characterisation, ....

Rosenthal, J.S. (1995) Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90, 558-566.

....(cf. Tierney [38] it will converge in distribution to , facilitating approximate sampling. One difficulty with these methods is that it is difficult to assess convergence to stationarity. This necessitates the use of difficult theoretical analysis (e.g. Meyn and Tweedie [27] Rosenthal [35]) or problematic convergence diagnostics (Cowles and Carlin [5] Brooks, et al. 2] to draw reliable samples and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [32] see also [33] and [34] and has been studied ....

Rosenthal, J. S. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 (1995), 558--566.

....(cf. Tierney [40] it will converge in distribution to , facilitating approximate sampling. One difficulty with these methods is that it is difficult to assess convergence to stationarity. This necessitates the use of difficult theoretical analysis (e.g. Meyn and Tweedie [29] Rosenthal [37]) or problematic convergence diagnostics (Cowles and Carlin [5] Brooks, et al. 2] to draw reliable samples and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [34] see also [35] and [36] and has been studied ....

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90 558--566.

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J.S. Rosenthal (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558-566.

No context found.

J.S. Rosenthal (1993), Minorization conditions and convergence rates for Markov chain Monte Carlo. Technical Report 9321, Dept. of Statistics, University of Toronto. J. Amer. Stat. Assoc., to appear.

No context found.

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90 558-566.

No context found.

J.S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995b) 558-566.

....V , which is a solution of the Foster Lyapunov drift condition (5) see Meyn and Tweedie [12, Theorem 16.1.4] Remark 2. Explicit expressions of the rate r and of the constant R as a function of the terms in (4) and (5) can be found in Meyn and Tweedie [13] Mengersen and Tweedie [11] Rosenthal [20], Roberts and Tweedie [18] Fort and Moulines [6] Douc et al. 3] and Fort [4] Under (A2) it is easily shown that P RS (x; has a nontrivial continuous component w.r.t. the Lebesgue measure and that this continuous component is bounded from below on a ball around x. From this, the ....

J. S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90(430):558-566, 1995.

....biases (cf. 6] In addition, it may be possible to do more sophisticated analysis (e.g. of autocorrelations) on the multiple parallel runs, though this may require greater communication between the di erent computers. 4.3. Theoretical quantitative bounds. It is sometimes possible (cf. 34] [46], 47] to use theoretical analysis to compute a burn in time B such that the distribution of the Markov chain after B steps is provably within of its stationary distribution. Such theoretical analysis is too dicult to be used routinely, however it has been successfully applied to some ....

....purposes, let us suppose we are interested in estimating the expected value E[ Z i ) i.e. the expected value E[g(Z i ) where g(z) z . We therefore set X i = 19 g(Z i ) Z i ) and use the estimator E j = X i on the j computer. Now, it is well known (see e.g. 49] [46]) that this Markov chain has as its stationary distribution the standard normal distribution N(0; 1) For illustrative purposes, we pretend that it is dicult and time consuming to sample directly from this distribution. We thus proceed (as in Observation 11 above) by having each computer j begin ....

J.S. Rosenthal (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558-566.

No context found.

Rosenthal, J.S. (1995b). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558-566.

No context found.

Rosenthal, J.S. (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558--566.

....0 equivalent to V , which is a solution of the Foster Lyapunov drift condition (5) see [11] Theorem 16.1.4) Remark 2. Explicit expressions of the rate r and of the constant R as a function of the terms in (4) and (5) can be found in Meyn and Tweedie [12] Mengersen and Tweedie [10] Rosenthal [19], Roberts and Tweedie [18] Fort and Moulines [4] and Douc et al. 3] Under (A2) it is easily shown that P d RS (x; has a nontrivial continuous component with respect to the Lebesgue measure and that this continuous component is bounded from below on a ball around x. From this, the ....

J. S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90:558-566, 1995.

No context found.

J.S. Rosenthal (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo.J. Amer. Stat. Assoc. 90, 558-566.

No context found.

J.S. Rosenthal (1995a), Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo. J. Amer. Stat. Assoc. 90 (1995), 558-566.

....conditions (cf. Tierney [43] it will converge in distribution to #, facilitating approximate sampling. One di#culty with these methods is that it is di#cult to assess convergence to stationarity. This necessitates the use of di#cult theoretical analysis (e.g. Meyn and Tweedie [32] Rosenthal [40]) or problematic convergence diagnostics (Cowles and Carlin [5] Brooks, et al. 2] to draw reliable samples and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [37] see also [38] and [39] and has been studied ....

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90 558--566.

....conditions (cf. Tierney [40] it will converge in distribution to #, facilitating approximate sampling. One di#culty with these methods is that it is di#cult to assess convergence to stationarity. This necessitates the use of di#cult theoretical analysis (e.g. Meyn and Tweedie [29] Rosenthal [37]) or problematic convergence diagnostics (Cowles and Carlin [5] Brooks, et al. 2] to draw reliable samples and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [34] see also [35] and [36] and has been studied ....

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90 558--566.

....conditions (cf. Tierney [38] it will converge in distribution to #, facilitating approximate sampling. One di#culty with these methods is that it is di#cult to assess convergence to stationarity. This necessitates the use of di#cult theoretical analysis (e.g. Meyn and Tweedie [27] Rosenthal [35]) or problematic convergence diagnostics (Cowles and Carlin [5] Brooks, et al. 2] to draw reliable samples and do proper inference. An interesting alternative algorithm, called coupling from the past (CFTP) was introduced by Propp and Wilson [32] see also [33] and [34] and has been studied ....

Rosenthal, J. S. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 (1995), 558--566.

....information about them. Such quantitative information is very important for practical applications. An initial attempt to provide useful quantitative bounds using the equations (1) and (2) was presented by Meyn and Tweedie (1994) using analytic techniques. Tighter bounds were obtained by Rosenthal (1995), using coupling methodology. Roberts and Tweedie (1999, 1998) later gave refined results, which were of the form kP n (x; Delta) Gamma kvar (A Bn)ae n ; where A and B are computable functions of , b, ffl, n 0 , V (x) and d j sup y2C V (y) It is guaranteed that ae 1 provided that ....

....0 (y) is non increasing for y Y : 8) In terms of this condition, we can state a general result (Roberts and Rosenthal, 1999b, Theorem 7) about slice sampler convergence. This result uses general quantitative bound results developed in Roberts and Tweedie (1999, 1998) building on the work of Rosenthal (1995) and Lund and Tweedie (1996) Recall that we are assuming (5) i.e. that sup x2R d f 1 (x) 1. Proposition 2. Consider the simple slice sampler for the target density (x) with any factorisation (x) f 0 (x) f 1 (x) If (8) holds for some 0 Y 1, then for all x 2 R d , and for all n , ....

Rosenthal, J.S. (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558--566.

....of Canada. The simple slice sampler s tractability to theoretical study stems from a stochastic monotonicity property. This allows good bounds on total variation distance from convergence to be obtained using the techniques developed in Roberts and Tweedie (1999, 1998) based on the work of Rosenthal (1995) and Lund and Tweedie (1996) Furthermore, convergence bounds can be given which are uniform over large classes of possible target density. Roberts and Rosenthal (1999) show that for any target distribution satisfying a particular condition on its level sets (restated as (9) below) convergence to ....

....theoretical bounds on the rates of convergence of the polar slice sampler algorithms. To do this, we require the following result from Roberts and Rosenthal (1999, Theorem 9) This result uses general quantitative bound results developed in Roberts and Tweedie (1999, 1998) based on the work of Rosenthal (1995) and Lund and Tweedie (1996) Recall also that the renormalisation condition (2) which states that sup x2R d f(x) 1, is assumed throughout. Proposition 3. Consider the simple slice sampler for the target density (x) with any factorisation (x) f 0 (x) f 1 (x) If (8) holds for some 0 ....

Rosenthal, J.S. (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558--566.

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J.S. Rosenthal (1995), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558-566.

....and Smith, 1990; Gelfand et al. 1990; Smith and Roberts, 1993; Tierney, 1994; Gilks, 1996) in that they guarantee an exact sample from the stationary distribution. In particular, they sidestep the need for provable bounds on convergence times of the algorithms (as in Meyn and Tweedie, 1994; Rosenthal, 1995). On the other hand, they require substantial computation to achieve just a single sample from the stationary distribution; to estimate functionals it may be necessary to obtain many such samples, each one requiring substantial additional computation. We are interested in ways to obtain more ....

Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association, 90:558--566.

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J.S. Rosenthal (1995b), Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Stat. Assoc. 90, 558--566.

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Rosenthal, J.S. (1995b), "Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo." Journal of the American Statistical Association, 90, 558--566. Correction, p. 1136.

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J. S. Rosenthal, \Minorization conditions and convergence rates for Markov chain Monte Carlo," Journal of the American Statistical Association, vol. 90, no. 430, pp. 558-566, 1995.

No context found.

J. S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90(430):558-566, 1995.

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J.S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995b) 558-566.

No context found.

Rosenthal, J. S. (1993). Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo. Tech. Rep. 9321, Dept. of Statistics, University of Toronto.

No context found.

Rosenthal, J.S. (1995), "Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo," Journal of the American Statistical Association, 90, 558-566.

No context found.

J.S Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90:558-566, 1995.

No context found.

J.S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Statist. Assoc. 90 (1995b) 558-566.

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: 387--405. Rosenthal, J. S. (1995b). Minorization condition and convergence rates for Markov chain Monte Carlo, Journal of the American Statistical Association

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J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association, 90:558--566, 1995.

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