G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:1325, 1997. Paper no. 2.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....: S(x n ) is the vector of random e ects associated with the n locations where we have observations. Suppose that S = S(x n 1 ) S(x n q ) are q 0 additional locations of interest for prediction. Diggle et al. 1998) use a xed scan hybrid algorithm (in the terminology of Roberts and Rosenthal, 1997) where the covariance parameters, the regression parameters, and each of the random e ects S 1 ; S n q are updated in turn in each scan. The update of a random e ect S i is computationally demanding, since it involves the calculation of the conditional variance given the n q 1 other ....

....is ( 0 ) 1 f( 0 jy) exp( 1 2h k ( 0 )k 2 ) f( jy) exp( 1 2h k ( k 2 ) 11) Using the gradient to adapt the proposal kernel to the target density may lead to much better convergence and mixing properties than for an ordinary random walk Metropolis chain. By Roberts et al. 1997) and Roberts and Rosenthal (1998b) the number of iterations required to obtain convergence is O(d 1 ) for the random walk algorithm and O(d 1=3 ) for the Langevin Hastings algorithm, so the bene t of using Langevin Hastings increases as the dimension increases. 3.1.3 Geometric ergodicity ....

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Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Probab. 2, 13-25.

....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 and Sokal [24] took an idea from the literature of ....

....discuss the corresponding continuous time process of a discrete time transition kernel. 3. For probability measures , we have k k 2 2 = k k 2 2 1: 4. We used the L 2 norm to measure the rate of convergence. For the relation between L 2 norm and other norms, see e.g. Roberts and Rosenthal [31]. 2.2 Path Bounds for 1 (P ) In this section, we prove a class of lower bounds for 0 (L) de ned in (1.6) and an upper bound of the Cheeger s k constant for a reversible Markov chain (discrete or continuous time) on R n described in 1.2. In both cases, we obtain upper bounds for 1 (P ) de ....

G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997b) 13-25.

....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 di erential geometry and proved the Cheeger s inequality for ....

G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997b) 13-25.

....F (B)j = 0 for all x 2 S. The latter condition is known to hold for aperiodic positive recurrent Harris chains; see Theorem 13.0.1 in Meyn and Tweedie (1993) Other sucient conditions for (B0) are discussed below in Remark 2.2. The measurability of x 7 sup B2S jM n (x; B) F (B)j was proved in Roberts and Rosenthal (1997). 2.1. Lemma. Suppose (B0) holds. Let p n , q n and r n be positive integers such that p n q n r n and q n p n 1. Then the sequences hQn i and h Qn i of distributions de ned by Q n = L(X 0 ; X pn ; X qn ; X rn j P ) and Qn = L(X 0 ; X pn ; X qn ; ....

Roberts, G.O. and Rosenthal, J. S. (1997), Geometric ergodicity and hybrid Markov chains. Electr.

....Then the invariant p.d.f. is q r; x) 4 r 2 sin 2 1 S r; x) uniform over S r; Take the uniform paths with b = b r; dre. By Corollary 2.4 (a) k ffl (r; k 0 (r; 4dre 4 r 2 sin 2 : 18 So, 1 Gamma 1 k ffl (r; the upper bounds for 1 (P ) in Theorem 2. 1, go to 1 as 0. Intuitively, as the angle becomes sharper, the convergence rate (for the continuous time Markov chain or the discrete time Markov chain with transition kernel 1 2 (I P ) becomes slower. Example 4.4. Two dimensional non convex case. a) Suppose S = f(x; y) 2 ( Gammaa; a) Theta R : a ....

G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997b) 13-25.

....1 X r=1 (f f) Q r (f f) 2.6) Suppose that the Markov chain is reversible. This means that Q is in detailed balance (1.1) with . Then the asymptotic variance of the empirical estimator can be written 6 in the following way; compare Mira and Geyer (1999) By Theorem 2. 1 of Roberts and Rosenthal (1997) and the V uniform ergodicity of Q, the transition distribution Q is L 2 ( geometrically ergodic. Further, detailed balance is equivalent to selfadjointness of Q as an operator on L 2 ( f Qg) Qf g) for f; g 2 L 2 ( 2.7) Write I(x; dy) x (dy) for the identity kernel. The ....

Roberts, G. O. and J. S. Rosenthal (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2, 13-25.

....So, without loss of generality, we can work with stationary Markov chains when dealing with the strong law of large number or the CLT. For general state spaces several conditions have been stated that guaranty the existence of a CLT such as uniform, geometric ergodicity or other mixing conditions [29, 22, 27]. In Section 3.2 a general sufficient condition for CLT will be given. A detailed discussion on this issue is available in [29] and [5] The variance in the CLT, v(f; P ) is the limit, as n tends to infinity, of oe 2 n = nVar [ n ] 1 n n X i=1 n X j=1 Cov [f(X i ) f(X j ) 1 ....

....that the spectrum of one operator is smaller than the spectrum of another operator, we can at most compare the suprema of the spectra and this is what we will do. For reversible geometrically ergodic chains, all the eigenvalues but the principal eigenvalue, 0P = 1, are bounded away from Sigma1 [27]. When considering a transition kernel as an operator on the subset L 2 0 ( of L 2 ( of zero mean functions, we eliminate from its spectrum the eigenvalue one associated with constant functions. Unless otherwise stated a transition kernel will be considered as an operator on L 2 0 ( ....

[Article contains additional citation context not shown here]

G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:13--25, 1997.

....Q r is the r step transition distribution of the underlying Markov chain. This central limit theorem holds for f 2 V if the chain is positive Harris recurrent and V uniformly ergodic (Meyn and Tweedie, 1993, Theorem 17.0. 1) and for square integrable f if the chain is also reversible (Roberts and Rosenthal, 1997, Corollary 2.1) Section 2 describes a result of McKeague and Wefelmeyer (1996) If the chain X 0 ; X n is reversible with transition distribution Q(x; dy) then the estimator 1 n P n i=1 Q(X i ; f) has smaller variance than the empirical estimator. To apply the result, we must ....

Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm.

.... Gamma t f (Q n;h ) j P n;h ) N (0; Omega Q Omega (m Gamma1) Am f) 2 ) h 2 H: Hence the empirical estimator is regular in the full model. 4. Reversible Markov chains Let us now treat the case when the Markov chain is known to be reversible, Q = Q. It follows from Theorem 2. 1 of Roberts and Rosenthal (1997) and the V uniform ergodicity of Q that Q is L 2 ( geometrically ergodic: kQ n Gamma Pik = supf ( Q n OE Gamma PiOE) 2 ) OE 2 L 2 ( OE 2 ) 1g Dae n for some finite D and some ae 1. This allows us to extend the operators U and A from LV to L 2 ( For m = 2 the ....

Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab., 2, 13-25.

....positive ffl, then p n(g n Gamma g) converges in distribution to a mean zero normal random variable whose variance can be written in terms of Cov (g(X 0 ) g(X n ) n = 0; 1; 2; when X 0 . For reversible chains, the ffl is unnecessary; that is, a finite second moment is sufficient (Roberts and Rosenthal, 1997). However, the Gibbs samplers currently under study are not reversible. If the function M( Delta) is bounded, the chain is called uniformly ergodic. Roberts and Polson (1994, Lemma 2) give a sufficient condition for uniform ergodicity of a Gibbs sampler (under a slightly different norm) However, ....

Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2: 13--25.

....every starting state X 0 j x p n(S n (g) Gamma (g) w N(0; oe 2 g ) 12) as n 1 (where N(0; 0) denotes the point mass at 0) This allows the derivation of con dence intervals for S n (g) as an estimate of (g) If P is geometrically ergodic and reversible w.r.t. then Corollary 2. 1 of (Roberts and Rosenthal, 1997) states that the CLT holds for every function g with (g 2 ) 1. CLTs for geometrically ergodic, but not necessarily reversible, Markov chains can be found in, e.g. Chan and Geyer, 1994) and chapter 17 of (Meyn and Tweedie, 1993) To establish geometric ergodicity in concrete situations the ....

Roberts, G. and Rosenthal, J. (1997). Geometric ergodicity and hybrid Markov chains.

....and normally distributed, therefore efficiency will be measured by the asymptotic variance of ergodic averages. For general state spaces several conditions have been stated that guaranty the existence of a central limit theorem, such as uniform, geometric ergodicity or other mixing conditions [83, 55, 73]. A detailed discussion on this issue is available in [83] and [8] The set of reversible transition kernels with respect to is a subset of P. In the first part of this chapter we restrict our attention to this subset while in the second part (Sections 2.5 2.6 2.7) we try to extend the ....

....the spectrum of another operator, we can at most compare the suprema of the CHAPTER 2. ORDERING MONTE CARLO MARKOV CHAINS 14 spectra and this is what we will do. For reversible geometrically ergodic chains, all the eigenvalues but the principal eigenvalue, 0P = 1, are bounded away from Sigma1 [73]. When considering a transition kernel as an operator on the subset L 2 0 ( of L 2 ( of zero mean functions, we eliminate from its spectrum the eigenvalue one associated with constant functions. Unless otherwise stated a transition kernel will be considered as an operator on L 2 0 ( ....

[Article contains additional citation context not shown here]

G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. 1997. Preprint at http://www.stats.bris.ac.uk/MCMC.

....So, without loss of generality, we can work with stationary Markov chains when dealing with the strong law of large number or the CLT. For general state spaces several conditions have been stated that guaranty the existence of a CLT such as uniform, geometric ergodicity or other mixing conditions [29, 22, 27]. In Section 3.2 a general sufficient condition for CLT will be given. A detailed discussion on this issue is available in [29] and [5] The variance in the CLT, v(f; P ) is the limit, as n tends to infinity, of oe 2 n = nVar [ n ] 1 n n X i=1 n X j=1 Cov [f(X i ) f(X j ) 1 ....

....that the spectrum of one operator is smaller than the spectrum of another operator, we can at most compare the suprema of the spectra and this is what we will do. For reversible geometrically ergodic chains, all the eigenvalues but the principal eigenvalue, 0P = 1, are bounded away from Sigma1 [27]. When considering a transition kernel as an operator on the subset L 2 0 ( of L 2 ( of zero mean functions, we eliminate from its spectrum the eigenvalue one associated with constant functions. Unless otherwise stated a transition kernel will be considered as an operator on L 2 0 ( ....

[Article contains additional citation context not shown here]

G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. 1997. Preprint at http://www.stats.bris.ac.uk/MCMC.

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G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997b) 13-25.

....P RS de ned by P RS : d (P 1 P d ) In this paper, we focus on the Random Scan Metropolis (RSM) algorithm, where X = R , and where each operator P i arises from a symmetric random walk Metropolis algorithm on the i th coordinate. This algorithm was studied by Roberts and Rosenthal [15, 16], and by Jarner and Hansen [9] One of the assumptions in Roberts and Rosenthal [16] is expressed in terms of the maximal curvature of all the geodesic curves on the contour manifold fy 2 R ; p(y) p(x)g as jxj 1. This condition is rather dicult to check even when d = 2; in addition, as ....

G. Roberts and J. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Comm. Prob., 2:13-25, 1997.

....de ned by PRS : d 1 (P 1 P d ) In this paper, we focus on the Random Scan Metropolis (RSM) algorithm, where X = R d , and where each operator P i arises from a symmetric random walk Metropolis algorithm on the i th coordinate. This algorithm was studied by Roberts and Rosenthal [14], 15] and by Jarner and Hansen [8] One of the assumptions in [15] is expressed in terms of the maximal curvature of all the geodesic curves on the contour manifold fy 2 R d ; p(y) p(x)g as jxj 1. This condition is rather dicult to check even when d = 2; in addition, as suggested in [8] ....

G.O. Roberts and J.S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Comm. Prob., 2:13-25, 1997.

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G.O. Roberts and J.S. Rosenthal (1997), Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability 2, Paper no. 2, 13-25.

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G.O. Roberts and J.S. Rosenthal (1997), Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability 2, Paper no. 2, 13-25.

.... 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 work has involved using the total variation distance (the L 1 norm) from stationarity of positive recurrent Markov chains, although there has been some work in other norms [8] The purpose of this paper is to show that geometric convergence in ....

....norm of a signed measure . Various necessary and sufficient conditions for such geometric ergodicity are given in [5, Chapters 15 and 16] Define the L 2 norm (with respect to ) of any signed measure as kk L 2 = Z ( d d ) 2 d and the space L 2 ( f : kk L 2 1g. Following [10], the chain is called L 2 ( geometrically ergodic if there exists c( 1 and ae 1 such that kP n ( Delta) Gamma k L 2 c( ae n (4) for every 2 L 2 ( In the reversible case this condition is equivalent to the existence of a spectral gap, and is implied by geometric ergodicity. ....

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G.O. Roberts and J.S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:2, 1997.

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G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:1325, 1997. Paper no. 2.

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Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability 2, 13-25.

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G.O. Roberts and J.S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Electronic Communications in Probability 2 (1997b) 13-25.

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G. O. Roberts and J. S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Elect. Comm. in Probab., 2:13-25, 1997.

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Roberts, G.O. and Rosenthal, J.S. (1997). Geometric ergodicity and hybrid Markov chains. Elect. Comm. in Probab. 2, 13--25.

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G.O. Roberts and J.S. Rosenthal. Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability, 2:13--25, 1997.

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