G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5-31.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some avors of the Gibbs sampler are reversible (Besag, Green, Higdon and Mengersen 1995) The use of RS for the purpose of constructing Monte Carlo standard errors is described in Section 5 and illustrated in Subsection 6.4. Geometric ergodicity is not necessary for CLTs (see e.g. Jarner and Roberts 2001) but a CLT may fail to hold even in very simple applications of subgeometric MCMC. For example, Roberts (1999) shows that for the independence Metropolis algorithm of Example 1, a CLT (for all functions h that are bounded away from zero at 1) will not hold if 2. In the next section, we de ....

Roberts, G. O. and Rosenthal, J. S. (1998a). Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion), Canadian Journal of Statistics 26: 5-31.

....et al. 2000) a full Bayesian analysis of the data set is performed. Below we x the regression parameters and the covariance parameters and 2 at values equal to the posterior means computed in Christensen et al. 2000) Roberts et al. 1997) Roberts and Rosenthal (1998b) and Breyer and Roberts (2000) show for certain classes of target densities that the proposal variance h should be tuned to obtain acceptance rates close to 0:23 for the random walk Metropolis algorithm and 0:57 for the Langevin Hastings algorithm. Strictly speaking these results do not cover truncated Langevin Hastings and ....

Roberts, G. O. and Rosenthal, J. S. (1998a). Markov chain Monte Carlo: some practical implications of theoretical results. Canad. J. Statist. 26, 5-31.

....and autocorrelations for di erent statistics. Some representative time series are shown in Figure 3. One may note that equilibrium is reached quickly and that has a heavy tailed posterior distribution. Theoretical results in Roberts et al. 1997) Roberts and Rosenthal (1998b) and Breyer and Roberts (2000) suggest that one should tune the proposal variances to obtain acceptance rates around 0:23 for random walk updates and 0:57 for Langevin Hastings updates. The overall acceptance rates for the updates of , and log were 0:57, 0:56, 0:23, and 0:27, respectively. We observed that the ....

Roberts, G. O. and Rosenthal, J. S. (1998a). Markov chain Monte Carlo: some practical implications of theoretical results. Canad. J. Statist. 26, 5-31.

....(Meyn 4 and Tweedie 1993, Chapter 15) Generally speaking, geometrically ergodic chains are good in the sense that they can be expected to quickly produce output that is similar to what one would get by sampling directly from the target distribution. The following example, introduced by Roberts and Rosenthal (1998), illustrates the di erence between geometric and non geometric convergence. Example 1. Suppose the target distribution is Exp(1) We say W Exp( if its density is e w I(w 0) Consider an independence Metropolis sampler with an Exp( candidate that evolves as follows: Let the current ....

Roberts, G. O. and Rosenthal, J. S. (1998). Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion), Canadian Journal of Statistics 26: 5-31.

....within other algorithms, e.g. within the E step of the EM algorithm (McCulloch [24] Gilks et al. 20] provide a wonderful introduction to applied MCMC. Readers interested in a more theoretical introduction should consult Besag et al. 3] Robert and Casella [31] and Roberts and Rosenthal 1 [32]. Speci c environmetrical applications where MCMC is useful include remote sensing (Green and Strawderman [21] capture recapture studies (Feinberg et al. 12] agricultural eld experiments (Besag et al. 3] animal breeding (Wang et al. 38] genetics (Wilson and Balding [41] and space time ....

G. O. Roberts and J. S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5-31, 1998.

.... computed in Christensen et al. 2000) 10 The proposal variance h is tuned to obtain acceptance rates close to the optimal values 0:23 for the random walk Metropolis algorithm and 0:57 for the Langevin Hastings algorithm, see Roberts et al. 1997) Roberts and Rosenthal (1998b) and Breyer and Roberts (2000). The truncation constant in Section 3.2 is chosen to be H = 50, which is roughly two times the maximal observed count. The matrix K in the decomposition = KK T is calculated using either Cholesky factorisation or the circulant embedding technique (Wood and Chan, 1994; Dietrich and Newsam, ....

Roberts, G. O. and Rosenthal, J. S. (1998a). Markov chain Monte Carlo: some practical implications of theoretical results. Canad. J. Statist. 26, 5-31.

....though the chains have coalesced at t = 1, we do not accept X 1 = 0 as a draw from . In CFTP, T and X 0 are dependent random variables. Therefore, a user who gets impatient or whose computer crashes and who therefore restarts 7 runs when T gets too large will generate biased samples. Another algorithm, due to Fill (1998), generates samples from in a way that is independent of the number of steps. 4 Fill s algorithm A simple version of Fill s algorithm (Fill) is: 1. Arbitrarily choose a time T and state x T = z. 2. Generate X T 1 jx T , X T 2 jx T 1 , X 0 jx 1 . 3. Generate [U 1 jx 0 ; x 1 ] U 2 jx ....

G.O. Roberts and J.S. Rosenthal. Markov chain monte carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5-32, 1998.

.... Mengersen (1995) and Gilks, Richardson Spiegelhalter (1996) An important issue in the implementation of MCMC algorithms is whether or not they actually converge to the distribution of interest, and if so, how quickly. For a discussion of these issues see, for example, Tierney (1994) and Roberts Rosenthal (1998). Convergence diagnostics (Cowles Carlin (1996) Brooks Roberts (1997) have been developed to monitor convergence of the algorithm while it is running, however none are completely satisfactory (Cowles, Roberts Rosenthal (1997) There has been much work on developing rigorous, a priori, ....

Roberts, G.O. & Rosenthal, J.S. (1998). Markov chain Monte Carlo: Some practical implications of theoretical results, with discussion. Canadian Journal of Statistics 26, 5--31.

.... to be artifacts of the approximations used in the analysis, and thus there is no reason to believe that geometric ergodicity does not hold when m 0 ( p 5 Gamma 2)m 00 and or a 1 (3K Gamma 2) 2K Gamma 2) Indeed, we have done some simulation experiments, similar to those described in Roberts and Rosenthal (1998, Section 4) which suggest that the central limit theorem holds in the a 1 (3K Gamma 2) 2K Gamma 2) case. On the other hand, geometric ergodicity is not necessary for central limit theorems. Our one way random effects model is a special case of the hierarchical general linear mixed model ....

Roberts, G. O. and Rosenthal, J. S. (1998). Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion), to appear in Canadian Journal of Statistics .

.... convergence control is to decide whether or not empirical averages of the quantities of interest, 1 T T X t=1 h( t) 1:1) have properly converged to the Bayes estimate IE [h( Since most MCMC algorithms produce chains which satisfy the Central Limit Theorem (see Geyer, 1992, or Roberts and Rosenthal, 1997, for theoretical issues) we link our assessment to a normality check for the standardised version of (1.1) The fundamental feature of the method is to subsample the original chain ( t) in such a way that the mean and variance of (1.1) can be simultaneously estimated. In the case of ....

....but, in latent variable models such as hidden Markov chains, a duality principle introduced in Diebolt and Robert (1994) can be invoked to produce samples on which to run the test. The validity of the Central Limit Theorem for MCMC algorithms is now well charted (see Meyn and Tweedie, 1993, and Roberts and Rosenthal, 1997, for theoretical founda3 tions) In most set ups, Omega T = 1 p T 2 h T X t=1 i h( t) Gamma IE [h( j ; 2:1) with 2 h = var (h( 0) 2 1 X t=1 cov (h( 0) h( t) is thus asymptotically distributed as a N (0; 1) random variable. ....

Roberts, G.O. & Rosenthal, J.S. (1997) Markov chain Monte Carlo: some practical implications of theoretical results. Tech. report, Stats. Lab., U. of Cambridge.

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G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5-31.

....To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [23] Tierney [24] Gilks et al. 11] Roberts and Rosenthal [21]. The Metropolis Algorithm proceeds by starting all the 2M N parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0; random variable to the parameter. It then accepts this new value with probability min(1; new) old) otherwise it rejects it ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5--31, 1998.

....To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm (for background see, e.g. Smith and Roberts [48] Tierney [49] Gilks et al. 22] Roberts and Rosenthal [45]) We denote this algorithm as BAYESIAN. The Metropolis Algorithm proceeds by starting all the 3n parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0, # ) random variable to the parameter. It then accepts this new value with probability min(1, ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5--31, 1998.

....at di erent speeds; they might have di erent user loads on them; one or more of them might be down; etc. Handling these issues correctly is crucial to the success of parallel Monte Carlo. In addition, Markov chain Monte Carlo algorithms are now very common (see for example [17] 51] 53] 22] [45]) and parallelising them presents additional diculties such as determining appropriate burn in time. We note that similar issues have been considered in various contexts in the operations research literature. In particular, in an excellent series of papers ( 23] 24] 25] 26] 27] Glynn ....

....on the speed at which the simulation happens to run, then it follows that a second run would not produce identical results even if started with the same pseudo random number seed. 4. Parallel Markov chain Monte Carlo. Markov chain Monte Carlo (MCMC) algorithms (see e.g. 17] 51] 53] 22] [45]) such as the Gibbs sampler and the Metropolis Hastings algorithm, have become extremely popular in statistics (especially Bayesian statistics) as a method of approximately computing dicult high dimensional integrals. They are also used in theoretical computer science for approximate counting ....

G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5-31.

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Roberts, G.O and Rosenthal, J.S. (1997). Markov chain Monte Carlo: Some practical implications of theoretical results. Canadian J. Statist., to appear.

....: To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see e.g. Smith and Roberts [14] Tierney [15] Gilks et al. 6] Roberts and Rosenthal [12]) 7 There is of course some arbitrariness in the specification of this Bayesian algorithm, e.g. in the form of the prior distributions and in the precise formula for the probability of a link from i to j. However, the model appears to work well in practice, as our experiments show. We note that ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5--31, 1998.

....To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [22] Tierney [23] Gilks et al. 11] Roberts and Rosenthal [20]. The Metropolis Algorithm proceeds by starting all the 2M N parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0; 2 ) random variable to the parameter. It then accepts this new value with probability min(1; new) old) otherwise it rejects it ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5--31, 1998.

....To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [13] Tierney [14] Gilks et al. 6] Roberts and Rosenthal [12]) There is, of course, some arbitrariness in the specification of this Bayesian algorithm, e.g. in the form of the prior distributions and in the precise formula for the probability of a link from i to j. However, the model appears to work well in practice, as our experiments show. We note that ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5--31, 1998.

....: To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see e.g. Smith and Roberts [14] Tierney [15] Gilks et al. 6] Roberts and Rosenthal [12]) 7 There is of course some arbitrariness in the speci cation of this Bayesian algorithm, e.g. in the form of the prior distributions and in the precise formula for the probability of a link from i to j. However, the model appears to work well in practice, as our experiments show. We note that ....

G.O. Roberts and J.S. Rosenthal. Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian Journal of Statistics, 26:5-31, 1998.

....be used to construct coalescence (see e.g. Murdoch and Green, 1998) whereby copies of the chain started at all elements of the state space all become equal simultaneously. This is especially important in exact sampling schemes such as coupling from the past (Propp and Wilson, 1996) and Fill s algorithm (Fill, 1998; Fill, Machida, Murdoch, and Rosenthal, 1999) In a di erent direction, small sets can be used to construct shift couplings (cf. Aldous and Thorrison, 1993; Roberts and Rosenthal, 1997) whereby two copies of the chain become equal at two di erent times. This can be used to provide bounds on the ....

G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5-31.

....[13] Meyn and Tweedie [10] Tierney [20] Smith and Roberts [19] Asmussen [1] Athreya et al. 2] and elsewhere. Some of these discussions were motivated partially by Markov chain Monte Carlo algorithms, which have recently received a great deal of attention (see e.g. 6] 19] 20] 7] [16]) and for which convergence issues are extremely important. Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 3G3. Email: jeff utstat.toronto.edu. Web: http: utstat.toronto.edu jeff . Supported in part by NSERC of Canada. Despite the numerous references to the ....

G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5--31.

No context found.

Roberts, G.O and Rosenthal, J.S. (1997). Markov chain Monte Carlo: Some practical implications of theoretical results. Canadian J. Statist., to appear.

....different speeds; they might have different user loads on them; one or more of them might be down; etc. Handling these issues correctly is crucial to the success of parallel Monte Carlo. In addition, Markov chain Monte Carlo algorithms are now very common (see for example [17] 52] 54] 22] [46]) and parallelising them presents additional difficulties such as determining appropriate burn in time. We note that similar issues have been considered in various contexts in the operations research literature. In particular, in an excellent series of papers ( 23] 24] 25] 26] 27] Glynn ....

....S 1 of computations before terminating. Thus, generally speaking, for parallel computation the usual S = 1 Unbiased Stopping Rule (as in Observation 7) is probably the best choice. 4. Parallel Markov chain Monte Carlo. Markov chain Monte Carlo (MCMC) algorithms (see e.g. 17] 52] 54] 22] [46]) such as the Gibbs sampler and the Metropolis Hastings algorithm, have become extremely popular in statistics (especially Bayesian statistics) as a method of approximately computing difficult high dimensional integrals. They are also used in theoretical computer science for approximate counting ....

G.O. Roberts and J.S. Rosenthal (1998), Markov chain Monte Carlo: Some practical implications of theoretical results (with discussion). Canadian J. Stat. 26, 5--31.

No context found.

Algor. 27, 170--217. Roberts, G.O. & Rosenthal, J.S. (1998). Markov chain Monte Carlo: Some practical implications of theoretical results, with discussion. Can. J. Statist. 26, 5--31.

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