M.K. Cowles and B.P. Carlin. Markov chain Monte Carlo convergence diagnostics: A comparative review. JASA, 91:883--904, 1996  Home/Search   Document Details and Download   Summary   Related Articles   Check

....It is therefore common to discard the early samples of each chain. The initial MC states computed in step (2) are drawn from a rough approximation to # # in order to minimize the number of burn in samples. However, as there is no universally good way to determine when one reaches equilibrium [6], we currently set by hand the number of burn in samples to be the same across all chains. Each particle is updated to keep the target posterior # # invariant. As explained below, the hybrid Monte Carlo algorithm requires that we are able to evaluate both the density and the gradient of the ....

....first frame only; i.e. we looked for a value of L that yields low MC autocorrelations, with rejection rates close to 15 . Finally, b should be chosen so that each chain has reached equilibrium after b updates. There is no guarantee that equilibrium will be reached, but available diagnostics [6] can help to adaptively determine b for each chain. 8 Discussion and Future Work In summary, we have seen how the hybrid Monte Carlo filter samples high dimensional distributions more efficiently than the particle filter. Unlike the particle filter, for which the number of particles grows ....

M.K. Cowles and B.P. Carlin. Markov chain Monte Carlo convergence diagnostics: A comparative review. JASA, 91:883--904, 1996

....K in advance. Hence, this method of choosing B j = K would appear to have limited appeal for parallel MCMC in general. 4.2. Convergence diagnostics. In the absence of good theoretical knowledge of appropriate burn in times B j , it is common to use convergence diagnostics (see e.g. 18] [5], 2] to determine the burn in time. Here the values B j are chosen on line, based on statistical analysis of the sample run 1 ; X 2 ; or perhaps of the underlying Markov chain run Z 2 ; in progress. Such convergence diagnostics often work well in practice. However, they have ....

....chain run Z 2 ; in progress. Such convergence diagnostics often work well in practice. However, they have at least two draw backs from the parallel MCMC perspective. First, they sometimes prematurely diagnose convergence by providing a burn in time B which is too small (see e.g. 33] [5]) leading to biases as in Subsection 4.1 above. Second, as shown in [6] by basing the burnin time on the sample in progress, convergence diagnostics sometimes introduce biases of their own (even if the Markov chain converges immediately) 13 We thus record: Observation 9. When running ....

M.K. Cowles and B.P. Carlin (1996), Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. J. Amer. Stat. Assoc. 91, 883-904.

....however none of these is efficient. Recently, a consistent solution for the maximum likelihood was introduced: Monte Carlo maximum likelihood. This algorithm requires no direct evaluation of the partition function, it is consistent and converges to the maximum likelihood with probability one [14, 15]. 3.2. Cluster based probability models Non parametric modeling allows more flexibility, but results in more complex algorithms and requires large training data sets. From the class of non parametric models the kernel estimate plays, in the last time, an important role in signal processing. ....

Cowles, M.K., Carlin, B.P. (1996) Markov Chain MonteCarlo convergence diagnostics: a comparative study, J. Amer. Statist. Soc. 91, pp. 883-904.

....is intended to give the parameters time to come close to their equilibrium distribution. Tests carried out using the R CODA package on the examples in section 5. 1 suggested that this was indeed effective in removing the transients, although we note that it is widely recognized (see, e.g. [2]) that determining when the equilibrium distribution has been reached is a difficult problem. Although the number of iterations used is much less than typically used for MCMC methods it should be remembered that (i) each iteration involves L = 20 leapfrog steps and (ii) that by using HMC we aim to ....

M. K. Cowles and B. P. Carlin. Markov-Chain Monte-Carlo Convergence Diagnostics---A Comparative Review. J. American Stat. Assoc., 91:883-- 904, 1996.

....analysis, but we are working to improve this. As part of a quality control effort, we studied the performance of our sampling scheme on a collection of voxel time series from several experiments. Graphical diagnostics, correlations among parameters, and various standard convergence diagnostics [14] based on parallel chains with different starting points suggest that the chains are mixing quite well, and equilibrating sufficiently and also that the Normal approximation is reasonable in most cases. However, more systematic study is needed. For the default configuration, our sampling algorithm ....

M. K. Cowles and B. P. Carlin. Markov chain monte carlo convergence diagnostics: A comparative review. Technical report, Harvard School of Public Health, 1995. 33

....its target distribution . Convergence diagnostics is still an area of active research. A recent review on methods used for establishing whether an MCMC algorithm has converged, i.e. whether its output can be regarded as samples from the target distribution of the Markov chain, has been given by Cowles and Carlin (1995). Some of these methods are implemented in the software CODA (Best et al. 1995) CODA is a menu driven collection of SPLUS functions for analysing the output obtained by BUGS. Besides trace plots and the usual tests for convergence, CODA calculates statistical summaries of the posterior and kernel ....

Cowles, M.K., and Carlin, B.P. 1995. Markov chain Monte Carlo convergence diagnostics: a comparative review. Technical Report, University of Minnesota.

.... Note that the energy when the chain was initialised was 2789 and the first 27 values are outside the range of the y axis, b) the angle of the front relative to north (OE f ) of the univariate sample paths since the generating parameters are known, although other diagnostics could be used (Cowles and Carlin, 1996). We find that the procedure is insensitive to the initial value of the GP parameters, but that the parameters describing the location of the front (OE f ; d f ) need to be initialised close to the correct values if the chain is to be run for sensible times. In application some preliminary ....

Cowles, M. K. and B. P. Carlin 1996. Markov-Chain Monte-Carlo Convergence Diagnostics---A Comparative Review. Journal of the American Statistical Association 91, 883--904.

...., and so determines the length of chain N required for acceptably accurate inference. Convergence will be more rapid if the chain exhibits good mixing behaviour. Many different formal methods have been proposed for choosing suitable values of m and N ; see for example the review article by Cowles and Carlin (1996). We have taken a less formal approach, assessing the mixing behaviour of our chains on the basis of graphical output of the results, and comparing results of chains run from more than one starting point where we felt further investigation seemed necessary. We fixed on running the chain for N = 20 ....

Cowles, M. K. and Carlin, B. P. (1996) Markov Chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91, 883--904.

....various diagnostics, convergence will have essentially occurred, after which additional draws can be assumed to come from the posterior distribution. Unfortunately, there is considerable disagreement within the statistics literature on how to assess convergence of this and other MCMC methods #Cowles and Carlin, 1996; Kass et al. 1997#. For multiple imputation problems, wehave the additional requirement that the draws we use for imputations must be statistically independent, which is not a characteristic of successive draws from Markov chain methods like IP. Some scholars reduce dependence by using every ....

Cowles, Mary Kathryn and Bradley P. Carlin. 1996. #Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 434 #June#: 883#904.

....various diagnostics, convergence will have essentially occurred, after which additional draws can be assumed to come from 12 the posterior distribution. Unfortunately, there is considerable disagreement within the statistics literature on how to assess convergence of this and other MCMC methods #Cowles and Carlin, 1996; Kass et al. 1997#. For multiple imputation problems, wehave the additional requirement that the draws we use for imputations must be statistically independent, which is not a characteristic of successive draws from Markovchain methods likeIP. Some scholars reduce dependence by using every dth ....

Cowles, Mary Kathryn and Bradley P. Carlin. 1996. #Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 434 #June#: 883#904.

....4.5 Convergence and Stopping Rules The two parameters of the SEM algorithm described above are the number of SEM steps and the number of Gibbs iterations performed at each SEM step while 65 generating W given (X, # (n) The choice of both parameters presents a problem. Cowles and Carlin [14] and Brooks and Roberts [8] give a review of convergence diagnostics for MCMC. Raftery and Lewis [53] suggest methods for determining the number of steps required for the burn in, and for diagnosing lack of convergence or slow convergence, based on fitting first and second order Markov models ....

Cowles, M. K., and Carlin, B. P. (1996), "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 883-904.

....which will be presented in the talk. 7 Discussion The algorithm assumes that the Markov chain has converged after a certain number of burn in sweeps. In this paper, the number of burn in sweeps is arrived at by inspection. Many convergence diagnostics have been proposed recently (see e.g. [15]) and should be incorporated into the algorithm. Related to the rate of convergence is the mixing property of the Markov chain. Our current algorithm proposes single node transitions. Transition that updates a block of nodes at a time are being investigated to improve the mixing property of the ....

M. Cowles and B. Carlin, "Markov chain Monte Carlo convergence diagnostics: A comparative review," Journal of the American Statistical Association, vol. 91, no. 434, pp. 883--904, 1996.

....on an i486 133MHzand UltraSPARC 167MHz, respectively. Note the successful tracking in Fig. 2 where a noticeable proportion of the live quads dies through time. The number of burn in sweeps of the MCMC is arrived at by inspection. Many convergence diagnostics have been proposed recently (see e.g. [9]) and will be incorporated into the algorithm. Related to the rate of convergence is the mixing property of the Markov chain. Our current algorithm proposes single node transitions. Transition that updates a block of nodes at a time should improve the mixing property of the chain. Since there is ....

M. Cowles and B. Carlin, "Markov chain Monte Carlo convergence diagnostics: A comparative review," J. of the American Statistical Association, 91:883--904, 1996.

....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 output of the algorithm. Brooks and Roberts [6] Cowles and Carlin [9], and Robert and Casella [31] contain thorough discussions about the use of convergence diagnostics. 4) A reasonable way to decide when to stop sampling is to wait until the ergodic average h n reaches some prespeci ed level of accuracy. Just as with iid sampling, if a central limit theorem ....

M. K. Cowles and B. P. Carlin. Markov chain Monte Carlo convergence diagnostics: A comparative review. Journal of the American Statistical Association, 91:883-904, 1996.

....Essentially the same approach can handle the general case (2.1) The convergence of the Gibbs sampler can be tested through some formal procedures, but we made the option to use multiple chains and graphical procedures. For some discussion on convergence problems see Gelman and Rubin(1992) and Cowles and Carlin(1996). 9 ....

Cowles, M.K. and Carlin, B.P.(1996). Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. Journal os the American Statistical Association, 91(434), 883--904.

....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 and used by a number of authors (including Kendall [26] Mller ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91 883--904.

....inspection of plots of the Monte Carlo output f t ; t = 1; ng. The Markov chains converge in less than 300 iterations in most examples according to visual inspection of the monitoring statistics. Here we set burn in m = 500. More formal methods for convergence diagnostics can be found in[12][13] Decision about the iteration number is an important and practical matter. The aim is to run the chain long enough to obtain adequate precision in the estimator. Here three chains are run in parallel with different starting values from Eq. 7) If they do not agree adequately, the iteration ....

M. Cowles and B. Carlin, "Markov Chain Monte Carlo convergence diagnostics: a comparative review," tech. rep., Division of Biostatistics, School of public health, University of Minnesota, 1994.

....important 1 issue in the implementation of Markov chain Monte Carlo 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, but none is completely satisfactory (Cowles, Roberts Rosenthal, 1997) There has been much work on developing rigorous, a priori, quantitative bounds on the convergence time; see for ....

Cowles, M.K. & Carlin, B.P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Am. Statist. Assoc. 91, 883--904.

....discarded in order to avoid biasing the results by inclusion of states that reflect the state in which the chain was started rather than its equilibrium distribution. The bewildering variety of methods for diagnosing convergence and discarding an appropriate burn in period have been reviewed by Cowles and Carlin (1996), Brooks and Roberts (1998) and Mengersen, et al. (1999) One convergence diagnostic, due to Johnson (1996) looks at multiple coupled chains that are started from different initial states, but that subsequently undergo transitions determined by the same random numbers. Rosenthal (1995b) ....

Cowles, M. K. and Carlin, B. P. (1996) "Markov chain Monte Carlo convergence diagnostics: a comparative study", Journal of the American Statistical Association, vol. 91, pp. 883-904.

....draws. Convergence of the MCMC chain Analysis of stock return volatility and trading volume 677 Fig. 3. Histograms of simulated parameters and sequence of draws: univariate S model P1 #;P2 #;P3 ## # . ##For an extensive review of current algorithms for checking MCMC output we refer to Cowles and Carlin (1996). is checked by comparing the draws of several chains simultaneously. ## The marginal distributions in the histograms are in line with our expectations. Table 2 provides summary statistics on these distributions. The constant in the transition equation, # (P1 Univariate) has a distribution that ....

Cowles, M. and Carlin, B. (1996) Markov Chain Monte Carlo convergence diagnostics: a comparative review, Journal of American Statistical Association, 91, 883---904.

....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 and used by a number of authors (including Kendall [24] Mller ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91 883--904.

....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 and used by a number of authors (including Kendall [23] Mller ....

Cowles, M. K. and Carlin, B. P. Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Amer. Statist. Assoc. 91 (1996), 883--904.

....at convergence. In addition, the simulation standard errors were similar to those corresponding to Table 1. We have implemented the Gibbs sampler using the algorithm given in Section 3.2. The convergence of the Gibbs sampler was checked using several diagnostic procedures as recommended by Cowles and Carlin (1996). We ran 10 multiple chains with dispersed initial values and computed potential scale reductions (PSR s) PSR values close to 1 are indicative of convergence of the Markov chain to the target distribution (See Gelman Rubin 1992) We also computed the autocorrelation coe#cients within chains to ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Amer. Statist. Assoc., 91, 883-904.

....Gelfand, Sahu, and Carlin (1996) we simultaneously standardized all covariates for both the KSCN data and NaSCN data in the implementation of the Gibbs sampler developed in Section 3.1. We checked convergence of the Gibbs sampler by using several convergence diagnostics procedures recommended by Cowles and Carlin (1996) and we found that the convergence occurred earlier than 500 iterations. Table 1 gives results for the top three models based on several values of ( a 0 ; oe a 0 ) In Table 1, we let T , T 2 , and ln T denote time, time square, and the natural logarithm of time. From Table 1, it can be seen ....

Cowles, M.K. and Carlin, B.P. (1996), "Markov Chain Monte Carlo convergence diagnostics: a comparative review," Journal of the American Statistical Association, 91, 883-904.

.... in high dimensions (similar to the example in Section 3 above) Note that this highlights the dangers of monitoring one dimensional summaries of Markov chain output for convergence assessment an issue which has implications for convergence diagnostic procedures for MCMC in general (see e.g. Cowles and Carlin, 1996; Robert, 1996; Brooks and Roberts, 1998) On the other hand, we see from Figures 5 and 6 that the polar slice sampler is mixing very well indeed, even in high dimensions. Furthermore its low auto correlation values are consistent across different choices of functional (x 2 1 , jxj 2 , and ....

Cowles, M.K. and Carlin, B.P. (1996), Markov chain Monte Carlo convergence diagnostics: A comparative Review. J. Amer. Stat. Assoc. 91, 883--904.

.... , 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 and used by a number of authors (including Kendall [23] M ller ....

Cowles, M. K. and Carlin, B. P. Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Amer. Statist. Assoc. 91 (1996), 883--904.

.... , 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 and used by a number of authors (including Kendall [24] M ller ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91 883--904.

....of convergence is to monitor the value of many one dimensional functions of the evolving process, once these appear stable the Markov chain is assumed to have converged. Clearly this approach is subjective, but usually works well. A discussion of formal convergence diagnostics can be found in Cowles and Carlin (1996). The next question is how many sweeps should be performed after the transient period has ended. Due to dependence within the Markov chain, the estimates produced from a sample of size M will have an asymptotic variance var( i ) 2 =M; where 2 is the sampling variance of i , the ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. J. Am. Stat. Soc. 91, 883--904.

....issue in using MCMC methods is to ascertain the convergence of the simulated Markov chain to the distribution of interest and, if possible, to come up with stopping rules or convergence controls pertaining to this goal. While the literature on convergence diagnostics is constantly expanding (see Cowles and Carlin, 1996; Brooks and Roberts, 1998; Mengersen et al. 1998) we show in Section #4 that Riemann sums can be used for convergence control, by providing evaluations of the mass of the stationary distribution explored by the Markov chain at a given iteration, as in Brooks (1998) as well as leading to ....

Cowles, M. and Carlin, B. (1996). Markov chain Monte Carlo convergence diagnostics: a comparison study. J. Amer. Statist. Assoc., 91:883904.

....needs to determine for how long the simulation should proceed in order to be close to equilibrium. Convergence diagnostics have been developed to assist the decision on the length of the burn in (that is to say, the number of initial iterations which should be discarded) for a recent review see [5]. Although these diagnostics might warn when convergence has not yet been reached, they cannot guarantee that the Markov chain is in or close to equilibrium. Recently Propp and Wilson [6] have devised an ingenious algorithm, which augments the simulation algorithm to produce a certificate whether ....

Cowles, M.K and Carlin, B.P. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review, Journal of the American Statistical Association, 91, 883 - 904.

....check convergence and mixing behavior of any MCMC algorithm. Theoretical considerations are typically limited to rather simple models; therefore empirical output analysis is more practical. This is still an active research area, the reader is referred to Raftery and Lewis (1996) Gelman (1996) Cowles and Carlin (1996) and the relevant parts of Gilks, Richardson and Spiegelhalter (1996) We always look at several plots such as time series plots of the sampled values and calculate routinely autocorrelation functions for every parameter. Figure 2 shows the time series plot of a specific parameter of our first ....

Cowles, M. K., and Carlin, B. P. 1996. "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review." Journal of the American Statistical Association 91: 883--904.

....detection of convergence is to monitor the value of one dimensional functions of the evolving process, once these appear stable the Markov chain is assumed to have converged. Clearly this approach is subjective, but usually works well. A discussion of formal convergence diagnostics can be found in Cowles and Carlin (1996). The next question is how many sweeps should be performed after the transient period has ended. Due to dependence within the Markov chain, the estimates produced will have an asymptotic variance var( w jk ) oe 2 =M; where oe 2 is the sampling variance of w jk , the integrated ....

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. J. Am. Stat. Soc. 91, 883--904.

.... the central limit theorem to justify an asymptotic normal approximation (Carlin and Louis 1996) Unfortunately these methods are difficult to use, particularly since statisticians still disagree about criteria for determining when a Markov chain has converged in distribution to the true posterior (Cowles and Carlin, 1996; Kass et al. 1997) We welcome research that implements these advanced techniques, but in this paper we recommend a simpler method that should prove more accessible to a wide range of social scientists. Another useful alternative is bootstrapping, a non parametric approach that relies on the ....

Cowles, Mary Kathryn and Bradley P. Carlin. 1996. "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 434 (June): 883--904.

....finally i (t 1) m is drawn from (i m ji (t 1) 1 ; i (t 1) 2 ; Delta Delta Delta ; i (t 1) m Gamma1 ; y) whence i has been fully updated. The convergence assessment methods of the Gibbs sampler rely on post processing simulated values. A review of convergence diagnostics is provided by Cowles and Carlin (1996). Also there are online routines available from the web address cited below and the convergence diagnostic home page 1 of Mengersen and Robert (1998) Although these investigations have value, they only provide empirical assurances of convergence and are not definitive proofs of convergence. ....

Cowles, M. K. and Carlin, B. P. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. J. amer. Staist. Assoc., 91, 883--904.

....sampling methods are going to be applied. However, this isn t very satisfying to the novice user. Let me go on to my next question and return later to this more di#cult issue. 2. 2 Assessing Convergence Kass: Together with Kate Cowles, Brad has written a nice review of convergence diagnostics (Cowles and Carlin 1996). But it left me thinking that knowledge about this topic is not as great as the number of papers that have been written about it might lead one to believe. Indeed, some experienced users simply examine the trace plots (for some collection of parameters) informally. So, what do each of you do to ....

Cowles, M.K., and Carlin, B.P. (1996), "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 883--904.

No context found.

M.K. Cowles and B.P. Carlin (1996), Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Amer. Stat. Assoc. 91, 883--904.

....most applied users of MCMC techniques have used convergence diagnostics (see for example, Roberts, 1992; Gelman and Rubin, 1992; Raftery and Lewis, 1992) to assess convergence. These diagnostics often work well in practice; however they are not completely understood and offer no guarantees. See Cowles and Carlin (1996) for a comprehensive review. In this paper, we present a way to make use of theoretical upper bounds (taken from Rosenthal, 1995b) without doing prohibitively difficult computations. Specifically, we consider the use of auxiliary simulations to numerically verify certain hypotheses (drift and ....

....a variance components model, long advocated (Gelfand and Smith, 1990; Gelfand et al. 1990) as an ideal candidate for the Gibbs sampler. Our third example (Section 5) is a Gibbs sampler for an ordinal probit model, as used in biostatistics contexts (Carlin and Polson, 1992; Albert and Chib, 1993; Cowles, 1996). In all three of these models, we use our auxiliary simulation method to obtain useful, quantitative bounds on the convergence time of the Markov chain being studied. We note that, in addition to burn in, there are other aspects of convergence that are relevant to applied use of MCMC methods ....

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Cowles, M.K. and Carlin, B.P. (1996), "Markov Chain Monte Carlo Convergence Diagnostics: a Comparative Review." Journal of the American Statistical Association, to appear.

....describes the file format required for such output to be read into CODA. 1.2 A cautionary note on convergence diagnostics Before proceeding further with CODA, all users should be aware that none of the convergence diagnostics implemented in this package (or indeed, anywhere else) are foolproof. Cowles and Carlin (1995) tested a number of these methods, and found examples when each failed to detect lack of convergence. We recommend using a combination of diagnostics plus visual inspection of the trace plots and summary statistics generated by CODA. In this way, the user should be able to assess the convergence ....

....approaches. We do not recommend any particular method as being superior, but suggest that users try a combination of diagnostics when checking for convergence. A brief summary of the theory and interpretation of each diagnostic is given below, and readers are referred to recent reviews by Cowles and Carlin (1995) and Brooks and Roberts (1995) plus the individual references to each method for further details. 4.1 Geweke Geweke (1992) proposes a convergence diagnostic based on standard time series methods. The test is appropriate for use with single chains when convergence of the mean of some function of ....

Cowles, M. K. and Carlin, B. P. (1995). Markov chain Monte Carlo convergence diagnostics: a comparative review. J Amer Statist Assoc, (to appear).

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M.K. Cowles and B.P. Carlin. Markov chain Monte Carlo convergence diagnostics: A comparative review. JASA, 91:883--904, 1996

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M. Cowles and B. Carlin, "Markov chain Monte Carlo convergence diagnostics: A comparative study," J. Amer. Statist. Assoc., vol. 91, pp. 883--904, 1996.

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M. Cowles, B. Carlin, Markov Chain Monte Carlo convergence diagnostics: a comparative review, Technical Report, Division of Biostatistics, School of Public Health, University of Minnesota, 1994.

No context found.

Cowles, M. K. and Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91 883-904.

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Cowles, M. K., Carlin, B.P., 1996. Markov chain Monte Carlo convergence diagnostics: A comparative review. Journal of American Statistical Association 91, 883-904.

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Cowles, K. and Carlin, B. P. (1998) Markov Chain Monte Carlo convergence diagnostics: a comparative review. J. Amer. Statist. Assoc., , .

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Cowles, M.K. and Carlin, B.P. (1994). "Markov chain Monte Carlo convergence diagnostics: a comparative review". Research Rpt. 94-008, University of Minnesota, Biostatistics, Minneapolis.

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Cowles, M. K. and Carlin, B. P. (1996), "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review," Journal of the American Statistical Association, 91, 883--904.

No context found.

Cowles, M. K. , and Carlin, B. P. (1996), \Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review", Journal of the American Statistical Association,

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Cowles, M. K. & Carlin, B. P. (1996). Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. Journal of the American Statis23 tical Association, 91, 883-904.

No context found.

Cowles M.K. and Carlin B.P. (1996). Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. Journal of the American Statistical Association, 91, 883-904.

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Cowles, M.K. and Carlin, B.O. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative study. J. Amer. Statist. Assoc. 91, 883-904.

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