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.

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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.

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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.

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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.

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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,

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