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## Markov chain monte carlo convergence diagnostics (1996)

Venue: | JASA |

Citations: | 364 - 6 self |

### Citations

5036 |
Stochastic relaxation, gibbs distributions, and the bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...a surprisingly short period of time, Markov chain Monte Carlo (MCMC) integration methods, especially the Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) and the Gibbs sampler (=-=Geman and Geman, 1984-=-; Gelfand and Smith, 1990) have emerged as extremely popular tools for the analysis of complex statistical models. This is especially true in the field of Bayesian analysis, which requires the evaluat... |

3538 | Equation of state calculation by fast computing machines - Metropolis, Rosenbluth, et al. |

2034 |
Monte Carlo sampling methods using Markov chains and their applications
- Hastings
- 1970
(Show Context)
Citation Context ...oved the manuscript. 1 Introduction In a surprisingly short period of time, Markov chain Monte Carlo (MCMC) integration methods, especially the Metropolis-Hastings algorithm (Metropolis et al., 1953; =-=Hastings, 1970-=-) and the Gibbs sampler (Geman and Geman, 1984; Gelfand and Smith, 1990) have emerged as extremely popular tools for the analysis of complex statistical models. This is especially true in the field of... |

1459 | Inference from iterative simulation using multiple sequences,” Stat - Gelman, Rubin - 1992 |

1214 |
Robust locally weighted regression and smoothing scatterplots
- Cleveland
- 1979
(Show Context)
Citation Context ...rations. Figure 3d shows Roberts' new (1994) diagnostic calculated from 11 chains started from dispersed initial values symmetrically arranged around the mean. The plotted lines are "lowess"=-= smooths (Cleveland, 1979-=-) of log-transformed I n and D n values. The fact that both sequences appear to stabilize and become equal after about fifty iterations suggests very rapid convergence. Table 3 shows Zellner and Min's... |

1210 |
Sampling-based approaches to calculating marginal densities
- Gelfand, Smith
- 1990
(Show Context)
Citation Context ...ly and for eachsof which explicit and easily sampled posterior condition-sals may be derived. Then, using marginals P(ae) and P(0)sestimated by "Rao-Blackwellization" of the Gibbs samplersoutput (see =-=Gelfand and Smith 1990-=-) for a particular valuesof the parameters a,i and /1, the "Gibbs sampler differencesconvergence criterion" may be calculated assA(?l)P(l lol) - PWOlP(ailoip) = 71sThe components needed for the "Gibbs... |

1107 | Markov chains for exploring posterior distributions
- Tierney
- 1994
(Show Context)
Citation Context ... so far separated that the state space would be effectively disconnected. We used this example to illustrate non-conjugate full conditionals as well as bimodality. A random-walk Metropolis algorithm (=-=Tierney, 1994-=-) was used to generate from each unnormalized full conditional. Nine parallel chains were run with starting values chosen at equal intervals from above the upper mode to below the lower mode. Plots fo... |

580 | Evaluating the accuracy of the sampling-based approaches to the calculation of posterior moments - Geweke - 1992 |

557 |
Stochastic simulation
- Ripley
- 1987
(Show Context)
Citation Context ...d be drawn to produce estimates with variance small enough to inspire confidence in their accuracy. For those diagnostics not addressing this latter issue, an alternative is to use batching (see e.g. =-=Ripley, 1987-=-, Section 6.2), or perhaps more sophisticated time series methods (as in Geyer, 1992). Sections 3 and 4 apply our collection of diagnostics to some relatively simple statistical models. In so doing, w... |

503 |
Sampling based approach to calculating marginal densities
- Gelfand, Smith
- 1990
(Show Context)
Citation Context ...riod of time, Markov chain Monte Carlo (MCMC) integration methods, especially the Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) and the Gibbs sampler (Geman and Geman, 1984; =-=Gelfand and Smith, 1990-=-) have emerged as extremely popular tools for the analysis of complex statistical models. This is especially true in the field of Bayesian analysis, which requires the evaluation of complex and often ... |

367 |
Nonuniversal critical dynamics in Monte Carlo simulations
- Swendsen, Wang
- 1987
(Show Context)
Citation Context ...dom effects models. More sophisticated MCMC algorithms can also offer impressive reductions in time to convergence. Promising ideas in this regard include the use of auxiliary variables (Swendsen and =-=Wang, 1987; Besag an-=-d Green, 1993), resampling and adaptive switching of the transition kernel (Gelfand and Sahu, 1994), and multichain annealing or "tempering" (Geyer and Thompson, 1995). In summary, a consens... |

365 | Explaining the Gibbs sampler - Casella, George - 1992 |

245 | CODA: Convergence Diagnosis and Output Analysis Software for Gibbs Sampling
- Best, Cowles, et al.
- 1995
(Show Context)
Citation Context ... subsequent iterations must be run to obtain the desired precision, as given by the NSE. Geweke's method attempts to address the issues of both bias and variance. It is available in the CODA package (=-=Best, Cowles, and Vines, 1995-=-). Like Gelman and Rubin's convergence diagnostic, Geweke's is essentially univariate, but if g(qq) were taken to be-2 times the log of the posterior density, then it also might be used to investigate... |

233 |
Practical Markov chain Monte Carlo
- Geyer
- 1992
(Show Context)
Citation Context ...heir accuracy. For those diagnostics not addressing this latter issue, an alternative is to use batching (see e.g. Ripley, 1987, Section 6.2), or perhaps more sophisticated time series methods (as in =-=Geyer, 1992-=-). Sections 3 and 4 apply our collection of diagnostics to some relatively simple statistical models. In so doing, we investigate whether use is indeed appropriate within the realm of common statistic... |

217 | Minorization conditions and convergence rates for Markov chain Monte
- Rosenthal
- 1995
(Show Context)
Citation Context ...sing these tools, the bounds obtained are quite loose, suggesting numbers of iterations that are several orders of magnitude beyond what would be considered reasonable or feasible in practice (though =-=Rosenthal, 1995-=-a, obtains tight bounds in a hierarchical normal means model related to James-Stein estimation). As a result, almost all of the applied work involving MCMC methods has relied on the second approach to... |

213 |
Annealing Markov chain Monte Carlo with applications to ancestral inference
- Geyer, Thompson
- 1995
(Show Context)
Citation Context ... of auxiliary variables (Swendsen and Wang, 1987; Besag and Green, 1993), resampling and adaptive switching of the transition kernel (Gelfand and Sahu, 1994), and multichain annealing or "temperi=-=ng" (Geyer and Thompson, 1995-=-). In summary, a consensus appears to be emerging that the proper approach to MCMC monitoring lies somewhere between the two extremes recently advocated by Geyer (one long chain, perhaps plotted versu... |

164 | A language and program for complex Bayesian modelling. The Statistician 43 - Gilks, Thomas, et al. - 1994 |

157 | Simulation run length control in the presence of an initial transient - Heidelberger, Welch - 1983 |

155 | How Many Iterations in the Gibbs Sampler - Raftery, Lewis - 1992 |

109 | Regeneration in Markov Chain Samplers - Mykland, Tierney, et al. - 1995 |

108 |
Facilitating the Gibbs sampler: The Gibbs stopper and the griddy-Gibbs sampler
- Ritter, Tanner
- 1992
(Show Context)
Citation Context ...c is near zero,sand the larger the number of replicate chains, the slowersthe entire computational process becomes.s2.5 Ritter and TannersLike Roberts's method, Ritter and Tanner's "Gibbs Stop-sper" (=-=Ritter and Tanner 1992-=-) is an effort to assess distri-sbutional convergence. The method may be applied eitherswith multiple parallel chains or by dividing the output of assingle long chain into batches. An importance weigh... |

107 | BUGS: Bayesian inference using Gibbs sampling, Version 0.6. Biostatistics Unit
- Spiegelhalter, Thomas, et al.
- 1997
(Show Context)
Citation Context ..., 1995) for the second stage of this process, with the first stage being accomplished by BUGS, the recently-developed software package for Bayesian analysis using Gibbs sampling (Thomas et al., 1992; =-=Spiegelhalter et al. 1994-=-,1995). Both of these programs and their manuals are freely available from the MRC Biostatistics Unit at the University of Cambridge (e-mail address: bugs@mrc_bsu.cam.ac.uk). Another emerging approach... |

95 | Covariance structure and convergence rate of the Gibbs sampler with various scans - Liu, Wong, et al. - 1995 |

75 |
Bugs: A program to perform Bayesian inference using Gibbs sampling
- Thomas, Spiegelhalter, et al.
- 1992
(Show Context)
Citation Context ...st, Cowles, and Vines, 1995) for the second stage of this process, with the first stage being accomplished by BUGS, the recently-developed software package for Bayesian analysis using Gibbs sampling (=-=Thomas et al., 1992-=-; Spiegelhalter et al. 1994,1995). Both of these programs and their manuals are freely available from the MRC Biostatistics Unit at the University of Cambridge (e-mail address: bugs@mrc_bsu.cam.ac.uk)... |

70 |
Spatial statistics and Bayesian computation (with discussion
- Besag, Green
- 1993
(Show Context)
Citation Context ...models. More sophisticated MCMC algorithms can also offer impressive reductions in time to convergence. Promising ideas in this regard include the use of auxiliary variables (Swendsen and Wang, 1987; =-=Besag and Green, 1993-=-), resampling and adaptive switching of the transition kernel (Gelfand and Sahu, 1994), and multichain annealing or "tempering" (Geyer and Thompson, 1995). In summary, a consensus appears to be emergi... |

53 |
On the convergence of successive substitution sampling
- Schervish, Carlin
- 1992
(Show Context)
Citation Context ...me specified subset E of the state space, so that ( ) r ( ) ( ) i i E Z �� is the posterior probability of E at iteration i. Assuming that the transition operator is self-adjoint and Hilbert-Schmi=-=dt (Schervish and Carlin, 1992), we ma-=-y write ( ) r r l l ( ) , i i i a O = + + 2 2 3 where r r = ���� lim , ( ) i i a 2 is some real number, and |l 3 |s|l 2 |sl 1 = 1 are the three largest eigenvalues of the kernel density. Note ... |

45 | Discussion of Markov chains for exploring posterior distributions - Chan, Geyer - 1994 |

40 | Rates of convergence for Gibbs sampling for variance component models
- Rosenthal
- 1995
(Show Context)
Citation Context ...sing these tools, the bounds obtained are quite loose, suggesting numbers of iterations that are several orders of magnitude beyond what would be considered reasonable or feasible in practice (though =-=Rosenthal, 1995-=-a, obtains tight bounds in a hierarchical normal means model related to James-Stein estimation). As a result, almost all of the applied work involving MCMC methods has relied on the second approach to... |

39 | Hierarchical Bayes models for the progression of HIV infection using longitudinal CD4 T-cell numbers (with discussion
- Lange, Carlin, et al.
- 1994
(Show Context)
Citation Context ...ed summary is given by Tierney (1995). The statistical applications of MCMC in just the last five years are far too numerous to list, covering such disparate areas as the modeling of HIV progression (=-=Lange et al., 1992-=-), archaeological shape estimation (Buck, Litton, and Stephens, 1993), determination of fuel economy potential in automobiles (Andrews et al., 1993), and the analysis of home run hitters in major leag... |

39 | Analysis of the Gibbs sampler for a model related to James-Stein estimators
- Rosenthal
- 1996
(Show Context)
Citation Context ...sing these tools, the bounds obtained are quite loose, suggesting numbers of iterations that are several orders of magnitude beyond what would be considered reasonable or feasible in practice (though =-=Rosenthal, 1995-=-a, obtains tight bounds in a hierarchical normal means model related to James-Stein estimation). As a result, almost all of the applied work involving MCMC methods has relied on the second approach to... |

38 | Detecting initialization bias in simulation output - Schruben - 1982 |

37 | Computation on Bayesian graphical models - DJ, Thomas, et al. - 1996 |

36 | Efficient parameterizations for normal linear mixed models - Gelfand, Sahu, et al. - 1995 |

32 | Convergence of Markov chain Monte Carlo algorithm - Polson - 1996 |

30 | On Markov chain Monte Carlo acceleration
- Gelfand, Sahu
- 1994
(Show Context)
Citation Context ...ime to convergence.sPromising ideas in this regard include the use of auxil-siary variables (Besag and Green 1993; Swendsen and Wangs1987), resampling and adaptive switching of the transitionskernel (=-=Gelfand and Sahu 1994-=-), and multichain annealingsor "tempering" (Geyer and Thompson 1995).sIn summary, a consensus appears to be emerging that thesproper approach to MCMC monitoring lies somewhere be-stween the two extrem... |

29 | Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths - Johnson - 1996 |

27 | Convergence diagnostics of the Gibbs sampler - Roberts - 1992 |

24 | Subsampling the Gibbs sampler - MacEachern, Berliner - 1994 |

24 | Rates of convergence for data augmentation on finite sample spaces - Rosenthal - 1993 |

24 | Optimal Tests for Initialization Bias in Simulation Output - Schruben, Singh, et al. - 1983 |

22 |
Maximum likelihood estimation for constrained or missing data models
- GELFAND, CARLIN
- 1993
(Show Context)
Citation Context ...cussions andssuggestions that greatly improved the manuscript.sin missing- and dependent-data settings where the likeli-shood itself involves complicated high-dimensional integralss(see, for example, =-=Gelfand and Carlin 1993-=-, and Geyer andsThompson 1992). Excellent tutorials on the methodologyshave recently been provided by Albert (1993) and Casellasand George (1992); a more complete and advanced sum-smary was given by T... |

22 | Ecient parametrizations for normal linear mixed models - Gelfand, K, et al. - 1995 |

22 | Looking at Markov samplers through cusum path plots: A simple diagnostic idea’, Statistics and Computing - Yu, Mykland - 1998 |

15 | Gibbs sampler convergence criteria - Zellner, Chung-ki |

14 | Teaching Bayesian statistics using sampling methods and MINITAB - Albert - 1993 |

13 | Efficient parametrizations for generalized linear mixed models - Gelfand, E, et al. - 1996 |

12 |
1992.“A Bayesian Analysis of a Poisson Random Effects Model for Home Run Hitters.” The American Statistician
- Albert
(Show Context)
Citation Context ...ical shape estimation (Buck, Litton, and Stephens, 1993), determination of fuel economy potential in automobiles (Andrews et al., 1993), and the analysis of home run hitters in major league baseball (=-=Albert, 1992-=-). While MCMC algorithms allow an enormous expansion of the class of candidate models for a given dataset, they also suffer from a well-known and potentially serious drawback: it is often difficult to... |

12 | Posterior computations for censored regression data - Wei, Tanner - 1990 |

10 | A Variational Control Variable for Assessing the Convergence of the Gibbs Sampler - Liu, Liu, et al. - 1992 |

9 |
Spatial Statistics and Bayesian Computation" (with discussion
- Besag, Green
- 1993
(Show Context)
Citation Context ...models. More sophisticated MCMC algorithms can also offer impressive reductions in time to convergence. Promising ideas in this regard include the use of auxiliary variables (Swendsen and Wang, 1987; =-=Besag and Green, 1993), resampl-=-ing and adaptive switching of the transition kernel (Gelfand and Sahu, 1994), and multichain annealing or "tempering" (Geyer and Thompson, 1995). In summary, a consensus appears to be emergi... |

9 | Comment: Monitoring Convergence of the Gibbs Sampler: Further Experience With the Gibbs Stopper - Cui, Tanner, et al. - 1992 |

9 |
Parameterization issues in Bayesian inference (with discussion
- Hills, Smith
- 1992
(Show Context)
Citation Context ...n, rather than on thesless soluble problem of convergence diagnosis. Clever repa-srameterization can often substantially improve correlationsstructure within a model, and hence speed convergence (sees=-=Hills and Smith 1992-=- for a general discussion and Gelfandset al. 1995a,b for treatments pecific to hierarchical randomseffects models). More sophisticated MCMC algorithms cansalso offer impressive reductions in time to c... |

7 | Convergence Diagnostics for Markov Chain Samplers - Garren, Smith - 1994 |

7 | Parametrization Issues in Bayesian Inference" (with discussion), in Bayesian Statistics 4 - Hills, Smith - 1992 |

5 |
Bayesian Estimation of Fuel Economy Potential due to Technology Improvements" (with discussion
- Andrews, Berger, et al.
- 1993
(Show Context)
Citation Context ...ch disparate areas as the modeling of HIV progression (Lange et al., 1992), archaeological shape estimation (Buck, Litton, and Stephens, 1993), determination of fuel economy potential in automobiles (=-=Andrews et al., 1993-=-), and the analysis of home run hitters in major league baseball (Albert, 1992). While MCMC algorithms allow an enormous expansion of the class of candidate models for a given dataset, they also suffe... |

5 | Discussion of "Markov chains for exploring posterior distributions," by L - Chan, Geyer - 1994 |

5 | Assessing distributional convergence of the Gibbs sampler - Roberts, Hills - 1992 |

4 | Methods for estimating L convergence of Markov chain Monte Carlo - Roberts - 1994 |

4 | Methods for Estimating L 2 Convergence of Markov Chain Monte Carlo," to appear - Roberts - 1994 |

3 |
Detecting a Change in the Shape of a Prehistoric Corbelled Tomb
- Buck, Litton, et al.
- 1993
(Show Context)
Citation Context ... applications of MCMC in just the last five years are far too numerous to list, covering such disparate areas as the modeling of HIV progression (Lange et al., 1992), archaeological shape estimation (=-=Buck, Litton, and Stephens, 1993-=-), determination of fuel economy potential in automobiles (Andrews et al., 1993), and the analysis of home run hitters in major league baseball (Albert, 1992). While MCMC algorithms allow an enormous ... |

3 | Parametrization Issues in Bayesian Inference" (with discussion), in Bayesian Statistics 4 - Hills, Smith - 1992 |

3 |
Monitoring the Convergence of Markov Samplers Based on Estimated L1 Error
- Yu
- 1994
(Show Context)
Citation Context ...hain at intervals, using a proposal density that generates candidate values from an overdispersed distribution independently of the current state of the Gibbs sampler chain (see Mykland, Tierney, and =-=Yu, 1994-=-). When such candidates are accepted and produce regenerations in the chain, diagnostics requiring multiple independent chains may be applied to the tours. At the same time, such a hybrid sampler form... |

3 | Convergence Diagnostics of the Gibbs Sampler," in Bayesian Statistics 4 - Roberts - 1992 |

2 |
On Markov Chain Monte Carlo Acceleration," to appear
- Gelfand, Sahu
- 1994
(Show Context)
Citation Context ...ime to convergence. Promising ideas in this regard include the use of auxiliary variables (Swendsen and Wang, 1987; Besag and Green, 1993), resampling and adaptive switching of the transition kernel (=-=Gelfand and Sahu, 1994), and mul-=-tichain annealing or "tempering" (Geyer and Thompson, 1995). In summary, a consensus appears to be emerging that the proper approach to MCMC monitoring lies somewhere between the two extreme... |

2 | Rejoinder: Replication without Contrition - Gelman, Rubin - 1992 |

2 |
Constrained Monte Carlo Likelihood for Dependent Data" (with discussion
- Geyer, Thompson
- 1992
(Show Context)
Citation Context ...ely used in Bayesian analysis, they have also been employed by frequentists in missing and dependent data settings where the likelihood itself involves complicated highdimensional integrals (see e.g. =-=Geyer and Thompson, 1992-=-; Gelfand and Carlin, 1993). Excellent tutorials on the methodology have been provided recently by Casella and George (1992) and Albert (1993); a more complete and advanced summary is given by Tierney... |

2 | A Language and Program for Co plex Bayesian Modelling - Gilks, Thomas, et al. - 1994 |

2 |
Equations of State Calcualtions by Fast Computing Machines
- Metropolis, Rosenbluth, et al.
- 1953
(Show Context)
Citation Context ...stions which greatly improved the manuscript. 1 Introduction In a surprisingly short period of time, Markov chain Monte Carlo (MCMC) integration methods, especially the Metropolis-Hastings algorithm (=-=Metropolis et al., 1953-=-; Hastings, 1970) and the Gibbs sampler (Geman and Geman, 1984; Gelfand and Smith, 1990) have emerged as extremely popular tools for the analysis of complex statistical models. This is especially true... |

2 | Computation on Bayesian Graphical Models," to appear with discussion - Spiegelhalter, Thomas, et al. - 1995 |

1 |
On Markov Chain Monte Carlo Acceleration," to appear
- Gelfand, Sahu
- 1994
(Show Context)
Citation Context ...ime to convergence. Promising ideas in this regard include the use of auxiliary variables (Swendsen and Wang, 1987; Besag and Green, 1993), resampling and adaptive switching of the transition kernel (=-=Gelfand and Sahu, 1994-=-), and multichain annealing or "tempering" (Geyer and Thompson, 1995). In summary, a consensus appears to be emerging that the proper approach to MCMC monitoring lies somewhere between the two extreme... |

1 | Computation on Bayesian Graphical Models," to appear with discussion - Thomas, A, et al. - 1995 |

1 | This content downloaded from 162.105.192.183 on Tue, 22 Oct 2013 22:50:59 PM All use subject to JSTOR Terms and Conditions 904 - Geyer, Thompson - 1996 |