Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science 7 473--483.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... since Gibbs sampling sometimes converges very slowly; see for example [M] Now, it may be possible to use convergence diagnostics to check if the distribution after (say) 1000 steps is indeed close to the distribution to which the chain appears to converge; see [G] Rob] On the other hand, see [GR] for warnings about possible problems. In any case, it would be comforting to have theoretical results regarding how many iterations are required before the chain has in fact converged. There has been limited analysis of this question to date (though it can be expected that there will be more in ....

A. Gelman and D.B. Rubin (1993), Inference from iterative simulation using multiple sequences, Statistical Science, to appear.

....constant 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 ....

....a form of diagnostic. Speci cally, if the estimates E j from the di erent computers are all very di erent, then this may suggest that the Markov chain has not yet converged, or that there is a problem with the algorithm. This is somewhat related to the multiple runs computer diagnostics of e.g. [18]. However, such diagnostic method should be used with care, to avoid introducing additional biases (cf. 6] In addition, it may be possible to do more sophisticated analysis (e.g. of autocorrelations) on the multiple parallel runs, though this may require greater communication between the ....

A. Gelman and D.B. Rubin (1992), Inference from iterative simulation using multiple sequences. Stat. Sci., Vol. 7, No. 4, 457-472.

....The length of the MD trajectory was chosen randomly between 40 to 80 integration steps. 5 independent Markov chains were started in every subset of metastable conformations. Convergence of HMC was controlled by the Gellmann and Rubin criterion, which evaluates the mixing of the chains [11]. All simulations have generated between 20000 and 200000 molecular states to reach convergence. The temperatures of the hierarchical embedding protocol were 1500K, 1000K, 600K and 300K. All simulations were performed in vacuum. The discretization per dihedral was performed under a resolution of ....

A. Gelman and D.B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7:457--511, 1992.

....established, time series methods based on the autocorrelation properties of the chain [99] or on batch means [35] can be used to determine the uncertainty introduced into any estimates caused by the use of a finite number of samples. A number of other practical considerations are important. In [31] the case was made for using multiple chains, initialised from an overdispersed distribution, to ensure that the distribution was adequately explored. This has the disadvantage that the burn in samples must be ignored for each chain. In [50] the use of coupled chains, running at different ....

A. Gelman and D.B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7(4):457--511, 1992.

....and explained in detail in Fig. 7. In order to control the statistical error of the required samplings of the bridge densities, we have to control the simulation length of each sampling. Since appropriate simulation lengths may vary drastically, we use the convergence estimator described in [13,15] to automatically stop the simulation. For this estimator, multiple realizations of a Markov chain X k are generated to compute estimates depending on the variances between these realizations. In view of the various samplings of bridge densities the hierarchical approach has another bene t: ....

A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci., 7(4):457-511, 1992.

....b] the interval chosen to contain most of the distribution for X. The mean squared biases and mean squared errors were computed over this grid. It is impossible to assess convergence of the MCMC chain for all simulated data sets. instead, for a few selected data sets, we used tests of convergence (Gelman and Rubin, 1992) separately for each parameter and also for the estimated function on a few selected grid points. The rst ve cases considered were the following: Case 1: The regression function, m is given in (7) with n = 100, a = 2:0, b = 2:0 2 = 0:3 2 , 2 u = 0:8 2 , x = 0 and 2 x = 1. ....

Gelman, A. and Rubin, D. B. (1992),\Inference from Iterative Simulation Using Multiple Sequences ", Statistical Science, 7, 457-472..

....When can samples be trusted Typically, the first k samples must be discarded to allow the sampler to burn in . The rest represent the posterior; but what is k . The usual approach is to start different sequences at different points, and then confirm that they give comparable answers (e.g. Gelman and Rubin, 1993; Geweke, 1992; Roberts, 1992) Another approach is to prove that the proposal process has rapid mixing properties (which is extremely difficult, e.g. Jerrum and Sinclair, 1996) Rapid mixing is desirable, because the faster the sampler mixes, the lower the variance of expectations estimated using ....

Gelman, A. and Rubin, D.B. 1993. Inference from iterative simulation using multiple sequences. Statistical Science,7: 457--511.

....Poisson processes with the following intensity functions: the Musa Okumoto with 1 (t) fi 1 = t ff 1 ) the Weibull with 2 (t) ff 2 fi 2 t ff 2 Gamma1 , and the superposition model with (t) 1 (t) 2 (t) respectively. We monitor the convergence of the Gibbs sampler using the Gelman and Rubin (1992) method that uses the analysis of variance technique to determine whether further iterations are needed. We found 20,000 iterations to be enough for the priors being considered. All the following numerical results are obtained with 20,000 iterations and 50 replications in the Gibbs sampler. Table ....

Gelman, A.E., and Rubin D. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci., 7, 457-472.

....and explained in detail in Fig. 6. In order to control the statistical error of the required samplings of the bridge densities, we have to control the simulation length of each sampling. Since appropriate simulation lengths may vary drastically, we use the convergence estimator described in [13,15] to automatically stop the simulation. For this estimator, multiple realizations of a Markov chain X k are generated to compute estimates depending on the variances between these realizations. In view of the various samplings of bridge densities the hierarchical approach has another bene t: ....

A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci., 7(4):457-511, 1992.

....configuration X reasonable is chosen. Calculating these functions for each s and plotting them as a function of s, the number of iterations needed may be found through visual inspection. More sophisticated methods for estimation of convergence may however also be performed (see Gelman and Rubin [9], Geyer [13] Smith and Roberts [34] and the references therein) Norwegian Computing Center, P.B. 114 Blindern, N 0314 Oslo, Norway Tel. 47) 22 85 25 00 Fax: 47) 22 69 76 60 Chapter 3 Simulation from marked point processes In Chapter 2, we discussed the use of Markov chains for ....

A. Gelman and D. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7(4):457--511, 1993. With discussion.

....posterior simulation, we simulated 10 chains with independent starting values and a burn in period of 10,000 draws per chain, and we stored the subsequent 100,000 observations. This number was su#cient to achieve convergence in the sense that # R was smaller than 1. 1 for all model parameters (Gelman and Rubin, 1992). For computing # R, the parameter labels need to be identified. This permutation problem can be handled by computing # R from the Gibbs output after having reordered the draws such that they all come from the same model region. The reordering can be done by applying a Q means type of 10 ....

Gelman, A., and Rubin, D.B. (1992). Inferences from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457--511.

.... (t) 1) distribution, if necessary with the constraint that 1=2 (which can be achieved by resampling from the beta distribution until a value less than 1 2 has been drawn) Repeat these steps independently for the several parallel simulations until the sequences appear mixed (see, e.g. Gelman and Rubin, 1992). 4. Select L (for example, 1000) random draws of (z; from the simulated sequences to represent posterior simulation draws. 5. From each posterior simulation draw (z l ; l ) simulate a replicated data set y rep l : first, compute y(z l ) second, simulate the components of y rep l ....

....two error probability model. For each of these models, we find that 2000 simulations were sufficient for approximate convergence, in the sense that the potential scale reduction factor is less than 1. 2 for all of the components of (u; v) all of the variance components, and the log likelihood (see Gelman and Rubin, 1992). We discard the first half of the simulations and save every fifth iteration (to save computer storage) For each of the two models fit to each of the two data sets, we then have 1000 draws of the vector z l = u; v) l from the posterior distribution; we use these for all further ....

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Gelman, A., and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457--511.

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Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science 7 473--483.

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Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7:457--511.

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Andrew Gelman and Donald B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7:457--472, 1992.

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Andrew Gelman and Donald B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7:457--472, 1992.

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A. Gelman and D. B. Rubin. Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7, 457:472, 1992.

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Gelman, A. and Rubin, D. B. (1992) Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457-72. 15

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A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences (with discussion). Statist. Sci., 7(4):457-511, 1992.

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A. Gelman and D.B. Rubin (1993), Inference from iterative simulation using multiple sequences, Statistical Science, to appear.

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Gelman, A., and Rubin, D. B. (1992). Inferences from Iterative Simulation using Multiple Sequences. Statistical Science, 7, 457--472.

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Gelman, A. and Rubin, D.B. `Inference from iterative simulation using multiple sequences #with discussion#', Statistical Science, 7, 457-511 #1992#.

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A. Gelman and D. B. Rubin, "Inference from iterative simulation using multiple sequences (with discussion)," Statist. Sci. 7, pp. 457--511, 1992.

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Gelman, A. and Rubin, D. (1992), "Inference from iterative simulation using multiple sequences", Statist. Sci., 7, 457-476.

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Gelman, A. and Rubin, D.B. (1992). "Inference from iterative simulation using multiple sequences (with discussion)". Statistical Science, 7, 457-511.

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