A. Gelman, "Inference and monitoring convergence," in Markov Chain Monte Carlo in Practice, pp. 132--143. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Eds.. London, U.K.: Chapman & Hall, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....were done with the FBM. To make convergence diagnostics and estimation of credible intervals (CI) easier, ten independent RJMCMC chains (with different starting points) were run for each case. For convergence diagnostics, we used visual inspection of trends, the potential scale reduction method (Gelman, 1996) and the Kolmogorov Smirnov test (Robert and Casella, 1999) Between model convergence in the RJMCMC was diagnosed with the methods discussed in (Brooks et al. 2001) As the number of visits to each model was typically very low, we mainly analysed the visits to each subpopulation having equal ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks, W. R., Richardson, S., and Spiegelhalter, D. J., editors, Markov Chain Monte Carlo in Practice, pages 131--144. Chapman & Hall.

....the HMC, and per case variances are sampled using the Gibbs sampling. The MCMC sampling was done with the FBM 1 software and Matlab code partly derived from the FBM and Netlab 2 toolbox. For convergence diagnostics, we used a visual inspection of trends, the potential scale reduction method (Gelman, 1996) and the Kolmogorov Smirnov test (Robert and Casella, 1999) 1 http: www.cs.toronto.edu radford fbm.software.html 2 http: www.ncrg.aston.ac.uk netlab Bayesian Model Assessment and Comparison Using Cross Validation Predictive Densities 13 3.2 Toy problem: MacKay s robot arm In this ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks, W. R., Richardson, S., and Spiegelhalter, D. J., editors, Markov Chain Monte Carlo in Practice, pages 131--144. Chapman & Hall.

....of the professors as advisor. We specified the success (or otherwise) of 4n students and queried the success of the last. Figure 6 shows the inference cost as a function of the total network size. Because we cannot compute the exact values for all n, we use a standard convergence diagnostic due to [Gelman, 1996] , checking against the exact values for small n. The resulting graph appears to be linear in the size of the network. It would appear that, for this type of network, the sampling algorithm scales well. Note also that cost is measured in terms of single variable state transitions performed by the ....

Andrew Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 131--143. Chapman and Hall, London, 1996.

....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. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 131-143. Chapman & Hall, 1996.

....of the algorithms are not repeated here, see (Neal, 1996, 1997, 1999) or the reference list in the web page of the FBM software 1 . See Appendix for the MCMC parameter values used in examples. For convergence diagnostics, we used visual inspection of trends, the potential scale reduction method (Gelman, 1996) and the Kolmogorv Smirnov test (Robert and Casella, 1999) In all cases, chains were run probably much longer than necessary to be on safe side. 3.2 Toy problem: MacKay s robot arm To illustrate some basic issues of the expected utilities computed from the cross validation predictive densities, ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks, W. R., Richardson, S., and Spiegelhalter, D. J., editors, Markov Chain Monte Carlo in Practice, pages 131--144. Chapman & Hall.

....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. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 131-143. Chapman & Hall, 1996. Alexander Fischer, Christof Schutte, Peter Deu hard, and Frank Cordes

....accepted (A) or rejected (R) 8. Results for Efficient Sampling In this section we show experimental results supporting the intuition that smart chain flipping leads to rapidly mixing chains. To assess the convergence of the sampler under different conditions, we use the approach discussed in (Gelman, 1996): we graph the time series for a single summary statistic in multiple, concurrently run MCMC simulations. Convergence can be assumed if all time series converge to the same value for the statistic. Displays such as this also give a qualitative understanding of the behavior of the different ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks et al. (Gilks et al., 1996), pp. 131--140.

....accepted (A) or rejected (R) 8. Results for Efficient Sampling In this section we show experimental results supporting the intuition that smart chain flipping leads to rapidly mixing chains. To assess the convergence of the sampler under different conditions, we use the approach discussed in (Gelman, 1996): we graph the time series for a single summary statistic in multiple, concurrently run MCMC simulations. Convergence can be assumed if all time series converge to the same value for the statistic. Displays such as this also give a qualitative understanding of the behavior of the different ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks et al. (Gilks et al., 1996), pp. 131--140.

....computed, which have to be stopped automatically after a reasonable number N of iterations. Because N can vary drastically from one Markov chain to another depending on the almost invariant set A k and the corresponding propagator, we use the non convergence estimator from Gelman and Rubin [13, 11]. For this estimator, multiple realizations of a Markov chain X k are generated to compute estimates depending on the variances between these realizations. There exist other estimators using only one Markov chain (see the discussion in [13] but this one is especially suited for the UCMC ....

A. Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 131-143. Chapman & Hall, 1996. 21

....number of colors, and for checking convergence of the Gibbs sampler, as indicated below. 5.2 Convergence detection The iteration steps of the Gibbs sampler are simple. Detecting convergence, however, is not straightforward. This point is discussed, among others, by Gilks and Roberts (1996) and Gelman (1996). The reason is that the process converges not to a single value but to a stationary probability distribution. We propose, for our model, a measure for improving convergence and a measure for checking convergence. The measure for improving convergence is to start the simulations with an ....

....2, changing into epochs where colors 1 and 2 have changed places. Such color changes indicate good mixing as defined by Gilks and Roberts (1996) and therefore can be taken as signs of convergence of the Gibbs sampler. When one wishes to make comparisons between parallel simulation runs (cf. Gelman, 1996), the parallel runs can also be compared on the basis of the colorings only. Drawing the new values of (#, #) in the Gibbs sampler depends only on the coloring and not on the previous value of (#,#) so that the stochastic process X (p) for p = 1, is a Markov chain and it makes sense to ....

[Article contains additional citation context not shown here]

GELMAN, A. (1996), "Inference and monitoring convergence", in Gilks et al. (1996), pp. 131 -- 144.

....To simplify the notation let fl 0 = Var [f(X i ) and fl k = Cov [f(X i ) f(X i k ) 5) which is the lag k autocovariance of the stationary time series ff(X n )g 1 n=1 . If the CLT holds, we might expect the limiting variance to be the limit of (4) as n 1. If this is the case we have ([11], Chapter 3) v(f; P ) fl 0 2 1 X k=1 fl k : 6) The asymptotic variance in the CLT given in (6) defines our criterion for ranking transition kernels. A transition kernel is reversible with respect to if, for all bounded functions f and g, ZZ f(y)g(x) dx)P (x; dy) ZZ ....

....P 0, and Theorem 3.2 finishes the proof. 3. 3 A Counterexample The rate of convergence in total variation distance of P n (and of weak convergence of X n ) to (x) is governed by the spectral radius, max;P , which, in finite state spaces is the second largest eigenvalue in absolute value [4, 11]. We thus have conflicting requirements: fast total variation convergence to equilibrium is obtained by having all eigenvalues small in absolute value while good properties in terms of asymptotic variance of ergodic averages are obtained by having small positive and large negative eigenvalues, ....

A. Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1996.

....modelling [25] Using this kind of models a suitable partition (blocking) of the vector of model parameters is naturally obtained and it is also easy to derive the full conditional distribution. The convergence rate and the strategies for choosing the proposal distribution are described in [26] [27]. B. MCMC in Function Reconstruction In this sub section we will describe how the problems defined in Section II can be tackled using MCMC methods. In order to explain the probabilistic models used in the different sampling schemes, we will resort to a Bayesian Network (BN) representation. BNs are ....

A. Gelman, "Inference and monitoring convergence," in Markov Chain Monte Carlo in Practice, W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Eds., chapter 8, pp. 131--144. Chapman & Hall, Great Britain, UK, 1996.

....to carefully 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 ....

Gelman, A. 1996. "Inference and monitoring convergence." In Markov Chain Monte Carlo in Practice, edited by W. R. Gilks, S. Richardson and D. J. Spiegelhalter, 131--143.

....use. However, geometric convergence rates can be found in the important case where the desired distribution is Gaussian or approximately Gaussian (see (Roberts, 1996) Convergence of the chain is practically assessed by one or more (mostly ad hoc) convergence diagnostics (see, for example, (Gelman, 1996)) based on the samples generated by the chain. Since the state X is typically of high dimension, and hence hard to examine 5 The success of this initialisation scheme will, of course, depend on how informative the prior distributions are. For very vague priors, the method is equivalent to ....

Gelman, A. (1996). Inference and monitoring convergence. In Gilks, W. R., Richardson, S., and Spiegelhalter, D. J., editors, Markov Chain Monte Carlo in Practice, pages 75--88. Chapman and Hall.

....let fl 0 = Var [f(X i ) and fl k = Cov [f(X i ) f(X i k ) 2.5) which is the lag k autocovariance of the stationary time series ff(X n )g 1 n=1 . If the central limit theorem holds, we might expect the limiting variance to be the limit of (2.4) as n 1. If this is the case we have ([22], Chapter 3) v(f; P ) fl 0 2 1 X k=1 fl k : 2.6) The asymptotic variance in the central limit theorem given in (2.6) defines our criterion for ranking transition kernels. CHAPTER 2. ORDERING MONTE CARLO MARKOV CHAINS 8 A transition kernel is reversible with respect to if, for all ....

....and Theorem 2.3.2 finishes the proof. Q.E.D. 2.3. 3 A Counterexample The rate of convergence in total variation distance of P n (and of weak convergence of X n ) to (x) is governed by the spectral radius, max;P , which, in finite state spaces is the second largest eigenvalue in absolute value [6, 22]. We thus have conflicting requirements: fast total variation convergence to equilibrium is obtained by having all eigenvalues small in absolute value while good properties in terms of asymptotic variance of ergodic averages are obtained by having small positive and large negative eigenvalues, as ....

A. Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1996.

....To simplify the notation let fl 0 = Var [f(X i ) and fl k = Cov [f(X i ) f(X i k ) 5) which is the lag k autocovariance of the stationary time series ff(X n )g 1 n=1 . If the CLT holds, we might expect the limiting variance to be the limit of (4) as n 1. If this is the case we have ([11], Chapter 3) v(f; P ) fl 0 2 1 X k=1 fl k : 6) The asymptotic variance in the CLT given in (6) defines our criterion for ranking transition kernels. A transition kernel is reversible with respect to if, for all bounded functions f and g, ZZ f(y)g(x) dx)P (x; dy) ZZ ....

....P 0, and Theorem 3.2 finishes the proof. 3. 3 A Counterexample The rate of convergence in total variation distance of P n (and of weak convergence of X n ) to (x) is governed by the spectral radius, max;P , which, in finite state spaces is the second largest eigenvalue in absolute value [4, 11]. We thus have conflicting requirements: fast total variation convergence to equilibrium is obtained by having all eigenvalues small in absolute value while good properties in terms of asymptotic variance of ergodic averages are obtained by having small positive and large negative eigenvalues, as ....

A. Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1996.

....posterior density. We drew ten samples from the approximate density using importance resampling and then used those as starting points for ten parallel runs of the Gibbs sampler, which adequately converged (in the sense of potential scale reductions b R less than 1. 1 for all model parameters; see Gelman, 1994) after 200 steps, after discarding the first half of each simulated sequence. We were left with a set of 1000 simulation draws of the vector of model parameters. Details of the Gibbs sampler implementation and the convergence monitoring appear in Gelman et al. 1994) and Gelman and Rubin (1992) ....

....than 1.1 for all model parameters; see Gelman, 1994) after 200 steps, after discarding the first half of each simulated sequence. We were left with a set of 1000 simulation draws of the vector of model parameters. Details of the Gibbs sampler implementation and the convergence monitoring appear in Gelman et al. 1994) and Gelman and Rubin (1992) respectively. Model checking using posterior predictive simulations The model was chosen to accurately fit the unequal means and variances in the two groups of subjects in the study, but there was still some question about the fit to individuals. In particular, the ....

Gelman, A. (1994). Inference and monitoring convergence. In this volume. Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1994). Bayesian Data Analysis. Chapman and Hall, to appear.

....Chain Monte Carlo in Practice: A Roundtable Discussion Moderator: Robert E. KASS Panelists: Bradley P. CARLIN, Andrew GELMAN, and Radford M. NEAL Markov chain Monte Carlo (MCMC) methods make possible the use of flexible Bayesian models that would otherwise be computationally infeasible. In recent years, a great variety of such applications have been described in the literature. Applied statisticians who are new to these ....

....allows me to detect convergence failure. Gelman: Let me echo Brad here. I ve encountered slow convergence in distributions that are essentially unimodal but are full of ridges in high dimensions. I ve also found obvious programming and modeling bugs using multiple starting points (see, e.g. Gelman 1996, p. 134) Neal: The problems I usually work with (e.g. neural network models, mixture models) may be a bit di#erent from those that Andrew and Brad work with. The maximum likelihood estimates for these models are often ridiculous. For example, maximum likelihood for a mixture of normals will ....

Gelman, A. (1996), "Inference and Monitoring Convergence," in Markov Chain Monte Carlo in Practice, eds. W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, London: Chapman and Hall, pp. 131--143.

.... Geyer, 1992) In our experience with Bayesian posterior simulation, however, we have found that the added information obtained from replication in terms of confidence in simulation results and protection from falselyprecise inferences (see, for example, the figures in Gelman and Rubin, 1992a, and Gelman, 1996) outweighs any additional costs in computer time required for multiple rather than single simulations. It is desirable to choose starting points that are widely dispersed in the target distribution. Overdispersed starting points are an important design feature for two major reasons. First, ....

....points by comparing several sequences drawn from different starting points and checking that they are indistinguishable. The potential scale reduction factor. For each scalar summary of interest (that is, all parameters and predictions of interest in the model) Gelman and Rubin (1992b) and Gelman (1996) recommend the following strategy: first discarding the first half of the simulated sequences to reduce the influence of the starting points; and then computing the potential scale reduction factor, labeled p b R, which is essentially the square root of the variances of the values of the scalar ....

Gelman, A. (1996). Inference and monitoring convergence. In Practical Markov Chain Monte Carlo, ed. W. Gilks, S. Richardson, and D. Spiegelhalter, 131--143. New York: Chapman & Hall.

....Chain Monte Carlo in Practice: A Roundtable Discussion Moderator: Robert E. Kass 1 Panelists: Bradley P. Carlin, Andrew Gelman, and Radford M. Neal August 4, 1997 Abstract. Markov chain Monte Carlo (MCMC) methods make possible the use of flexible Bayesian models that would otherwise be computationally infeasible. In recent years, a great variety of such applications have been described in the literature. Applied ....

....me to detect convergence failure. Gelman: Let me echo Brad here. I ve encountered slow convergence in distributions that are essentially unimodal but are full of ridges in high dimensions. I ve also found obvious programming and modeling bugs using multiple starting points (see, e.g. p. 134 of Gelman, 1996). Neal: The problems I usually work with (e.g. neural network models, mixture models) may be a bit different from those that Andrew and Brad work with. The maximum likelihood estimates for these models are often ridiculous. For example, maximum likelihood for a mixture of normals will place ....

Gelman, A. (1996), "Inference and Monitoring Convergence," in Markov Chain Monte Carlo in Practice, eds. W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, London: Chapman and Hall, pp. 131--143.

No context found.

A. Gelman, "Inference and monitoring convergence," in Markov Chain Monte Carlo in Practice, pp. 132--143. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Eds.. London, U.K.: Chapman & Hall, 1996.

No context found.

A. Gelman. Inference and monitoring convergence. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors, Markov Chain Monte Carlo in Practice, pages 131-143. Chapman & Hall, 1996. 21

No context found.

Gelman, A., 1996: Inference and monitoring convergence. In Markov Chain Monte Carlo in Practice. W.R. Gilks, S. Richardson, and D.J. Spiegelhalter (Eds.), Chapman and Hall, London, pp. 131-140.

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