M.K. Cowles and J.S. Rosenthal (1996), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Stat. and Comput. 8 (1998), 115-124.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....proves their existence. This type of convergence ensures that a central limit theorem for ergodic averages is valid [27] which is the basis of some convergence diagnostics procedures [23] This result is also the starting point of numerical studies that lead to estimates of these quantities; see [8]. V. APPLICATION TO NEUTRON DETECTION In this section, we present the results obtained from our algorithm applied to synthetic data and real data provided by the Commissariat l nergie Atomique (CEA, French nuclear civil research center) In order to assess the performance of the algorithm, we ....

M. K. Cowles and J. S. Rosenthal, "A simulation approach to convergence rates for Markov chain Monte Carlo algorithms," Stat. Comput., vol. 8, pp. 115--124, 1998.

....that kP n ( 0 ; 0 ) k 0:01 (31) where ( 0 ; 0 ) is the starting value. In each case, we used 0 = y 1 ; y K ; y) T . As is clear from (30) a starting value for is not required. This convergence criterion ( 0:01) has become fairly standard (Rosenthal 1996, Cowles and Rosenthal 1998, Roberts and Rosenthal 1999) Table 7 contains the results. For example, under the rst hyperparameter setting, after 3 10 8 iterations of the block Gibbs sampler, the total variation distance to stationarity is at most 0.00429. It takes about 2 minutes to run 1 million iterations of our block ....

Cowles, M. K. and Rosenthal, J. S. (1998). A simulation approach to convergence rates for Markov chain Monte Carlo algorithms, Statistics and Computing 8: 115-124.

....) k 0:01 (29) where ( 0 ; 0 ) is the starting value. In each case, we used 0 = y 1 ; y K ; y) T where y = M 1 T P K i=1 P m i j=1 y ij . A starting value for is not required. This convergence criterion ( 0:01) has become fairly standard (Rosenthal 1996, Cowles and Rosenthal 1998, Roberts and Rosenthal 1999a) A brief description of each data set and the corresponding results follows. Table 1: Hyperparameter Con gurations Con guration a 1 a 2 b 1 b 2 0 0 1 1 1 1 1 1 0 2 1 1 1 1 1e 12 0 3 2 1 1 1e 6 1e 12 0 4 0.5 1e 6 1 1e 6 1e 12 0 5 1e 6 1e 6 1e 6 1e 6 1e 12 0 ....

....total variation distance corresponding to one million iterations is still essentially 1. Of course, it might be possible to improve the situation by, for example, using a di erent energy function or trying to nd sharper inequalities than those used in Jones and Hobert (2000) On the other hand, Cowles and Rosenthal (1998) used simulation to estimate Rosenthal s bounds for the dye stu data (with the hyperparameters set at values similar to our con guration 33 Table 6: Styrene Exposure Data Worker 1 2 3 4 5 6 7 m i 3 3 3 3 3 3 3 y i 3.302 4.587 5.052 5.089 4.498 5.186 4.915 Worker 8 9 10 11 12 13 m i 3 3 3 3 ....

Cowles, M. K. and Rosenthal, J. S. (1998). A simulation approach to convergence rates for Markov chain Monte Carlo algorithms, Statistics and Computing 8: 115-124.

....that such methods will sometime declare convergence too soon, since the Markov chain settling down is only a necessary condition for convergence, but not sufficient. Another method of bridging the gap between theory and practice is to use auxiliary simulation to estimate the required constants (Cowles and Rosenthal, 1996). This is the approach taken here. We use theoretical bounds as a basis for a stopping criteria, and then use auxiliary simulation to estimate a theoretical constant that is required for this bound. By combining these two ingredients we obtain an approximate stopping criterion that can be applied ....

Cowles, M.K., and J.S. Rosenthal. 1996. A simulation approach to convergence rates for Markov Chain Monte Carlo. Technical report, Harvard.

....within of its stationary distribution. Such theoretical analysis is too dicult to be used routinely, however it has been successfully applied to some reasonably complicated examples of the Gibbs sampler (see e.g. 47] Furthermore, certain auxiliary simulation methods have been proposed ([7], 4] to estimate such theoretical burn in times B in more general situations. If we are able to obtain (or estimate) such a theoretical burn in time B, then implementing parallel MCMC is straightforward. Indeed, we simply set B j = B for each computer j, and then compute E j and E as above. The ....

M.K. Cowles and J.S. Rosenthal (1996), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Stat. and Comput. 8 (1998), 115-124.

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M.K. Cowles and J.S. Rosenthal (1998), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Stat. and Computing 8, 115-124.

.... (Y ) assuming that f 1 (x) 0:01 and that (8) is satisfied for this value of Y ) That is, n (Y ) represents a number of iterations that is sufficient for the sampler to have converged to within 0:01 of its stationary distribution in total variation distance (a convergence criterion suggested in Cowles and Rosenthal, 1998). Thus, using Proposition 3, we compute convergence times n (Y ) as a function of Y from (8) to be as follows: Y n (Y ) 1.0 525 0.9 615 0.8 728 0.5 1,400 0.33 2,395 0.25 3,475 0.1 10,850 0.01 160,000 0.001 2,075,000 Table 1. Convergence times n (Y ) as a function of Y . To make the meaning ....

Cowles, M.K. and Rosenthal, J.S. (1998), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Stat. and Comput. 8, 115--124.

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M.K. Cowles and J.S. Rosenthal (1996), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Stat. and Comput., to appear.

.... (0:985015) n (n Gamma 15:7043) For example, with n = 530, we obtain kP 530 ssl (x; Delta) Gamma ( Delta)k 0:0095 : Hence, for this example, just 530 iterations suffices to make the total variation distance to stationarity provably less than 1 (a convergence criterion suggested in Cowles and Rosenthal, 1996). Now, it follows immediately from Proposition 1 that this same bound applies when (x) e Gammaax is the (un normalised) density of the exponential distribution Exp(a) for any a 0, not just for a = 1. Specifically, writing the transition kernel of the simple slice sampler for Exp(a) as P ....

Cowles, M.K. and Rosenthal, J.S. (1996), A simulation approach to convergence rates for Markov chain Monte Carlo algorithms. Preprint.

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