## The Number of Iterations, Convergence Diagnostics and Generic Metropolis Algorithms (1995)

Venue: | In Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter and |

Citations: | 31 - 3 self |

### BibTeX

@INPROCEEDINGS{Raftery95thenumber,

author = {Adrian E. Raftery and Steven M. Lewis},

title = {The Number of Iterations, Convergence Diagnostics and Generic Metropolis Algorithms},

booktitle = {In Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter and},

year = {1995},

pages = {115--130},

publisher = {Chapman and Hall}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction In order to use Markov chain Monte Carlo, MCMC, it is necessary to determine how long the simulation needs to be run. It is also a good idea to discard a number of initial "burnin " simulations, since from an arbitrary starting point it would be unlikely that the initial simulations came from the stationary distribution intended for the Markov chain. Also, consecutive simulations from Markov chains are dependent, sometimes highly so. Since saving all simulations can require a large amount of storage, researchers using MCMC sometimes prefer saving only every third, fifth, tenth, etc. simulation, especially if the chain is highly dependent. This is sometimes referred to as thinning the chain. While neither burn-in nor thinning are mandatory practices, they both reduce the amount of data saved from a MCMC run. In this chapter, we outline a way of determining in advance the number of iterations needed for a given level of precision in a MCMC algorithm.

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Citation Context ... misleading answers, and this is certainly true. However, the creation of an overdispersed starting distribution can be a major chore and can add substantially to the complexity of an MCMC algorithm (=-=Gelman and Rubin, 1992-=-b: Section 2.1). There is clearly a trade-off between the extra work and cost required to produce and use an overdispersed starting distribution and whatever penalty or cost might be experienced on th... |

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Citation Context ...tter often arises in hierarchical models when the algorithm enters parts of the parameter space where the random effects variance is small, and can require redesigning the MCMC algorithm itself (e.g. =-=Besag and Green, 1993-=-). Acknowledgment This research was supported by ONR contract no. N00014-91-J-1074 and by NIH grant 5R01HD26330. We are grateful to Andrew Gelman for stimulating discussions and to Wally Gilks for hel... |

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Citation Context ...he initial N min iterations can also suggest ways to improve the algorithm. The gibbsit program outputs can be used for diagnostic purposes. These outputs can be combined to calculate I = M +N N min (=-=Raftery and Lewis, 1992-=-b). This statistic measures the increase in the number of iterations due to dependence in the sequence. Values of I much greater than 1 indicate a high level of dependence. We have found that values o... |

46 |
One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo
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(Show Context)
Citation Context ...he initial N min iterations can also suggest ways to improve the algorithm. The gibbsit program outputs can be used for diagnostic purposes. These outputs can be combined to calculate I = M +N N min (=-=Raftery and Lewis, 1992-=-b). This statistic measures the increase in the number of iterations due to dependence in the sequence. Values of I much greater than 1 indicate a high level of dependence. We have found that values o... |

31 |
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Citation Context ... misleading answers, and this is certainly true. However, the creation of an overdispersed starting distribution can be a major chore and can add substantially to the complexity of an MCMC algorithm (=-=Gelman and Rubin, 1992-=-b: Section 2.1). There is clearly a trade-off between the extra work and cost required to produce and use an overdispersed starting distribution and whatever penalty or cost might be experienced on th... |

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Citation Context ...s the current estimated variance matrix. Others have suggested using a variable schedule, based in one way or another on how well the chain is behaving, to determine what the value of oe j should be (=-=Muller, 1991; Clifford-=-, 1994). Gelman (1993) has studied what the "optimal" variance should be for the case where one is simulating a univariate standard normal distribution using a Metropolis algorithm with a no... |

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Citation Context ...are designed to protect against) is only one of the possible causes of slow convergence in MCMC. Others include high posterior correlations (which can be removed by approximate orthogonalization; see =-=Hills and Smith, 1992), and &qu-=-ot;stickiness" of the chain. The latter often arises in hierarchical models when the algorithm enters parts of the parameter space where the random effects variance is small, and can require rede... |

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Citation Context ... log ( it = (1 \Gammasit )) = j + ffi i ffi i iidsN (0; \Sigma) : (1) The prior on j is Gaussian and the prior on \Sigma is inverted gamma with a shape parameter of 0:5 and a scale parameter of 0:02 (=-=Lewis, 1994-=-). The ffi i 's are random effects representing unobserved sources of heterogeneity in fertility such as fecundability and coital frequency. There are also measured covariates in the model, but these ... |

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Citation Context ...roblem arises in the example which follows. Example We illustrate these ideas with an example from the analysis of longitudinal World Fertility Survey data (Raftery, Lewis, Aghajanian and Kahn, 1993; =-=Lewis, 1993-=-). The data are complete birth histories for about 5,000 Iranian women, and here we focus on the estimation of unobserved heterogeneity. Letsit be the probability that woman i had a child in calendar ... |

1 |
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(Show Context)
Citation Context ...estimated variance matrix. Others have suggested using a variable schedule, based in one way or another on how well the chain is behaving, to determine what the value of oe j should be (Muller, 1991; =-=Clifford, 1994). Gelman -=-(1993) has studied what the "optimal" variance should be for the case where one is simulating a univariate standard normal distribution using a Metropolis algorithm with a normal distributio... |