Abstract
In this paper, we introduce the notion of efficiency (consistency) and examine some asymptotic properties of Markov chain Monte Carlo methods. We apply these results to the data augmentation (DA) procedure for independent and identically distributed observations. More precisely, we show that if both the sample size and the running time of the DA procedure tend to infinity, the empirical distribution of the DA procedure tends to the posterior distribution. This is a local property of the DA procedure, which may be, in some cases, more helpful than the global properties to describe its behavior. The advantages of using the local properties are the simplicity and the generality of the results. The local properties provide useful insight into the problem of how to construct efficient algorithms.
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Acknowledgments
This is, essentially, the second part of the author’s Ph.D. thesis at Graduate School of Mathematical Sciences, the University of Tokyo. The author wishes to express his thanks to the Ph.D. supervisor, Prof. Nakahiro Yoshida for his several helpful comments and suggestions. The author also thank to the Associate Editor and anonymous referee for constructive comments which helped a lot to improve the paper.
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Supported in part by Grant-in-Aid for JSPS Fellows (19-3140) and Grant-in-Aid for Young Scientists (B) 22740055.
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Kamatani, K. Local consistency of Markov chain Monte Carlo methods. Ann Inst Stat Math 66, 63–74 (2014). https://doi.org/10.1007/s10463-013-0403-3
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DOI: https://doi.org/10.1007/s10463-013-0403-3