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May 1998 Simulating normalizing constants: from importance sampling to bridge sampling to path sampling
Andrew Gelman, Xiao-Li Meng
Statist. Sci. 13(2): 163-185 (May 1998). DOI: 10.1214/ss/1028905934


Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, especially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to explore these methods from a more statistical perspective, through establishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong. Thermodynamic integration, which is also known in the numerical analysis literature as Ogata's method for high-dimensional integration, corresponds to the use of infinitely many and continuously connected bridges (and thus a "path"). Our path sampling formulation offers more flexibility and thus potential efficiency to thermodynamic integration, and the search of optimal paths turns out to have close connections with the Jeffreys prior density and the Rao and Hellinger distances between two densities. We provide an informative theoretical example as well as two empirical examples (involving 17- to 70-dimensional integrations) to illustrate the potential and implementation of path sampling. We also discuss some open problems.


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Andrew Gelman. Xiao-Li Meng. "Simulating normalizing constants: from importance sampling to bridge sampling to path sampling." Statist. Sci. 13 (2) 163 - 185, May 1998.


Published: May 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0966.65004
MathSciNet: MR1647507
Digital Object Identifier: 10.1214/ss/1028905934

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.13 • No. 2 • May 1998
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