Journal of Applied Probability

Random construction of interpolating sets for high-dimensional integration

Mark Huber and Sarah Schott

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Computing the value of a high-dimensional integral can often be reduced to the problem of finding the ratio between the measures of two sets. Monte Carlo methods are often used to approximate this ratio, but often one set will be exponentially larger than the other, which leads to an exponentially large variance. A standard method of dealing with this problem is to interpolate between the sets with a sequence of nested sets where neighboring sets have relative measures bounded above by a constant. Choosing such a well-balanced sequence can rarely be done without extensive study of a problem. Here a new approach that automatically obtains such sets is presented. These well-balanced sets allow for faster approximation algorithms for integrals and sums using fewer samples, and better tempering and annealing Markov chains for generating random samples. Applications, such as finding the partition function of the Ising model and normalizing constants for posterior distributions in Bayesian methods, are discussed.

Article information

J. Appl. Probab., Volume 51, Number 1 (2014), 92-105.

First available in Project Euclid: 25 March 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods 65D30: Numerical integration

Integration Monte Carlo method cooling schedule self-reducible


Huber, Mark; Schott, Sarah. Random construction of interpolating sets for high-dimensional integration. J. Appl. Probab. 51 (2014), no. 1, 92--105. doi:10.1239/jap/1395771416.

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