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Biometrika Advance Access originally published online on August 11, 2009
Biometrika 2009 96(3):635-644; doi:10.1093/biomet/asp036
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© 2009 Biometrika Trust

Article

Approximating the {alpha}-permanent

S. C. Kou

Department of Statistics, Harvard University, Cambridge, Massachusetts 02138, U.S.A. kou{at}stat.harvard.edu

P. McCullagh

Department of Statistics, University of Chicago, 5734 University Avenue, Chicago 60637, U.S.A. pmcc{at}galton.uchicago.edu

Received for publication 1 May 2008. Revision received 1 December 2008.
  Abstract

The standard matrix permanent is the solution to a number of combinatorial and graph-theoretic problems, and the {alpha}-weighted permanent is the density function for a class of Cox processes called boson processes. The exact computation of the ordinary permanent is known to be #P-complete, and the same appears to be the case for the {alpha}-permanent for most values of {alpha}. At present, the lack of a satisfactory algorithm for approximating the {alpha}-permanent is a formidable obstacle to the use of boson processes in applied work. This paper proposes an importance-sampling estimator using nonuniform random permutations generated in a cycle format. Empirical investigation reveals that the estimator works well for the sorts of matrices that arise in point-process applications, involving up to a few hundred points. We conclude with a numerical illustration of the Bayes estimate of the intensity function of a boson point process, which is a ratio of {alpha}-permanents.

Key Words: Boson point process • Conditional intensity • Density estimation • Sequential importance sampling


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