Summary
A new method for the numerical integration of very high dimensional functions is introduced and implemented based on the Metropolis' Monte Carlo algorithm. The logarithm of the high dimensional integral is reduced to a 1-dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval. The improvement in accuracy is found to be substantial comparing to the conventional crude Monte Carlo integration. Several numerical demonstrations are made, and variability of the estimates are shown.
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Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys.21, 1087–1092 (1953)
Ogata, Y.: A Monte Carlo method for the objective Bayesian procedure. Research Memorandum No. 347, The Institute of Statistical Mathematics, Tokyo 1988
Ogata, Y., Tanemura, M.: Approximation of likelihood function in estimating the interaction potentials from spatial point patterns. Research Memorandum No. 216, The Institute of Statistical Mathematics, Tokyo 1981
Ogata, Y., Tanemura, M.: Likelihood analysis of spatial point patterns. Research Memorandum No. 241, The Institute of Statistical Mathematics, Tokyo 1984a
Ogata, Y., Tanemura, M.: Likelihood analysis of spatial point patterns. J.R. Statist. Soc. B46, 496–518 (1984b)
Ogata, Y., Tanemura, M.: Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns. Ann. Inst. Statist. Math. (1988)
Hammersley, J.M., Handscomb, D.C.: Monte Carlo Methods. London: Methuen & Co Ltd (1964)
Ripley, B.D.: Simulating spatial patterns: dependent samples from a multivariate density. Appl. Statist.28, 109–112 (1979)
Wood, W.W.: Monte Carlo studies of simple liquid models. In: Temperley, H.N.V., Rowlinson, J.S., Rushbrooke, G.S. (eds.) Physics of simple liquids, Chap. 5, pp. 115–230, Amsterdam: NorthHolland (1968)
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Ogata, Y. A Monte Carlo method for high dimensional integration. Numer. Math. 55, 137–157 (1989). https://doi.org/10.1007/BF01406511
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DOI: https://doi.org/10.1007/BF01406511