Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-17T23:36:10.707Z Has data issue: false hasContentIssue false

Sequential Monte Carlo

Published online by Cambridge University Press:  24 October 2008

J. H. Halton
Affiliation:
Balliol CollegeOxford

Abstract

This paper defines the concept of sequential Monte Carlo and outlines the principal modes of approach which may be expected to yield useful sequential processes. Three workable sequential processes, derived from a non-sequential method of J. von Neumann and S. M. Ulam for solving systems of linear algebraic equations, are described and analysed in detail.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1962

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cutkosky, R. E.A Monte Carlo method for solving a class of integral equations. J. Res. Nat. Bur. Standards, 47 (1951), 113–5.CrossRefGoogle Scholar
(2)De Groot, M. H.Unbiased sequential estimation for binomial populations. Ann. Math. Statist. 30 (1959), 80101.CrossRefGoogle Scholar
(3)Fieller, E. C. and Hartley, H. O.Sampling with control variables. Biometrika, 41 (1954), 494501.CrossRefGoogle Scholar
(4)Forsythe, G. E. and Leibler, R. A.Matrix inversion by a Monte Carlo method. Math. Tables Aids Comput. 4 (1950), 127–9.CrossRefGoogle Scholar
(5)Halmos, P. R.Measure theory (New York, 1956).Google Scholar
(6)Halton, J. H.On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2 (1960), 8490. (Also On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. p. 196.)CrossRefGoogle Scholar
(7)Halton, J. H. and Handscomb, D. C.A method for increasing the efficiency of Monte Carlo integration. J. Assoc. Comput. Mach. 4 (1957), 329–40.CrossRefGoogle Scholar
(8)Hammersley, J. M.Monte Carlo methods for solving multi-variables problem. Ann. New York Acad. Sci. 86 (1960), 844–74.CrossRefGoogle Scholar
(9)Hammersley, J. M. and Mauldon, J. G.General principles of antithetic variates. Proc. Cambridge Philos. Soc. 52 (1956), 476–81.CrossRefGoogle Scholar
(10)Hammersley, J. M. and Morton, K. W.A new Monte Carlo technique: antithetic variates. Proc. Cambridge Philos. Soc. 52 (1956), 449–75.CrossRefGoogle Scholar
(11)Handscomb, D. C.Proof the antithetic variates theorem for n > 2. Proc. Cambridge Philos. Soc. 54 (1958), 300–1.CrossRefGoogle Scholar
(12)Kahn, H. The use of different Monte Carlo sampling techniques. University of Florida symposium on Monte Carlo methods, pp. 146–90 (New York, 1956).Google Scholar
(13)Kiefer, J. and Weiss, L.Some properties of generalized sequential probability ratio tests. Ann. Math. Statist. 28 (1957), 5774.CrossRefGoogle Scholar
(14)Lévy, P.Théorie de l'addition des variables aléatoires (Paris, 1937).Google Scholar
(15)Marshall, A. W. The use of multi-stage sampling schemes in Monte Carlo computation. University of Florida symposium on Monte Carlo methods, pp. 123–40 (New York, 1956).Google Scholar
(16)Opler, A.Monte Carlo matrix calculations with punched card machines. Math. Tables Aids Comput. 5 (1951), 115–20.CrossRefGoogle Scholar
(17)Page, E. S.The Monte Carlo solution of some integral equations. Proc. Cambridge Philos. Soc. 50 (1954), 414–25.CrossRefGoogle Scholar
(18)Stein, C. and Wald, A.Sequential confidence intervals for the mean of a normal dis-tribution with known variance. Ann. Math. Statist. 18 (1947), 427–33.CrossRefGoogle Scholar
(19)Wald, A.Sequential analysis (New York, 1950).Google Scholar
(20)Wald, A. and Wolfowitz, J.Bayes solutions of sequential decision problems. Ann. Math. Statist. 21 (1950), 82–9.CrossRefGoogle Scholar
(21)Wasow, W.Random walks and the eigenvalues of elliptic difference equations. J. Res. Nat. Bur. Standards, 46 (1951), 6573.CrossRefGoogle Scholar
(22)Wasow, W.A note on the inversion of matrices by random walks. Math. Tables Aids Comput. 6 (1952), 7881.CrossRefGoogle Scholar
(23)Wolfowitz, J.On sequential binomial estimation. Ann. Math. Statist. 17 (1946), 489–93.CrossRefGoogle Scholar
(24)Wolfowitz, J.The efficiency of sequential estimates and Wald's equation for sequential processes. Ann. Math. Statist. 18 (1947), 215–30.CrossRefGoogle Scholar