Abstract
This paper describes how embedded sequences of positive interpolatory integration rules (PIIRs) obtained from Gauss-Hermite product rules can be applied in Bayesian analysis. These embedded sequences are very promising for two major reasons. First, they provide a rich class of spatially distributed rules which are particularly useful in high dimensions. Second, they provide a way of producing more efficient integration strategies by enabling approximations to be updated sequentially through the addition of new nodes at each step rather than through changing to a completely new set of nodes. Moreover, as points are added successive rules change naturally from spatially distributed non-product rules to product rules. This feature is particularly attractive when the rules are used for the evaluation of marginal posterior densities. We illustrate the use of embedded sequences of PIIRs in two examples. These illustrate how embedded sequences can be applied to improve the efficiency of the adaptive integration strategy currently in use.
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References
Berntsen, J. and Espelid, T. (1984) On the use of Gauss-quadrature in adaptive integration schemes.BIT,24, 239–242.
Davis, P. J. and Rabinowitz, P. (1984)Methods of Numerical Integration (2nd edn), Academic Press, Orlando, Florida.
Dellaportas, P. (1990) Imbedded integration rules and their applications in Bayesian analysis. Unpublished PhD thesis, Department of Mathematics and Statistics, Polytechnic South West, Plymouth.
Genz, A. (1986) Fully symmetric interpolatory rules for multiple integrals.SIAM J. Numer. Anal. 23, 6, 1273–1283.
Grieve, A. P. (1987) Applications of Bayesian software: two examples.The Statistician,36, 283–288.
Keast, P. and Lyness, J. N. (1979) On the structure of fully symmetric multidimensional quadrature rules.SIAM J. Numer. Anal. 16, 11–29.
Laurie, D. P. (1985) Practical error estimation in numerical integration.Journal of Computational and Applied Mathematics,12–13, 425–431.
Lyness, J. N. (1965) Symmetric integration rules for hypercubes.Math. Comp.,19, 260–276, 394–407, 625–637.
Mantel, P. and Rabinowitz, P. (1977) The application of integer programming to the computation of fully symmetric integration formulas in two and three dimensions.SIAM J. Numer. Anal. 14, 425–431.
McNamee, J. and Stenger, F. (1967) Construction of fully symmetric numerical integration formulas.Numer. Math. 10, 327–344.
Monegato, G. (1978) Some remarks on the construction of extended Gaussian quadrature rules.Mathematics of Computation,32, 141, 247–252.
Naylor, J. C. (1982) Some numerical aspects of Bayesian inference. Unpublished PhD thesis, University of Nottingham.
Naylor, J. C. (1987) Bayesian alternatives tot-tests.The Statistician,36, 241–246.
Naylor, J. C. and Shaw, J. E. H. (1985) Bayes Four—User guide. Nottingham Statistics Group, Department of Mathematics, University of Nottingham.
Naylor, J. C. and Smith, A. F. M. (1982) Applications of a method for the efficient computation of posterior distributions.Appl. Statist. 31, 3, 214–225.
Naylor, J. C. and Smith, A. F. M. (1983) A contamination model in clinical chemistry.The Statistician,32, 214–225.
Naylor, J. C. and Smith, A. F. M. (1988a) An archaeological inference problem.J. Amer. Statist. Assoc.,83, 403, 588–595.
Naylor, J. C. and Smith, A. F. M. (1988b) Econometric illustrations of novel numerical integration strategies for Bayesian inference.Journal of Economietrics,38, 103–126.
Patterson, T. N. L. (1968a) The optimum addition of points to Quadrature formulae.Math. Comp.,22, 847–856.
Patterson, T. N. L. (1968b) On some Gauss and Lobatto based Quadrature formulae.Math. Comp.,22, 877–881.
Patterson, T. (1973) Algorithm 468: Algorithm for numerical integration over a finite interval.Communications of the ACM,16, 694–699.
Piessens, R. (1973) An algorithm for automatic integration.Angewandte Informatik,9, 399–401.
Piessens, R. and Randers, M. B. (1974) A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type.Mathematics and Computation,28, 125, 135–139.
Rabinowitz, P., Kautsky, J., Elhay, S. and Butcher, J. C. (1987) On sequences of imbedded integration rules, inNumerical Integration: Recent Developments, P. Keast and G. Fairweather (eds), NATO ASI series, Reidel, Dordrecht, pp. 113–139.
Rabinowitz, P. and Richer, N. (1969) Perfectly symmetric two-dimensional integration formulas with minimal number of points.Math. Comp.,23, 765–780.
Shaw, J. E. H. (1987) Aspects of numerical integration and summarisation. Research report 103, Dept. of Statistics, University of Warwick.
Shaw, J. E. H. (1988). A quasirandom approach to integration in Bayesian statistics.Ann. Statist.,16, 895–914.
Smith, R. L. and Naylor, J. (1987) A comparison of maximum likelihood and Bayesian estimators for the three parameter Weibull distribution.Appl. Statist. 36, 358–369.
Smith, A. F. M., Skene, A. M., Shaw, J. E. H.et al. (1985) The implementation of the Bayesian paradigm.Commun. Statist. Theor. Meth.,14, 1079–1102.
Smith, A. F. M., Skene, A. M., Shaw, J. E. H. and Naylor, J. C. (1987) Progress with numerical and graphical methods for practical Bayesian statistics.The Statistician,36, 75–82.
Stroud, A. H. (1971)Approximate Calculation of Multiple Integrals, Englewood Cliffs, NJ, Prentice Hall.
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Dellaportas, P., Wright, D. Positive embedded integration in Bayesian analysis. Stat Comput 1, 1–12 (1991). https://doi.org/10.1007/BF01890832
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DOI: https://doi.org/10.1007/BF01890832