Positive embedded integration in Bayesian analysis

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.

[1]  J. Naylor,et al.  A Contamination Model in Clinical Chemistry: An Illustration of a Method for the Efficient Computation of Posterior Distributions , 1983 .

[2]  J. N. Lyness,et al.  On the Structure of Fully Symmetric Multidimensional Quadrature Rules , 1979 .

[3]  Giovanni Monegato,et al.  Some remarks on the construction of extended Gaussian quadrature rules , 1978 .

[4]  Andrew P. Grieve,et al.  Applications of Bayesian software: two examples , 1987 .

[5]  J. Naylor,et al.  Applications of a Method for the Efficient Computation of Posterior Distributions , 1982 .

[6]  Terje O. Espelid,et al.  On the use of Gauss quadrature in adaptive automatic integration schemes , 1984 .

[7]  Richard L. Smith,et al.  A Comparison of Maximum Likelihood and Bayesian Estimators for the Three‐Parameter Weibull Distribution , 1987 .

[8]  J. E. H. Shaw,et al.  A Quasirandom Approach to Integration in Bayesian Statistics , 1988 .

[9]  Alfred Houle.  The Genealogical Tree of Bayesians , 1983 .

[10]  Philip Rabinowitz,et al.  The Application of Integer Programming to the Computation of Fully Symmetric Integration Formulas in Two and Three Dimensions , 1977 .

[11]  T. Patterson.  On some Gauss and Lobatto based integration formulae. , 1968 .

[12]  J. C. Naylor,et al.  Econometric illustrations of novel numerical integration strategies for Bayesian inference , 1988 .

[13]  Petros Dellaportas.  Imbedded integration rules and their applications in Bayesian analysis , 1990 .

[14]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[15]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[16]  T. Patterson,et al.  The optimum addition of points to quadrature formulae. , 1968 .

[17]  J. McNamee,et al.  Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas , 1967 .

[18]  J. N. Lyness.  Symmetric integration rules for hypercubes. I. Error coefficients , 1965 .

[19]  J. Butcher,et al.  On Sequences of Imbedded Integration Rules , 1987 .

[20]  J. E. H. Shaw,et al.  The implementation of the bayesian paradigm , 1985 .

[21]  Philip Rabinowitz,et al.  Perfectly symmetric two-dimensional integration formulas with minimal numbers of points , 1969 .

[22]  A. F. M. Smith,et al.  An Archaeological Inference Problem , 1988 .

[23]  Robert Piessens.  Algorithm for automatic integration , 1973 .

[24]  R. Piessens,et al.  A note on the optimal addition of abscissas to quadrature formulas of Gauss and Lobatto type , 1974 .

[25]  Dirk Laurie,et al.  Practical error estimation in numerical integration , 1985 .

[26]  J. C. Naylor.  Bayesian Alternatives to T‐Tests , 1987 .

[27]  Thomas N. L. Patterson,et al.  Algorithm 468: algorithm for automatic numerical integration over a finite interval [D1] , 1973, CACM.

[28]  Alan Genz,et al.  Fully symmetric interpolatory rules for multiple integrals , 1986 .

[29]  A. F. M. Smith,et al.  Progress with numerical and graphical methods for practical Bayesian statistics , 1987 .