James M. Dickey

Bayesian Methods for Censored Categorical Data

Abstract Bayesian methods are given for finite-category sampling when some of the observations suffer missing category distinctions. Dickeys (1983) generalization of the Dirichlet family of prior distributions is found to be closed under such censored sampling. The posterior moments and predictive probabilities are proportional to ratios of B. C. Carlsons multiple hypergeometric functions. Closed-form expressions are developed for the case of nested reported sets, when Bayesian estimates can be computed easily from relative frequencies. Effective computational methods are also given in the general case. An example involving surveys of death-penalty attitudes is used throughout to illustrate the theory. A simple special case of categorical missing data is a two-way contingency table with cross-classified count data xij (i = 1, …, r; j = 1, …, c), together with supplementary trials counted only in the margin distinguishing the rows, yi (i = 1, …, r). There could also be further supplementary trials report...

QUANTIFYING AND USING EXPERT OPINION FOR VARIABLE-SELECTION PROBLEMS IN REGRESSION

Abstract This paper gives a general method for use in the chemical industry for eliciting and quantifying an experts subjective opinion concerning a normal linear regression model. The intention is to ask the expert assessment questions that he or she can meaningfully answer and to use the elicited values to determine a probability distribution on the regression parameters that quantifies and expresses the experts opinions. A regression model may represent a chemical production process, for example, and the corresponding elicited distribution would embody the experts opinion concerning the effects on product output of independent variables for process control and environmental factors. It may be uncertain what independent variables should be featured in the regression, so the experts opinion is represented by a mixture of multivariate distributions, where each distribution in the mixture corresponds to a different subset of independent variables. Among the uses to which an elicited distribution might be put is design of experiments, discussed here with regard to Bayesian design criteria. An example is given of the elicitation and use of a subjective distribution in which an industrial chemist quantified his opinion about a chemical process.

Computation of Carlson's Multiple Hypergeometric Function R for Bayesian Applications

Abstract Carlsons multiple hypergeometric functions arise in Bayesian inference, including methods for multinomial data with missing category distinctions and for local smoothing of histograms. To use these methods one needs to calculate Carlson functions and their ratios. We discuss properties of the functions and explore computational methods for them, including closed form methods, expansion methods, Laplace approximations, and Monte Carlo methods. Examples are given to illustrate and compare methods.

Numerical application of the fundamental theorem of prevision

The fundamental theorem of the operational subjective theory of probability, from the viewpoint established by Bruno de Finetti, provides the solutions to probability problems in terms of the results of a pair of specific linear or nonlinear programming problems. We state a general version of this theorem in a computationally feasible form and present numerical examples of its use. The examples display interesting extensions of the Bienayme-Chebyshev inequality and a variation on the Kolmogorov inequality in the context of finite discrete quantities. The Bienayme-Chebyshev application is extended to exemplify the use of a nonlinear programming algorithm to resolve a common question regarding coherent inference. In concluding discussion, we comment on the sizes of realistic problems and suggest a variety of applications for such computations, among them the safety assessment of complex engi-neering systems, the analysis of agricultural production statistics, and a synthesis of subjective judgments in macro...

Filtered-Variate Prior Distributions for Histogram Smoothing

Abstract We develop prior distributions for histogram inference favoring smooth population frequencies; that is, probability vectors with small differences for neighboring categories. We give a theory of prior-random probability vectors representable as a linear transform, or “filter,” of a standard random probability vector, or equivalently, a random weighted average of nonrandom smooth probability vectors. Promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors, the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous-type filtered-variate prior distribution.

Distribution of functionals of a Ferguson-Dirichlet process over an n-dimensional ball

The c-characteristic function has been shown to have properties similar to those of the Fourier transformation. We now give a new property of the c-characteristic function of the spherically symmetric distribution. With this property, we can easily determine whether a distribution is spherically symmetric. The exact probability density function of the random mean of a spherically symmetric Ferguson-Dirichlet process with parameter measure over an n-dimensional spherical surface and that over an n-dimensional ball are given. We further give the exact probability density function of the random mean of a Ferguson-Dirichlet process with parameter measure over an n-dimensional ellipsoidal surface and that over an n-dimensional ellipsoidal solid.

Moments of the poly-Cauchy density with applications in estimation

Simple mathematical formulae for the mean and variance of a poly-Cauchy density (proportional to a product of two Cauchy densities) are derived here and then applied to obtain Bayesian estimators for the mean of a normal population and the difference between means of two normal populations. The proposed estimators are arguably superior to the traditional estimators and to the usual Bayesian estimators, and may be highly robust.