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In a logical probability approach to inference, distributions on a parameter space are interpretable as representing states of knowledge, and any prevailing state of knowledge may be taken to have been arrived at from a previous state of ignorance (indifference) followed by an accumulation of prior data. In this paper an indifference procedure is introduced that requires postulating what size and what kind of samples will and will not (in a special sense) permit statistical inference and prediction--e.g., one observation from a two-parameter normal model is not (in our special sense) sufficient to permit inference about the variance but two observations are. In essence, the procedure stipulates that prior indifference distributions be improper but become proper after an appropriate minimal sample. With some limitation on the family of priors considered, this procedure permits unique specification of indifference for the more commonly encountered statistical models. Furthermore, these specifications are affected neither by change of the scale of measurement of the observations, nor by the sampling rule. [ABSTRACT FROM AUTHOR]

In this paper we obtain a multivariate generalization of the inverted beta distribution. Properties of this distribution and its connection with the multivariate Student-t distribution are discussed. Two asymptotic formulae for approximating the related probability integral are developed and illustrated with numerical examples. Application of this distribution to a problem in Bayesian inference is given. [ABSTRACT FROM AUTHOR]

In this paper we review the derivation of the predictive density function for the normal multiple regression model, state and prove a general theorem on optimal point prediction, and show how the predictive density can be employed in the analysis of an illustrative investment problem. Then we derive the predictive density function for the multivariate normal regression model and indicate how it can be used in the analysis of several problems. [ABSTRACT FROM AUTHOR]

The posterior distribution for parameters of a data distribution is usually the major objective of a Bayesian statistical analysis. Relatively little attention has been given to the fact that either a prior or a posterior distribution implies a marginal distribution, which may be called a "predictive distribution," for outcomes of any sample not yet observed. Predictive distributions have been mainly applied to design problems, such as determination of optimal sample size. In this paper tentative suggestions are made for applications to statistical inference, especially problems of appropriateness, selection, interpretation, and validation of formal models. [ABSTRACT FROM AUTHOR]

Reliability data often include information that the failure event has not yet occurred for some items, while observations of complete lifetimes are available for other items. Bayesian analysis requires a conjugate posterior density function after any combination of complete and incomplete observations. This paper considers density functions having failure rate function consisting of a known function multiplied by an unknown scale factor. It is shown that a gamma family of priors is conjugate for both complete and incomplete experiments. A very flexible and convenient model results from the assumption of a piecewise constant failure rate function. [ABSTRACT FROM AUTHOR]