Summary A generalized form of the cross-validation criterion is applied to the choice and assessment of prediction using the data-analytic concept of a prescription. The examples used to illustrate the application are drawn from the problem areas of univariate estimation, linear regression and analysis of variance.

A basic polynomial dependence on sample size is used to provide economical derivations of the formulae for sampling moments in simple random sampling.

Sanov's statement of first-order asymptotic behaviour of probabilities of large deviations of an empirical distribution function is here established for empirical probability measures, with attendant simplification of conditions. For the case of distribution functions, our theorem is strictly more general than a specialisation of results of Hoadley.

In a previous paper, the authors introduced a new criterion of expectation consistency between probability distributions of discrete data given discrete parameter values and arbitrary posterior probability distributions for the parameter. It is here shown, under very weak assumptions, that expectation consistency implies that the posterior distributions are generalized Bayes. However when the posterior distributions are generalized Bayes, the implied prior distribution need not be unique. The class of implied distributions is characterized in terms of a partition of parameter space.

Summary We describe a range of routine statistical problems in which marginal posterior distributions derived from improper prior measures are found to have an unBayesian property—one that could not occur if proper prior measures were employed. This paradoxical possibility is shown to have several facets that can be successfully analysed in the framework of a general group structure. The results cast a shadow on the uncritical use of improper prior measures. A separate examination of a particular application of Fraser's structural theory shows that it is intrinsically paradoxical under marginalization.

Journal Article Expectation consistency of inverse probability distributions Get access A. P. DAWID, A. P. DAWID University College London Search for other works by this author on: Oxford Academic Google Scholar M. STONE M. STONE University College London Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 59, Issue 2, August 1972, Pages 486–489, https://doi.org/10.1093/biomet/59.2.486 Published: 01 August 1972 Article history Received: 01 May 1971 Published: 01 August 1972

For two routine statistical problems, inference about the ratio of two exponential means and inference about the coefficient of variation of a normal random variable, a serious pathology of Bayesian inference based on improper priors is uncovered and investigated.