Prior near ignorance for inferences in the k-parameter exponential family

Abstract

This paper proposes a model of prior ignorance about a multivariate variable based on a set of distributions . In particular, we discuss four minimal properties that a model of prior ignorance should satisfy: invariance, near ignorance, learning and convergence. Near ignorance and invariance ensure that our prior model behaves as a vacuous model with respect to some statistical inferences (e.g. mean, credible intervals, etc.) and some transformation of the parameter space. Learning and convergence ensure that our prior model can learn from data and, in particular, that the influence of on the posterior inferences vanishes with increasing numbers of observations. We show that these four properties can all be satisfied by a set of conjugate priors in the multivariate exponential families if the set includes finitely additive probabilities obtained as limits of truncated exponential functions. The obtained set is a model of prior ignorance with respect to the functions (queries) that are commonly used for statistical inferences and, because of conjugacy, it is tractable and easy to elicit. Applications of the model to some practical statistical problems show the effectiveness of the approach.