Bayes spaces: use of improper distributions and exponential families

Bayes spaces are vector spaces of sigma-additive positive measures. Proportional measures are considered equivalent and can be represented by densities with respect to a fixed dominating measure. The addition in these spaces is perturbation. It corresponds to Bayes theorem, which appears as a linear operation. Bayes spaces, with continuous dominating measures, contain finite and infinite measures. Finite measures are equivalent to probability measures. Infinite measures include what in Bayesian statistics are called improper priors and non-integrable likelihood functions, justifying the use of such improper densities in Bayes theorem. Many concepts of probability theory can be handled in a natural way in the context of Bayes spaces. Particularly, an exponential family of probability densities appears as a cone contained in an affine subspace of the Bayes space. The framework of Bayes spaces allows an easy handling of exponential families and their extensions to improper distributions. Furthermore, the vector space structure of Bayes spaces allows the definition of derivatives of densities. In Bayesian statistics, these derivatives are a new tool to examine sensitivity of posterior distributions with respect to both observed data and prior changes.