Locally uniform prior distributions

Abstract

Suppose that Xσ | θ ∼ N(θ, σ2) and that σ → O. For which prior distributions on θ is the posterior distribution of θ given Xσ asymptotically N(Xσ, σ2) when in fact Xσ ∼ N(θ0, σ2)? It is well known that the stated convergence occurs when θ has a prior density that is positive and continuous at θ0. It turns out that the necessary and sufficient conditions for convergence allow a wider class of prior distributions - the locally uniform and tail-bounded prior distributions. This class includes certain discrete prior distributions that may be used to reproduce minimum description length approaches to estimation and model selection.