Locally uniform prior distributions

Abstract

Suppose that $X_{\sigma} | \mathbf{\theta} \sim N(\mathbf{\theta}, \sigma^2)$ and that $\sigma \to 0$. For which prior distributions on $\mathbf{\theta}$ is the posterior distribution of $\mathbf{\theta}$ given $X_{\sigma}$ asymptotically $N(X_{\sigma}, \sigma^2)$ when in fact $X_{\sigma} \sim N(\theta_0, \sigma^2)$? It is well known that the stated convergence occurs when $\mathbf{\theta}$ has a prior density that is positive and continuous at $\theta_0$. 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.