LOCALLY UNIFORM PRIOR DISTRIBUTIONS

Suppose that X(sigma)\theta similar to N(theta, sigma(2)) and that sig ma --> 0. For which prior distributions on theta is the posterior dist ribution of theta given X(sigma) asymptotically N(X(sigma), sigma(2)) when in fact X(sigma) similar to N(theta(0), sigma(2))? It is well kno wn that the stated convergence occurs when theta has a prior density t hat is positive and continuous at theta(0). It turns out that the nece ssary and sufficient conditions for convergence allow a wider class of prior distributions-the locally uniform and tail-bounded prior distri butions. This class includes certain discrete prior distributions that may be used to reproduce minimum description length approaches to est imation and model selection.