Construction of tests of power one from smooth priors and the law of the iterated logarithm for posterior distributions

Abstract

The simple Bayes rules defined in Section 1 are constructed from a prior which has a point mass at θ=0 and a smooth normal part on {θ≠0}. Here it is pointed out that it is also possible to construct a stopping time with the properties (1.1) and (1.2) from a smooth prior. This can be done by stopping when the posterior mass of a neighbourhood of θ=0 becomes too small.