Alternatively, Sebastian, you could find that an infinite uniform prior (or rather, the limit of a series of finite priors that approach uniformity) is manageable given the evidence you already have, whose likelihood function produces a nonuniform (limit) posterior. However, it is VERY IMPORTANT that you do all your calculations on finite distributions and then calculate how such finite distributions approach a limit, rather than assuming the limit already accomplished and trying to calculate directly with infinities. Otherwise you will have paradoxes up the wazoo.

I receive my policy on this from the teachings of the master, E. T. Jaynes, whose holy words may be found in "Probability Theory: The Logic of Science", particularly Chapter 15.