 For the urn example, we can compute the posterior probability \(p(\theta\mid n_w)\) using Bayes’ rule, and the likelihood given by the binomial distribution above. In order to do so, we also need to assign a prior probability distribution to the parameter \(\theta\). One natural choice, which was made early on by Bayes himself and by Laplace, is to put a uniform prior over the parameter \(\theta\). Bayes’ own rationale for this choice was that then if you work out the probability of each value for the number of whites in the sample based only on the prior, before any data is observed, all those probabilities are equal. Laplace had a different justification, based on the Principle of Indifference. This principle states that if you don’t have any reason to favor one hypothesis over another, you should assign them all equal probabilities.