Bayesian Posteriors For Arbitrarily Rare Events

Each period, either a blue die or a red die is tossed. The two dice land on side \bar{k} with unknown probabilities $p_{\bar{k}}$ and $q_{\bar{k}}$, which can be arbitrarily low. Given a data-generating process where $p_{\bar{k}}\ge q_{\bar{k}}$, we are interested in how much data is required to guarantee that with high probability the observer's Bayesian posterior mean for $p_{\bar{k}}$ exceeds that for $q_{\bar{k}}$. If the prior is positive on the interior of the simplex and vanishes no faster than polynomially to zero at the simplex boundaries, then for every $\epsilon>0$, there exists $N\in\mathbb{N}$ so that the observer obtains such an inference after n periods with probability at least $1-\epsilon$ whenever $np_{\bar{k}}\ge N$. This result can fail if the prior vanishes to zero exponentially fast at the boundary.