Bayesian Posteriors for Small Multinomial Probabilities

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 $p_{\bar{k}}\ge c q_{\bar{k}}$, 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 $(1-\delta)c$ times that for $q_{\bar{k}}$. If the prior densities for the two dice are positive on the interior of the probability simplex and behave like power functions at the boundary, 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 $np_{\bar{k}}\ge N$. The condition on $n$ and $p_{\bar k}$ is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.