On the uniform consistency of Bayes estimates for multinomial probabilities (Q1813488)

Let \(\pi\) be a random binomial probability to be estimated on the basis of \(n\) trials which result in \(j\) successes. The authors show that the posterior odds ratio for the inequality \(|\pi- j/n|< h\) versus the reverse inequality is bounded below by \(\psi(h)\exp(2nh^ 2)\) with a constant \(\psi\) depending only on the ``degree of positivity'' of the prior. This result implies uniform concentration of the posterior distribution around the frequency \(j/n\) for priors which are uniformly positive. The same result holds for the multinomial distribution.