New insights into the impact of the choice of the prior for the success parameter of Binomial distributions

DOI: 10.5176/2251-1911_CMCGS18.19

Authors: Fatemeh Ghaderinezhad

Abstract:There exist various choices for the prior distribution of the success parameter in Binomial distributions. Popular choices are the Beta distribution, which is the conjugate prior, the Jeffreys’ prior and the Haldane prior. In the present paper, we propose a measure of the impact of each prior on the posterior distribution by comparing, at fixed sample size ?, each resulting posterior to the data likelihood, which can be seen as the posterior resulting from the Uniform prior. More precisely, we provide tight lower and upper bounds for the Wasserstein distance between each of the three posteriors and the likelihood. This allows us to formally confirm the general belief that, for a reasonable choice of prior distribution and a large enough sample size, the impact of the prior on the posterior inference is small. Moreover, we see that the impact of the Jeffreys’ and Haldane prior, unlike for the Beta prior, depends on the data observed.

Keywords: Bayesian statistics, Binomial distribution, Posterior distribution, Prior distribution, Stein’s method for nested densities, Wasserstein distance