Posterior Moments of the Cauchy Distribution

The posterior moments of parameters specifying distributions are minimum mean square Bayesian estimators for the corresponding moments of those parameters, and as such are ubiquitous in the Bayesian approach to statistical inference of distributions. The Cauchy distribution is most notable for its wide tails, decided absence of high-order moments, and non-existence of less-than-data dimension suucient statistics. Thus it vastly diiers qualitatively from the Gaussian distribution, where tails are small, moments of all orders exist, and dimension-two suucient statistics always exist. In this paper the posterior moments of the position parameter of the Cauchy distribution are found in closed form. (Estimating the other parameter, the width or distance parameter of the Cauchy is done using the same mathematics.) The interplay between the amount of data acquired for the estimation of the position parameter and the existence of higher order moments of the inferred posterior distribution for the postition parameter is made explicit.