A Bayesian perspective on estimating mean, variance, and standard-deviation from data

Abstract

After reviewing some classical estimators for mean, variance, and standard-deviation and showing that un-biased estimates are not usually desirable, a Bayesian perspective is employed to determine what is known about mean, variance, and standard deviation given only that a data set in-fact has a common mean and variance. Maximum-entropy is used to argue that the likelihood function in this situation should be the same as if the data were independent and identically distributed Gaussian. A noninformative prior is derived for the mean and variance and Bayes rule is used to compute the posterior Probability Density Function (PDF) of (; ) as well as ; 2 in terms of the sucient statistics x = 1 n P i xi and C = 1 n P i (xi x)2 : From the joint distribution marginals are determined. It is shown that x pC pn 1 is distributed as Student-t with n 1 degrees of freedom, p 2 nC is distributed as generalized-gamma with c = 2 and a = n 1 2 ; and 2 2 nC is distributed as inverted-gamma with a = n 1 2 : It is suggested to report the mean of these distributions as the estimate (or the peak if n is too small for the mean to be de ned) and a con dence interval surrounding the median.