Objective Prior for the Number of Degrees of Freedom of a t Distribution

Abstract

In this paper, we construct an objective prior for the degrees of freedom of a $t$ distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dominate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the $t$ distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with $t$-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days.