Another X validated question on which I spent more time than expected. Because of the somewhat unusual parameterisation used in BDA.for the inverse χ² distribution. The interest behind the question is in the induced distribution on the parameter associated with the degrees of freedom ν of the t-distribution (question that coincided with my last modifications of my undergraduate mathematical statistics exam, involving a t sample). Whichever the prior chosen on ν, the posterior involves a nasty term

as the Gamma function there is quickly explosive (as can be checked Stirling’s formula). Unless the prior π(ν) cancels this term, which is rather fishy as the prior would then depend on the sample size n. Even though the whole posterior is well-defined (and hence non-explosive). Rather than seeking a special prior π(ν) for computation purposes, I would thus favour a modelling restricted to integer valued ν’s as there is not much motivation in inferring about non-integer degrees of freedom.