Somebody asked a question about a conjugate prior distribution for Student-t distribution with unknown degrees of freedom. It was answered that there are no conjugate prior distribution in that case.
However, in my task I know this parameter, but the mean and the scale of Student-t distribution are not given. And I want to use a conjugate prior distribution to approximate the mean and the scale.
So what is the conjugate prior distribution for Student-t distribution in case when the number of degrees of freedom is known?
Conjugate Prior for Student T distribution with known degrees of freedom
conjugate-priordistributionsprobabilityt-distribution
Best Answer
And indeed there is no natural conjugate for the parameters of a Student distribution since as already pointed out in the link the distribution is not from the exponential family.
That's being said, I think your question is not hopeless, and few clarifications should be provided in this context.
My first question is: Why do you need a conjugate prior ?. For developing a Gibbs sampler ? in a specific context such as variational Bayes ?
If you do not have any specific reason to have a conjugate prior, standard estimation procedures implemented in many softwares (NUTS), do not require conjugacy and are pretty efficient in many problems.
However, if conjugacy is mandatory for your research question, I often recommend to rewrite your prior through a hierarchical representation using full conjugacy priors. For Student distribution you can check section 4.1 of this paper: DOI:10.1214/11-BA631. It presents some interesting strategies in the context of variational Bayes for such scenarios.
I can elaborate on specific aspects if you precise your question.