Conjugate Prior for Student T distribution with known degrees of freedom


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?

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.

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