When fitting a Student-t model, how should the posterior distribution of the degrees-of-freedom parameter (ν) affect interpretation of the mean and variance? If most of the posterior for ν is above 2, but some probability remains below 2, can variance-based quantities still be interpreted reliably, or should they be treated cautiously?
More generally, should the mean and variance only be considered well defined if essentially all of the posterior lies above their respective thresholds (ν>1 for the mean and ν>2 for the variance)? For example, if 95% of the posterior for ν is above 1 but the credible interval overlaps 2, would it be appropriate to interpret the mean while being cautious about the variance?
Since the estimand of interest requires a finite variance I contstrained nu > 1.99
prior.1 <-
prior(gamma(2, 0.1), class = "nu", lb = 1.99)
All Rhats are 1.
So since the majority of the mass is above 0 is this ok?
This was with no constraint
@Solomon
@paul.buerkner
If you want a finite mean, set prior(..., class = nu, lb = 1). If you also want a finite variance, set prior(..., class = nu, lb = 2). Personally, I’m a fan of lb = 2; that can still let the rest of the \nu posterior get very small, if needed. This is what I would do, for example, if I wanted to fit a Student-t model to experimental data, for which I’d want to compute a standardized mean difference effect size.
Ok great, so this is what I will try.
Do you also have any suggestions how how to backtransform estimates if I did a log student t? or is it not possible becaue the jensen inequality formula for a lognormal doesnt apply.
Thanks
What do you mean by doing a log student t?
I mean a model where I first transform the response as log(y) and then fit a Student-t likelihood to that transformed response.
This is analogous to fitting a Gaussian model to log(y), which corresponds to a lognormal model for y. In the lognormal case, posterior_epred() can return the expectation on the original response scale while accounting for Jensen’s inequality.
What I am asking is whether there is an analogous way to obtain E[Y] on the original response scale if I first log transform my response using a Student-t likelihood.
Ah. I’d just find a more natural likelihood, like the gamma or the Fréchet. See here.
