Is it possible to construct a model with a parameter with a probability distribution that is not fitted to the data (i.e., posterior = prior)? E.g., in a linear regression model, I know the posterior probability distribution from external measurements, and I want to fit only the slope to the data (+ I’m interested in the the joint probability)? How would I write such a model in stan?
My first guess is to define the distribution of the parameter that should not change with strong hyper-priors?
yes exactly. The part that I find tricky is setting the parameter that stays the same to a normal ~ (mu, sd) distribution rather than to a single values
I get you. I expect some other people on here might know of a proper way to do this - I think in particular if you code directly in Stan (which I do not) it might be possible to set a parameter as fixed. But I think your idea of setting a tight prior on the parameter is reasonable to get started. This is probably a hack or could go wrong but you could potentially try to set it as uniform and only allow it to be between e.g., .99 and 1.01 or something (if you wanted it set to ~1)
The conceptually appealing way to do this is with a cut: Cuts in Bayesian graphical models | SpringerLink, but algorithms to sample the cut posterior are pretty non-trivial to implement (to say the least).