# Combining non-independent priors

I've been working on a stats package that includes Bayesian survival models. The user is allowed to write priors directly for all the parameters involved. However, I think it's pretty difficult for typical users to come up with priors for baseline parameters; if you were to ask me to put a prior distribution on the shape and rate for a Weibull distribution, I think even I might have a lot of trouble in some problems. Keep in mind that the baseline parameters reflect parameter variables where all the covariates are zero, which can be totally non-sensical, i.e. BMI = 0.

What I would prefer to do is allow a user to do is specify a prior on a more straightforward measure; i.e. maybe specify the priors of the median, 25th and 75th percentile for subjects with a given set of covariates. Better yet, priors for different subjects with different sets of covariates.

The problem with this is that it is totally unreasonable to believe these priors are independent. As such, it's not clear to me how to combine these priors in a manner that doesn't overstate the information in the prior. Technically, one could do something like specify a multi-variate normal prior and specify correlations of the prior...but this seems way worse than specifying priors on the baseline parameters themselves. I'd much prefer a method to combine independently elicited priors that are not statistically independent in a manner that is minimally informative.

I'm sure there must be some discussion of this idea in the literature, but I wouldn't know where to start.

• Just a side note, if this is not know: independent priors may not lead to independence in the posterior. In fact, most independent priors lead to a dependence structure in the posterior. So if the goal is to obtain dependence in the posterior, non-independent priors need not be required. – Greenparker Jun 3 '17 at 6:42
• @Greenparker: my goal is not dependence in the posterior. Rather, I would like to solicit priors about several different functions of the parameters from a user. However, all these priors are definitely not independent and if I treat all these priors as independent, I will have greatly overstated the prior certainty. So my problem is how do I combine all this solicited priors without overstating the certainty? If anything, I would prefer to understate the certainty (i.e. provide something like a lower bound on the prior information). – Cliff AB Jun 3 '17 at 16:40