Hi, I’m curious as to what prior is recommended for hierarchical logistic. I’ve read that Cauchy is good for non-hierarchical. Thanks.
Hi Blake, I think prior choice is always heavily dependent on the context. What parameters do you want to set the priors on? As far as I know, in default logistic regression, the parameters are on a log-odds scale and your priors should reflect that. For example, if you had a model that estimated the odds of someone getting cancer 2 to 1 (~66% probability), the log odds would be 0.69. If the model estimated the odds of getting the cancer to be 10 to 1 (~91% probability), the log odds would be 2.3. If the model estimated the odds of getting cancer to be 100 to 1 (~99% probability), the log odds would be 4.6.
The point I’m trying to make is that the log odds space is pretty flat (in a pretty huge leaps of odds from 2 to 10 to 100, we go only small steps in log odds from 0.69 to 2.3 to 4.6). Your choice of priors for a logistic models should reflect that. For example, if you have an outcome that occurs 50 percent of the time (e.g. whether a child is born female, you have a dataset with 480 boys and 520 girls), it would be pretty nonsensical to put a normal(0, 10) prior on your intercept, because that would put quite a lot prior density on the outcome being either extremely rare or extremely frequent (-20 and 20 log odds are two sd’s away from the mean) when ignoring the effect of covariates. I can’t imagine any covariate that could change the probability of a child being born female by more than a few percentage points, and so a more reasonable choice would be a tighter prior on the Intercept and the slopes (e.g. normal(0, 0.5) Intercept and normal(0, 0.1) on slopes).