Stanimals,
In the course of applying regularized horseshoe priors to models in which two output variables respond to the same set of input variables (a la Piironen & Vehtari 2017), I noticed that the coefficients of two different output variables (number of wildfire events and expected wildfire size) are positively related:
I am currently using separate regularized horseshoe priors for these two outputs, but I’ve been considering explicitly modeling a correlation in coefficients among responses by generalizing the univariate horseshoe prior to a multivariate horseshoe prior. This would be consistent with our intuition for this system (conditions that increase the probability of fire occurrence are expected to also increase the size of a fire), and it seems like it might be useful to allow information sharing among the M output dimensions.
So, I wanted to bounce this idea off the group and see if there are any obvious pitfalls that I’m overlooking before diving too deep. Currently, what I’m doing for my M = 2 output variables is something to the effect of:
for each output dimension (m = 1, 2) and each input variable (j = 1, …, D), where lambda could be \tilde{\lambda} for a regularized horseshoe prior.
An equivalent specification places an uncorrelated multivariate normal prior on each length M vector of coefficients for each input j:
for each input variable j =1, …, D.
Written this way, one might be tempted to introduce a parameter rho that allows correlation among the coefficients:
Does this seem reasonable, or am I missing something obviously wrong with this approach?
Edit: maybe you could also share information among the M output dimensions by placing a multivariate Cauchy prior on the lambdas…