Use simpler regularize horseshoe parameterization

Summary:

There are different regularized horseshoe parameterizations. Two of them with Stan code are presented in Appendix C of https://projecteuclid.org/euclid.ejs/1513306866

The current rstanarm code is using parameterization shown in Appendix C.2. This parameterization was chosen as it seemed to be useful for non-regularized horseshoe.

Appendix C.1 has a simpler parameterization which has D+1 less parameters where D is the number of regression coefficients. This simpler parameterization is faster as the posterior dimensionality is smaller. We also now have lot of experimental experience that it works well with regularized horseshoe and thus it would be good to switch to the simpler parameterization.

Start by locating the current regularized horseshoe implementation in rstanarm and compare it with the Stan code in Appendix C.1 and C.2 of the linked paper. Confirm the simpler parameterization preserves the model’s behavior while reducing posterior dimensionality by D+1 parameters, then run the relevant existing tests.