In addition to the two implementations given in the Lee and Wagenmakers book, 
this folder contains three further implementationso of the hierarchical model.
These further implementations are more in line with the suggested way such 
models should be programmed in Stan. Some features of the original hierarchical 
implementation were specifically added to allow for a better convergence using 
Gibbs sampling such as implemented in JAGS or Bugs. These features do not help 
the convergence using the NUTS/HMC sampler used by Stan and in fact, the 
original model shows convergence issues with divergent transitions that can only
be handeled by relatively strongly changing the Stan samplers' default settings.

The new implementations avoid the Gibbs specific features. Specifically, these 
implementations do not use the Wishart as prior for the correlation matrix and 
do not use the parameter expansion. 
We include three implementations which follow the three recommendations given in
the Stan manual (section 6.12. "Multivariate Priors for Hierarchical Models") 
for the Multivariate Regression Example of Gelman and Hill. All three of these 
have no parameter expansion parameters (xichat, ...). The other changes are: 

1. MPT_3_Stan.R:
Simply replaces the Wishart prior for the correlation with a correlation matrix 
Omega with LKJ prior and scaling vector tau. 
This model still requires some tweaking of the sampler setting to run without 
any divergent transitions or other numerical problems. There also can occur some
correlation matrices during estimation that are not positive definite. However, 
it increases fitting speed dramatically. Final fits still does not look 
completely well behaved (when looking at the traceplots).

2. MPT_4_Stan.R:
Further replace the correlation matrix Omega with its Cholesky factor L_Omega 
and corresponding prior. Likewise, tau will be replaced by corresponding scaling
parameter sigma. Whereas this again noticably increases the speed of 
optimization it does still requires tweaking of the sampler setting to converge
without problems and the final fit is not completely well behaved.

3. MPT_5_Stan.R:
This contains a further reparametrization of the multivariate normal 
distribution using a univariate normal distribtuion and basically solves the 
remaining issues. Specifically, this uses a non-centered parametrization of the 
multivariate normal (also known as the 'Matt trick' on the Stan mailing list). 
It further dramatically increases the speed and at the same time dramatically
increases the number of effective samples using the same number of samples. It 
also requires only minimal tweaking of the sampler setting to remove the very 
few remaining issues. Finally, the traceplot show no problem whatsoever.
The non-centered parameterization is extensively discussed in section 21.2. of 
the Stan manual. But it also is the same as given in section 6.12. 


For further details see the correspondoing thread on the Stan mailing list:
https://groups.google.com/forum/#!topic/stan-users/ZTiu_ij_bC8