Setting up a Hierarchical Multinomial Processing Tree in Stan - stan

Question

I'm having trouble setting up a Hierarchical Multinomial Processing tree in Stan. As a starting point, I am trying to add a hierarchy to the simple model here:
https://github.com/stan-dev/example-models/blob/master/Bayesian_Cognitive_Modeling/CaseStudies/MPT/MPT_1_Stan.R
I'm not sure why the code does not work. Any help would be greatly appreciated.
Example Data (Based on Julia syntax):
Nsub = 2
Ntrials = 100
FCat = [20 60 20;30 50 20]
data {
// Number of subjects
int<lower=1> Nsub;
// Number of Trials
int<lower=1> Ntrials;
// Data
int<lower=0,upper=Ntrials> FCat[Nsub,4];
}
parameters {
vector<lower=0,upper=1>[Nsub] c;
vector<lower=0,upper=1>[Nsub] r;
vector<lower=0,upper=1>[Nsub] u;
real<lower=0> c_omega;
real<lower=0> r_omega;
real<lower=0> u_omega;
real<lower=0,upper=1> c_kappa;
real<lower=0,upper=1> r_kappa;
real<lower=0,upper=1> u_kappa;
}
transformed parameters {
simplex[4] theta[Nsub];
real<lower=0> c_A;
real<lower=0> c_B;
real<lower=0> r_A;
real<lower=0> r_B;
real<lower=0> u_A;
real<lower=0> u_B;
c_A <- c_kappa*c_omega;
c_B <- (1-c_kappa)*c_omega;
r_A <- r_kappa*r_omega;
r_B <- (1-r_kappa)*r_omega;
u_A <- u_kappa*u_omega;
u_B <- (1-u_kappa)*u_omega;
// Create category responses
for (i in 1:Nsub){
theta[i,1] <- c[i]*r[i];
theta[i,2] <- (1 - c[i])*sqrt(u[i]);
theta[i,3] <- (1 - c[i])*2*u[i]*(1 - u[i]);
theta[i,4] <- c[i]*(1 - r[i]) + (1 - c[i])*sqrt(1 - u[i]);
}
}
model {
// HyperPriors
c_omega ~ gamma(2,8);
r_omega ~ gamma(2,8);
u_omega ~ gamma(2,8);
c_kappa ~ beta(50,50);
r_kappa ~ beta(50,50);
u_kappa ~ beta(70,30);
// Priors
c ~ beta(c_A, c_B);
r ~ beta(r_A, r_B);
u ~ beta(u_A, u_B);
for (i in 1:Nsub){
FCat[i] ~ multinomial(theta[i]);
}
}

Answer 1

The sqrt() function just takes square roots of a scalar (real or int), returning a real.
You can use print() to see if your values sum to 1. Or you can do a test yourself and reject if they don't. The usual stick-breaking construction for a simplex is given in the manual in the simplex transform. The usual approach is to take the first value to be between 0 and 1, then the next to be some fraction of what's left (1 - first value), and so on. We also have softmax built in. You have to be careful with identifiability with a softmax parameterization --- the priors will matter unless you set one of the inputs to a constant.