Stan: Copula on observed variable and latent variable - stan

Question

Consider observed data y1 and y2. y1 is measured on a continuous scale and y2 is measured on a binary scale. A continuous latent variable z is assumed to generate y2 as: y2 = I(z > 0). (If z is normal then y2 is binary probit marginally). Furthermore, a copula is used to model the dependency between y1 and z. This model could be written hierarchically (with some abuse of notation) as:
y2 = I(z > 0)
(y1, z) ~ C(F_y1( |w), F_z( |w) | phi)
w, phi ~ priors
where w is the vector of marginal parameters for y1 and z, F_y1 and F_z are respective marginal cdfs for y1 and z, phi is the copula parameter.
How could this be modelled in Stan? I have written a custom probability function to sample y1 and z from the bivariate likelihood produced by the copula. What I don't know how to do is to account for (generate?) the latent variable(s) z, and how to specify the relationship between y2 and z.
I have already looked at Probit regression with data augmentation in stan, but this does not seem helpful due to the copula I have in my model.
Edit: I might be mistaken about the above link not being useful. I have written the following code, would appreciate comments on if it looks correct (theoretically).
functions {
real copulapdf_log(real[] y1, real[] z, vector mu1, vector mu2, real sigma1, real phi, int n){
real logl;
real s;
logl <- 0.0;
for (i in 1:n){
s <- log(dCphi_du1du2_s(normal_cdf(y1[i],mu1[i],sigma1), logistic_cdf(z[i],mu2[i],1), phi)) + normal_log(y1[i],mu1[i],sigma1) + logistic_log(z[i],mu2[i],1);
logl <- logl + s;
}
return logl;
}
}
data {
int<lower=0> n; // number of subjects
int<lower=0> k1; // number of predictors for y1
int<lower=0> k2; // number of predictors for y2
real y1[n]; // continuous data
real y2[n]; // 0/1 binary data
matrix[n, k1] x1; // predictor variables for y1
matrix[n, k2] x2; // predictor variables for y2
}
transformed data{
int<lower=-1, upper=1> sign[n];
for (i in 1:n) {
if (y2[i]==1)
sign[i] <- 1;
else
sign[i] <- -1;
}
}
parameters {
real phi; // frank copula param
vector[k1] b1; // beta coefficients for y1
vector[k2] b2; // beta coefficients for y2
real<lower=0> abs_z[n]; // abs value of latent variable
real<lower=0> sigma1; // sd for y1's normal distribution
}
transformed parameters {
real v[n];
vector[n] mu1; // location for y1
vector[n] mu2; // location for z
for (i in 1:n) {
v[i] <- sign[i] * abs_z[i];
}
mu1 <- x1 * b1;
mu2 <- x2 * b2;
}
model {
b1 ~ normal(0, 100);
b2 ~ normal(0, 100);
phi ~ normal(0, 10);
increment_log_prob(copulapdf_log(y1, v, mu1, mu2, sigma1, phi, n));
}

Answer 1

If you need the latent parameter formulation, that's just like the Albert and Chib characterization of probit regression. What you need to do is declare the truncation in the parameters. There's an example in the manual chapter on regression involving multivariate probit that shows how it's done. Basically the positive values get a lower=0 constraint and the negative ones an upper=0 constraint and then you put both sets of parameters together into a z vector (if you actually need to put it back together).