Hello everyone,

I am trying to fit the model in the picture on simulated data. I used the same model for generating data except that the covariance matrix was an identity matrix, and the beta bars were standard normal random variables. I have tried both centered and non-centered parameterization; I encountered with following problems.

Firstly, there were some divergent samples in the centered model. Before the switch to the non-centered version, I tried the handle the problem by increasing the parameter of LKJ distribution and decreasing the step size. It worked and posterior beta distributions almost centered around the generated beta values. But it was not the same for beta bars. Their posterior distributions are far away from generated beta bars. I wonder how I can handle this problem.

Secondly, I tried non-centered parameterization. It worked, but run time was ten times higher than centered parameterization for the same number of iterations. Is that normal? I thought run time would decrease.

Also, chains did not mix, and almost all iterations exceeded the max_treedepth even for max_treedepth=13.

I shared the data simulation code and centered and non-centered models in this order. I would be very happy if anyone could help me.

### data_generator

```
data{
/* Dimensions <em>/
int<lower=0> M;// number of varaibles
int<lower=0> N;// number of observations
/</em> Design Matrix <em>/
matrix[N,M] X;
/</em> offset <em>/
vector<lower=0>[N] offset;
/</em> Hyperparameters*/
vector<lower=0>[M] sigma; // diagonal values of the covariance matrix, equals to ones matrix.
}
generated quantities{
vector [M] betabar; // mean vector of multivariate normal distribution
vector[M] beta; // regression coefficients
real<lower=0> a; // prior for relative risk
vector<lower=0>[N] wi; //relative risk
vector[N] xi; // mean of poisson distribution
int<lower=0> y_rep[N]; // response variable
for(i in 1:M){betabar[i]=normal_rng(0,1);}
beta=multi_normal_rng(betabar,diag_matrix(sigma));
a=gamma_rng(1,0.01);
for(j in 1:N){wi[j]=gamma_rng(a,a);}
xi=exp(X*beta+log(offset)+log(wi));
for(z in 1:N){y_rep[z]=poisson_rng(xi[z]);
}
}
```

### centered

```
data{
/* Dimensions <em>/
int<lower=0> M;// number of varaibles
int<lower=0> N;// number of observations
/</em> Design Matrix <em>/
matrix[N,M] X;
/</em> offset <em>/
vector<lower=0>[N] offset;
/</em> Outcome <em>/
int<lower=0> y[N];
/</em> Hyperparameters*/
vector<lower=0>[M] sigma; //diagonal values of the covariance matrix, equals to 10s vector
}
parameters {
vector[M] beta; // coeffiencts in linear equation
real<lower=0> a; // prior for relative risk
vector<lower=0>[N] wi; // relative risk
vector [M] betabar; // mean vector of multivariate normal distribution
cholesky_factor_corr[M] Lcorr;// cholesky factor (L_u matrix for R)
}
transformed parameters{
vector[N] xi; // mean vector of poisson distribution
corr_matrix[M] R; // correlation matrix
cov_matrix[M] Sigma; // covariance matrix
xi=exp(X*betabar+log(offset)+log(wi));
R = multiply_lower_tri_self_transpose(Lcorr);
Sigma = quad_form_diag(R, sigma);
}
model {
target+=normal_lpdf(betabar|0,1);
target+=lkj_corr_cholesky_lpdf(Lcorr|4);
target+=multi_normal_lpdf(beta | betabar,Sigma);
target+=gamma_lpdf(a|1,0.01);
target+=gamma_lpdf(wi|a,a);
target+=poisson_lpmf(y|xi);
}
```

### noncentered

```
data{
/* Dimensions <em>/
int<lower=0> M;// number of varaibles
int<lower=0> N;// number of observations
/</em> Design Matrix <em>/
matrix[N,M] X;
/</em> offset <em>/
vector<lower=0>[N] offset;
/</em> Outcome <em>/
int<lower=0> y[N];
/</em> Hyperparameters*/
vector<lower=0>[M] sigma; //diagonal values of the covariance matrix, equals to 10s vector
}
parameters {
vector[M] delta; // std normal variable that multiplied with covariance matrix
vector[M] delta2; // std normal variable that multiplied with mean
real<lower=0> a;
vector<lower=0>[N] wi; // relative risk
cholesky_factor_corr[M] Lcorr;// cholesky factor (L_u matrix for R)
}
transformed parameters{
corr_matrix[M] R; // correlation matrix
vector[M] beta; // regression coefficients
vector[N] xi; // mean of poisson
R = multiply_lower_tri_self_transpose(Lcorr);
beta=delta2*10+quad_form_diag(R, sigma) <em>delta;
xi=exp(X</em> beta+log(offset)+log(wi));
}
model {
target+=normal_lpdf(delta2|0,1);
target+=lkj_corr_cholesky_lpdf(Lcorr|2);
target+=normal_lpdf(delta|0,1);
target+=gamma_lpdf(a|1,0.01);
target+=gamma_lpdf(wi|a,a);
target+=poisson_lpmf(y|xi);
}
```

Edited by @jsocolar for syntax highlighting.