Compound Poisson Gamma

Modeling
1

Hi everyone,

I need some help including discrete parameters in the model. I found some posts on Compound Poisson Gamma models but I couldn’t really follow. How can we implement this model in Stan?
In my case, the scale parameter of the Gamma distribution is N_j, which is Poisson distributed itself. I believe that I can’t write it as I am doing cause HMC is not optimized for discrete parameters. Since N_j is unbounded I couldn’t find a solution in the manual.
Is it possible to implement a continuous Poisson distribution?
f(x,\lambda)=a(\lambda)*\frac{e^{-\lambda}\lambda^x}{\Gamma(x+1)}
Here is my code:

data {
 int<lower=0> N;  //number of observations
 vector<lower=0>[N] S_j;  //Aggregate loss by j (response variable) 
 vector<lower=0>[N] N_j; //Aggregate number of observations by j  
 int<lower=0> K;  //number of states
 int<lower=0> M; //number of states-1
 matrix[N,K] design_matrix_1;
 matrix[K,M] design_matrix_2;
}
transformed data {
  }
parameters {
// vector<lower=0>[K] lambda_j;
 vector<lower=0>[K] theta_j;
 real beta_0;
 vector[M] beta;
 vector<lower=0>[M] alpha_j;
 real<lower=0> alpha_0;
 real<lower=0> sigma_mu;
 real<lower=0> sigma_alpha;
}
transformed parameters {
  
  vector[N] theta_index = design_matrix_1 * theta_j;
 // vector[N] lambda_index = design_matrix_1 * lambda_j;
  
  vector[K] mu_index = beta_0 + design_matrix_2 * beta * sigma_mu;
  vector[K] alpha_index = alpha_0 + design_matrix_2 * alpha_j * sigma_alpha;
 
}
model { 
target += cauchy_lpdf(alpha_0 | 0, 5);
target += cauchy_lpdf(sigma_mu | 0, 1);
target += cauchy_lpdf(sigma_alpha | 0, 25);
target += normal_lpdf(alpha_j | 0, 1);
target += normal_lpdf(beta | 0, 1);
target += normal_lpdf(beta_0 | -9, 1);

target += gamma_lpdf(theta_j | alpha_index, alpha_index .* exp(-mu_index));

//target += poisson_lpmf(N_j | lambda_index);

target += gamma_lpdf(S_j | N_j , theta_index);

}
generated quantities{
   real yrep[N]= gamma_rng( N_j , theta_index);
}

instead of

1 Like
2

Hi, would you be able to write your full model, from data to priors and likelihoods? It might be easier to me to help.

3

Here’s a previous discussion:

Does that help?

4

I checked that already @emiruz, but I found out that I was misinterpreting my problem and there is no need of applying such a model.
I found people working in an approximation for the Tweedie, however it is an on-going project and there is no official package yet. I might go back to that once I finish this model.

Thank you anyways!

1 Like