I have been trying to get this model to work for some time now, but it seems that in order for the model to return estimates that are close to my true generating ones, I have to set 15+ items per item type. I was told that if it requires this many items per type to work, then something is wrong with my model but I am not sure how to solve something like that. I have tried sum to 0 constraints but that does not seem to have a significant effect on the estimates.

The data that works well may be for instance, 200 people, 60 items, and 3 item types, such that there are 20 items of each type. However, if i do 200 people, 18 items, 3 item types, such that there are 6 items per type, there is too much error between the estimated and true parameters for beta and gamma. What can I do to solve this (if anything)?

```
data {
int<lower=1> I; // number of observations
int<lower=1> J; // number of items
int<lower=1> T; // total number of item types
int<lower=1> N; // total number of counts
int<lower=0> Y[N]; // observed counts
int<lower=1, upper=I> person_id[N]; // person id for each observation
int<lower=1, upper=J> item_id[N]; // item id for each observation
int<lower=1, upper=T> type[N]; // type id for each item type
}
parameters {
vector[I] theta;
vector[T] gamma;
vector[J] beta;
}
model {
// Priors
theta ~ normal(0,1);
beta ~ normal(0,1);
gamma ~ normal(0,1);
//sum(beta) ~ normal(0,.001*J);
for (n in 1:N) {
real lambda = exp(theta[person_id[n]] + beta[item_id[n]] + gamma[type[n]]);
target += poisson_lpmf(Y[n] | lambda);
}
}
```