# Hierarchical Poisson regression with a normal group level

I have a group of items. I model each with the Poisson regression (counts), but one of the regression coefficients is modelled on a group level using the normal distribution.

Assuming there are $$m$$ items, and for each item, there are $$n$$ data points. The observation $$i$$ for the item $$j$$ is denoted as $$y_{ji}$$, the features are $$x_{ji}$$, the $$j$$'s item distribution parameters independent from other items are $$\theta_{j}$$, and the parameter drawn from a common group is $$\phi_j$$. Then the probability density of the item is:

$$\prod_{i=1}^nPois(y_{ji}|x_{ji}, \theta_j, \phi_j)$$

Next, each coefficient $$\phi_j$$ comes from the common normal distribution:

$$p(\phi_j| a, b) = \frac{b}{2\pi}e^{-\frac{b(\phi_j - a)}{2}}$$

where $$a$$ is the mean, and $$b$$ is the precision.

Thus, the full hierarchical model can be written as:

$$p(Y | X) = \prod_{j = 1}^m\bigg\{\frac{b}{2\pi}e^{-\frac{b(\phi_j - a)}{2}}\prod_{i=1}^nPois(y_{ji}|x_{ji}, \theta_j, \phi_j)\bigg\}$$

I need to find the parameters $$\theta_j, \phi_j, a, b$$. I use MLE for this task.

In order to optimize the whole thing, I use the Newton method and coordinate ascent. It works by optimizing each item in turn, and then estimate the normal group-level parameters $$a$$ and $$b$$ analytically.

However, the log-likelihood optimization somehow sets the parameters $$\phi_j$$ for ALL items to THE SAME VALUE. As a result, the precision tends to infinity, and MLE fails! What process can cause this model to set all coefficients $$\phi_j$$ to the same value? Other parameters, $$\theta_j$$, and the group mean $$a$$ seem to be optimized correctly (look reasonable to me).