Bayesian Poisson Regression with Gamma Prior Formulas Are there closed form formulas for the posterior and evidence of a Poisson-Gamma Bayesian regression model? I was not able to find anything that is accessible online. I am not sure for which model can we find such an expression. I was thinking of the following formulation (notation follow):
Likelihood is Poisson: $P(y|X, \theta) = \prod_{i=1}^m \frac{1}{y_i!}(x_i^t \theta)^{y_i} \exp(x_i^t \theta)$ and prior is Gamma: $P(\theta) = \prod_{j=1}^n \frac{\beta^{\alpha}}{\Gamma (\alpha)} \theta_j^{\alpha-1}\exp(-\beta\theta_j)$.
Notation: $m$ number of observations, $n$ number of parameters, $X \in \mathbb{R}^{m\times n}$ a design matrix with rows $x_i \in \mathbb{R}^n$, $\theta \in \mathbb{R}^{n}$ the parameter I would like to infer and $y \in \mathbb{R}^m$ data.
My questions:

*

*Is there a closed form expression for the posterior / evidence for the above model?

*Is there a closed form expression if we formulate the model as a GLM with (e.g.) with a $\log$ link?

*If there are no closed form formulas - is there a regression model with a conjugate prior, for positive data and parameters?

Freely available references are great, a complete derivation is even better.

It depends on what you would call a "closed form expression". If you expand the powers of $y_i$ into of sum of monomials (in $\theta_j$) then you can write the likelihood as a sum of (potentially very large number of) components, where each component has the form of a product of Gamma distributions. So the posterior is also a mixture distribution of independent Gamma components.
Your prior is technically a conjugate prior in this case because it belongs to the same family (a mixture with one component).
Practically however this will not be very useful unless the numbers are really small  - I believe the number of components will be roughly exponential in $\sum y_i$.
On the other hand it is rather straightforward to calculate the gradient  of the likelihood/posterior w.r.t $\theta_j$, so you can use it to find the MLE/MAP using gradient-descent optimization methods.
