Prior of a product Let say, we have N servers. Every server is in production for a different amount of years. For every server, we know how many times this server crashed, the total for all the years. A number of crashes can be modelled by Poisson distribution.
$X_i \sim  Poisson(\lambda_i) $ let's define $\lambda_i = \theta_it_i$ where $\theta_i$ is number of failures per year and $t_i$ is number of years.
I can assume that prior for $\theta_i$ is the same for all servers. They all are similar pieces of equipment. Can I make the same assumption for $\lambda_i$? This multiplication makes me wonder. Should I consider $t_i$ as a random variable too? Multiplication of a random variable on a constant shouldn't change our assumption on a prior, but multiplication of two random variables could change it.
I'm trying to build a model looking on example but wonder if my setup is the same as one in the example.
 A: A simple model is:
$$X_i\sim\text{Poisson}\big(\exp(\alpha + \theta t_i)\big)$$
where $X_i$ and $t_i$ are treated as observed variables (or data), and $\alpha$ and $\theta$ are unobserved variables (or parameters to be estimated).
This model assumes:


*

*that all the servers functioned similarly, and

*the number of crashes grew exponentially with years of production.


$e^\alpha$ is the expected number of crashes when the server is 0 years. And $e^\theta$ will give you % change in crashes for each additional year of production.
And you only need to place priors on the constants, $\alpha$ and $\theta$.

A different model is:
$$X_i\sim\text{Poisson}\big(\exp(\alpha_i + \ln{(t_i)}\big)$$
This model assumes:


*

*the longer a server runs, the more it fails in a constant way, and

*the failure rates differs across servers as captured by $\alpha_i$.


This is likely a stronger model given your comments. And $e^{(\alpha_i)}$ will return the expected failure rate (failures/year) for each server $i$.
There is only need to place a prior on $\alpha_i$, $\alpha_i$ would be what is known as an observation-level random effect. And is similar in spirit to a negative-binomial model.

For fully observed variables such as $t_i$, there is no need to place a prior on them, since they are not outcomes.
