I'm trying to derive the posterior distribution for the failure rate (lambda) of a process with poisson distribution.
I have tried the use of an improper uniform distribution on lambda by letting the max tend to infinity. My posterior then just has the Poisson like form again (but as a likelihood function of continuous variable lambda).
I've also tried using various expert opinions as confidence intervals - ranging from what I consider optimistic to very conservative - and approximating by a gamma distribution.
From these approaches I get very similar results, but I'm not sure I have a sufficiently good argument to defend my use of priors.
So, I'm concerned my approach is not robust and on reading about Jeffrey's prior I'm wondering if I should learn more about it and how to use it.