Prior for Bayesian Inference on Failure Rate in Poisson Distribution

Question

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.

Any advice?

Answer 1

"I have tried the use of an improper uniform distribution on lambda by letting the max tend to infiniti. My posterior is then just the Poisson again." This cannot be possible since the posterior of lambda is continuous. You have to distinguish between the sampling model and the distributions of the parameters in a Bayesian framework.

The Gamma prior is a conjugate prior for the Poisson distribution. This is a reasonable choice is you have reliable prior information.

An alternative is the use of Reference (Jeffreys) priors which does not require the elicitation of hyperparameters. In addition, the posterior inferences obtained with this sort of prior are typically similar to those obtained with the classical approach.

See: http://www.uri.edu/artsci/ecn/burkett/545lect5.pdf

You can use a flat prior on the parameter $\lambda$. However, given that this is not a proper prior, you have to check conditions for the existence of the posterior distribution corresponding to this prior. This is,

$$\int_0^{\infty} L(\lambda)d\lambda<\infty,$$

where $L$ is the likelihood function, up to a proportionality constant. In this case

$$L(\lambda)\propto \lambda ^{n\bar{x}}e^{-n\lambda},$$

where $n$ is the sample size and $\bar{x}$ is the sample mean. Then

$$\int_0^{\infty} L(\lambda)d\lambda \propto \bar{x} n^{-n \bar{x}} \Gamma (n \bar{x}).\,\,\, if \,\,\, \bar{x}\neq 0,$$

and $1/n$ if $\bar{x}=0$. Therefore, the posterior is proper.