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In Bayesian inference, the Gamma-Poisson model uses usually a Gamma($\alpha$,$\beta$) prior on the $\lambda$ parameter of the Poisson distribution.

Are there any rules for setting appropriate values ​​to these $\alpha$ and $\beta$ parameters of the Gamma prior?

The prior is normally set before seeing the data; so, what information would we need to have, in order to have an idea of ​​the values ​​to give to $\alpha$ and $\beta$?

Would we need to have at least a rough idea of what is the mean $\lambda$ value that should be observed in the Poisson experiment we plan?

Or would we need to have at least a rough idea of what is the maximum value to be observed in the Poisson experiment we plan?

Or any other kind of information, so as not to immediately blindly give $\alpha$ and $\beta$ nonsense values?

Any return appreciated.

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One obvious way is to base these parameter on previous studies via the meta-analytic predictive approach. E.g. this paper is an example, even if refers to the re-parameterization in terms of a (log-)mean rate and a (log-)dispersion parameter (you get a negative binomial distribution when you integrate out the gamma distributed random effects, as discussed in the paper). In the absence of previous studies, there's probably some experts (or perhaps some observational database you can analyze) that you can get a rough idea of the mean rate and how much it varies.

Even with this general approach, you still have different philosophies. Using the meta-analytic predictive approach as is or various modifications that want to allow for the possibility that something has change from the historical data (i.e. dynamically down-weight the prior information, if the current data contradicts it). There's also weakly informative priors, where you take what you guess and then allow for a lot of uncertainty (more than you probably think you need, but so much that anyone reasonable saying "but couldn't it be as much as..." is likely still covered).

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  • $\begingroup$ Thanks Björn, although I don’t have access to your paper… It seems to me that if I have an idea of the lambda, say lambda~20, that will give me an indication of the alpha/beta ratio to use, which is the average of the Gamma(alpha,beta) prior. But I could just as easily decide that alpha=20 and beta=1, or alpha=100 and beta=5. As here, beta is the inverse of the scale, the less is beta, the more will be the spread of the gamma distribution. So, favor a low value for beta in the absence of other information, is it what you mean by saying: “allow for a lot of uncertainty”? Am I right? $\endgroup$
    – Andrew
    Commented May 30 at 15:11
  • $\begingroup$ Discussed in the paper (but in terms of the more natural parameterization). If you email the corresponding author, they are allowed to send you a private copy of their final submitted version That is much more appropriate approach to get access than using something of questionable legality like sci-hub. $\endgroup$
    – Björn
    Commented May 30 at 17:32

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