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I'd like to choose an uninformative prior for the scale and shape parameters of the Gamma distribution. Any help and suggestions will be appreciated.

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    $\begingroup$ There is no completely 'flat' gamma prior among proper distributions (with densities integrating to unity). The closest you can get is to use very small shape and rate parameters. (Then, for example, with a Poisson likelihood, the posterior depends very largely on the likelihood and hardly at all on the prior.) $\endgroup$ – BruceET Mar 4 at 2:33
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    $\begingroup$ The use of "uninformative" or "noninformative" should be abandoned in favour of "reference" as there is no such thing (see my answer there). One approach to reference priors is Jeffreys', which alas involves a digamma function for the shape parameter, unless one restricts the shape to be an integer. $\endgroup$ – Xi'an Mar 4 at 8:36

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