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In the Albert book on Bayesian computation with R, exercise 4.8.5 (p.83), it is suggested to use

$$ p(a, b) \sim (a \times b)^{-2} $$

as the non-informative prior for the Poisson/Gamma model:

$$ f(y\,|\, a,b) = \frac{\Gamma(y+a)}{\Gamma(a) \times y!}\frac{b^{a}}{(b+1)^{y+a}} $$

It does work, but could anyone give me an intuition on that - in other words, why is this prior non-informative?

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    $\begingroup$ What makes you say it does "work" ? $\endgroup$
    – Stéphane Laurent
    Mar 31, 2014 at 18:13
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    $\begingroup$ For both $a$ and $b$ it has a spike at $0$ to allow for very small values and has a Cauchy tail to allow for very large values. The behavior at $0$ makes it improper, but if we bounded away from $0$ it would still have infinite mean. These are somewhat desirable properties for a "noninformative" prior to have. $\endgroup$
    – guy
    Mar 31, 2014 at 18:14

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