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I need to update the failure rate (given as deterministic) based on new rate of failure about the same system (it is a deterministic one too). I read about conjugate priors and Gamma distribution as a conjugate for the Poisson process.

Also, I can equate the mean value of Gamma dist. ($\beta/\alpha$) to the new rate (as a mean value) but I do not have any other information such as standard deviation, Coefficient of Variation, 90th percentile value,...etc. Is there a magic way to manipulate that and find parameters for the prior Gamma hence I get the posterior which Gamma too?

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    $\begingroup$ Your question is not clear. Could you please edit the text and add a bit more context? $\endgroup$ – user28 Sep 15 '10 at 3:25
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    $\begingroup$ ... and maybe a better topic? $\endgroup$ – user88 Oct 5 '10 at 10:52
  • $\begingroup$ I attempted to make it a better title; feel free to change it to something more appropriate $\endgroup$ – Jeromy Anglim Nov 4 '10 at 6:25
  • $\begingroup$ What parameterization are you using for your Gamma? $\endgroup$ – Glen_b Aug 2 '13 at 5:12
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A Gamma distribution is not a conjugate prior for a Gamma distribution. There is a conjugate prior for the Gamma distribution developed by Miller (1980) whose details you can find on Wikipedia and also in the pdf linked in footnote 6. Checkout section 3.2 on page 25 of this paper, there is a prior with four parameters: p, q, r, & s

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I believe M. Tibbit's answer refers to the general case of a gamma with unknown shape and scale. If the shape α is known and the sampling distribution for x is gamma(α, β) and the prior distribution on β is gamma(α0, β0), the posterior distribution for β is gamma(α0 + nα, β0 + Σxi). See this diagram and the references at the bottom.

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  • $\begingroup$ Couldn't you just simulate the posterior Gamma distribution from the full conditionals defined by the conjugate priors of the Gamma alpha and beta respectively? Thanks. $\endgroup$ – Brash Equilibrium Sep 4 '13 at 1:20

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