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I'm referring to page 22 of this white paper.

On page 22, it says the following: given that

  • $s_i \sim \text{Exp}(\theta), i = 1,..,c$

  • $\theta \sim\text{Gamma}(k, \Theta)$,

Then the posterior distribution for $\theta$ given the data is

$ \theta | s \sim \text{Gamma}(k + c, \frac{\Theta}{1 + \Theta cs}) $

where $s$ is the mean of the data, $s = \frac{\sum_{i=1}^{c}s_i}{n}$

But this is inconsistent with my understanding of the conjugate Gamma-Exponential system, where after updating the posterior distribution is as follows:

$ \theta | s \sim \text{Gamma}(k + c, k + cs)$

Am I missing something super obvious here? Thank you!

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Two things.

  1. I think you have a typo: you're probably expecting the posterior to be $\text{Gamma}(k+c, \Theta + cs)$.

  2. Note that the author of the paper is using the alternative parametrization of the exponential distribution, which has pdf $f(x) = \frac{1}{\theta}e^{-x/\theta}$. If you're looking at a derivation that uses the standard characterization, you might have some flipped fractions. In particular, try looking at the reciprocal of $\frac{\Theta}{1 + \Theta cs}$.

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