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#1




Inverse Exponential / Gamma Conjugate Prior?
Suppose that loss amounts have an inverse exponential distribution with mode (theta/2), and the prior of theta is a Gamma distribution with alpha = 2 and theta = 5.
If two losses are observed at 10, 20...then what are the shortcut formulas that get you to the posterior distribution and the predictive distribution? First principles for these seem to take a while with a larger margin for computation mistakes...hoping there's some shortcut formulas for this. 
#2




Do you have the answer for the posterior alpha, and shape parameter (I wouldn't call it theta, as theta is a random variable)?

#3




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You can figure this out from other things that you probably know. I'll restate the problem this way: X  T is Inverse Exponential with parameter T, while T is a GammaRV with parameters alpha and theta. For a fixed but unknown value of T, you observe n values of X as x1, x2, ..., xn. Find the posterior distribution of T and the predictive distribution of X. First, let X' = 1 / X, with values x1' = 1 / x1, x2' = 1 / x2, ..., xn' = 1 / xn. You should know that X'  T is Exponential with mean 1 / T. But you probably know that if X'  T is Exponential with mean 1 / T and T is a GammaRV with parameters alpha and theta, and you observe values of X' that are x1', x2', ..., xn', then the posterior distribution of T is a GammaRV with parameters alpha* and theta* where alpha* = alpha + 1 + 1 + ... + 1 = alpha + n and 1 / theta* = 1 / theta + x1' + x2' + ... + xn'. So that answers one of your questions. You might also know that this makes the (predictive) distribution of X' a 2ParetoRV with parameters alpha*' = alpha* and theta*' = 1 / theta*. From that you should know that that makes the predictive distribution of X = 1 / X' an Inverse Pareto with parameters tau*'' and theta*'', where tau*''= alpha*' and theta*'' = 1 / theta*'. So that answers your second question. It may well be simpler to plod through the standard Bayesian analyses instead, but that requires you to recognize some of these unusual distributions from their density functions. Looks to me like a good problem to skip on the examfocus on what is commonly tested, not on what might conceivably be tested. For most people, trying to prepare for everything is in fact preparing to fail.
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