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Thread: Compound Distribution Hyper-Parameters

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    Question Compound Distribution Hyper-Parameters




    Hi,
    If a compound distribution is made up of a gamma dist with parameters (a,b), and b is gamma-distributed so that its parameters are (c,d), such that I have a compound distribution -> f(a,c,d) (i.e. b has been marginalised out).
    My question is:
    If I am using the log-likelihood of the compound dist and running optimization code on this function to estimate the parameters for a,c,d then are the values I get for a,c,d equivalent to the values I would get for a,c,d if I solved for (a,b) and (c,d) using their original gamma functions? I am asking because the compound function does not solve using standard optimization algorithms, but the individual gamma functions do, so I was wondering why one would use the compound dist in the first place? I was not sure how to go about proving this mathematically.
    Thanks for your responses!

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    Re: Compound Distribution Hyper-Parameters


    As you said, once you marginalized out the parameter b, then you left with a compound distribution with 3 parameters and this is your model.

    I am not quite sure if I get the meaning in the latter part of question. You do not observe any realization (data) of b (isn't it?) so I do not see how you estimate c, d based on the original gamma distribution.

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