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Math Help - Quick Bayesian Probability Question

  1. #1
    Guy
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    Quick Bayesian Probability Question

    Hey all,

    I need to check my understanding of this. Let \theta have some prior on it and observe X_1, ..., X_n, X_{n + 1}, which are independent conditional on \theta, such that EX_i|\theta = \theta. I'm asked to find the posterior mean of X_{n + 1} given X_1, ..., X_n. I'll spare the extra details.

    My reasoning is

    EX_{n + 1} | X_1, ..., X_n = E\left(EX_{n + 1} | \theta, X_1, ..., X_n \right) | X_1, ..., X_n
    = E\left(EX_{n + 1} | \theta \right) | X_1, ..., X_n (from conditional independnce)
    = E\theta | X_1, ..., X_n (since EX|theta = theta)

    So the posterior mean of X_{n + 1} is the posterior mean of \theta. This seems a little off to me for some reason I can't explain. Does this look okay? It seems like this is working out to an expectation over the joint distribution of \theta|X_1, ..., X_n and X_{n + 1} | \theta but I guess I'm not sure if that's what is meant by asking for the posterior mean.

    More broadly, if I'm asked for anything posterior, should I get something that is free of theta?
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  2. #2
    Guy
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    Nothing? After thinking about it I think this is fine, but I'd like to get some confirmation that I'm not making some fundamental mistake.
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