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Bayesian computation of joint density, marginal posterior

  1. Dec 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi, so I am having trouble understanding the steps to get to certain densities.

    For example, suppose i have data y1,...,yJ ~ Binomial (njj)

    We also have that θj ~ Beta (α,β)

    Now our joint posterior is:

    p(β,α,θ|y) ~ p(α,β) ∏ ([itex]\Gamma[/itex](α+β) / [itex]\Gamma[/itex](α)[itex]\Gamma[/itex](β)) θjα+yj-1(1-θj)β-1+nj-yj

    Next, we find the posterior of θ given (α,β,y), the "joint density".

    I do not understand this step.

    Here is what it is suppose to be:

    p(θ|α,β,y)= ∏ ([itex]\Gamma[/itex](α+β+nj) / [itex]\Gamma[/itex](α+yj)[itex]\Gamma[/itex](β+nj-yJ)) θjα+yj-1(1-θj)β-1+nj-yj

    How did they get this? In my class and from sources I have read it says you can obtain this by "dropping the terms that are not dependent on θ"... but I do not see where the nj and + yj, etc. came from.

    After this step we wish to find the marginal posterior of (α,β), p(α,β|y) ~ p(β,α,θ|y)/p(θ|α,β,y.

    Is there another way to do this as well? I know it can also be written as p(α,β|y)~ g(α,β) ∏ f(yj|α,β).

    But then, if done this way, what is f(yj|α,β).

    In another example:

    http://www-stat.wharton.upenn.edu/~edgeorge/Research_papers/GZpriors.pdf

    On the 6th page, it says, integrating out θ1,...,θp, we get...
    How did they integrate out exactly?


    I realize these are questions I should know from calculus but I just don't understand the steps to getting these results.

    Any help is appreciated!
     
  2. jcsd
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