# Bayesian computation of joint density, marginal posterior

1. Dec 4, 2012

### missavvy

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) ∏ ($\Gamma$(α+β) / $\Gamma$(α)$\Gamma$(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)= ∏ ($\Gamma$(α+β+nj) / $\Gamma$(α+yj)$\Gamma$(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!

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