Bayesian computation of joint density, marginal posterior

[missavvy]

Homework Statement

Hi, so I am having trouble understanding the steps to get to certain densities.

For example, suppose i have data y₁,...,y_J ~ Binomial (n_j,θ_j)

We also have that θ_j ~ Beta (α,β)

Now our joint posterior is:

p(β,α,θ|y) ~ p(α,β) ∏ ($\Gamma$(α+β) / $\Gamma$(α)$\Gamma$(β)) θ_j^α+y_j-1(1-θ_j)^β-1+n_j-y_j

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+n_j) / $\Gamma$(α+y_j)$\Gamma$(β+n_j-y_J)) θ_j^α+y_j-1(1-θ_j)^β-1+n_j-y_j

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 n_j and + y_j, 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(y_j|α,β).

But then, if done this way, what is f(y_j|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,...,θ_p, 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!

 

[lippyka]

Homework Equationsp(α,β|y)~ g(α,β) ∏ f(yj|α,β)The Attempt at a SolutionFor the first part, I think the term nj that appears in the joint density is due to the fact that θj is a Binomial random variable with parameter nj. The terms + yj arise because the Binomial distribution is defined such that the probability of success is given by θj and the probability of failure is 1-θj and the number of trials (in this case successes) is yj. For the second part, I think that integrating out θ1,...,θp just means that you are taking the integral of the joint density p(α,β,θ|y) over all possible values of θ1,...,θp. This will give you the marginal posterior of (α,β) which is what we are looking for.