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

For example, suppose i have data y_{1},...,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) ∏ ([itex]\Gamma[/itex](α+β) / [itex]\Gamma[/itex](α)[itex]\Gamma[/itex](β)) θ_{j}^{α+yj-1}(1-θ_{j})^{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+n_{j}) / [itex]\Gamma[/itex](α+y_{j})[itex]\Gamma[/itex](β+n_{j}-y_{J})) θ_{j}^{α+yj-1}(1-θ_{j})^{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 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 θ_{1},...,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!

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# Homework Help: Bayesian computation of joint density, marginal posterior

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