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enter image description hereI'm reading about Beta-Binomial. Assume a r.v. x~Bin(n, pi). (pi) in itself, is a r.v. drawn from Beta(alpha, beta) I was thinking, may be I can take the Expected value of pi (which is alpha/(alpha+beta) ) and substitute it in the original Binomial distribution and I'm done, I got P(x). My question is: what is the difference between the distribution that I got, and Beta-Binomial?

PS: Beta-Binomial finds the joint distribution then marginalizes (pi) to find the predictive distribution P(x).

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Your distribution seems to be a binomial distribution with parameters $n$ and $\frac{\alpha}{\alpha+\beta}$. This has support on $\{0,1,2,\ldots,n\}$ and mean $\frac{n\alpha}{\alpha+\beta}$ and variance $\frac{n\alpha\beta}{(\alpha+\beta)^2}$

The original Beta-binomial distribution also has support on $\{0,1,2,\ldots,n\}$ and mean $\frac{n\alpha}{\alpha+\beta}$, but its variance is $\frac{n\alpha\beta}{(\alpha+\beta)^2} \frac{\alpha+\beta+n}{\alpha+\beta+1}$, which is rather larger

Essentially, more extreme values near $0$ and $n$ are more likely in the original Beta-binomial distribution than in your distribution

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