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I have the following

$$x\sim NegativeBinomial(r,\theta)$$

$$\theta \sim beta(a,b)$$

and proved that posterior is

$$\theta|x \sim beta(\sum x_{i} +a,nr+b)$$

I would like to find the posterior predictive distribution so I calculate $$f(y|x)=\int_{0}^{1} f(y|\theta)p(\theta|x)d\theta$$ which is equal to $$\begin{pmatrix} y+r-1\\ y \end{pmatrix}\frac{B(\sum x_{i}+a+y,r+nr+b)}{B(\sum x_{i}+a,nr+b)}$$

and I don't truly know which distribution is this , but it seems similar to Beta-Binomial.

Any idea or help would be great.

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There is a distribution called a "Beta Negative Binomial" distribution, which properties are described on Wikipedia, which corresponds to this case. Its main properties are described on this page, but I wonder how useful it is to give it a name.

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