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An observation $x$ is taken from a negative binomial distribution $X \sim \text{Negative-Binomial}(k,\pi)$. The parameter, $\pi$, is allocated a beta prior $\pi \sim (\alpha,\beta)$.


My attempt:

\begin{align} P(\pi \mid x) &\propto p(\pi)L(\pi) \\ P(\pi \mid x) &\propto \pi^{\alpha-1}(1-\pi)^{\beta-1} \pi^k (1-\pi)^x \\ P(\pi \mid x) &\propto \pi^{\alpha+k-1}(1-\pi)^{\beta+x-1} \end{align}

So the posterior is ${\rm Beta}(\alpha+k,\beta+x)$

Am I correct?

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Yes, that is correct. However, be aware that you can write the negative binomial in a different manner, so it may look different depending on which version you use. Looks like you're using the distribution defined by Wikipedia (https://en.wikipedia.org/wiki/Negative_binomial_distribution).

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