# Negative binomial-beta conjugate model¶

## Posterior predictive distribution¶

If $$X|p \sim \mathcal{NB}(r, p)$$ with $$p \sim \mathcal{B}(\alpha, \beta)$$, then the posterior predictive probability density function, the expected value and variance of $$X$$ are

$f(x; r, \alpha, \beta) = \binom{x + r - 1}{r - 1}\frac{B(\alpha + r, \beta + x)}{B(\alpha, \beta)}, \quad x = 0, 1, 2, \ldots.$
$\mathrm{E}[X] = r\frac{\beta}{\alpha - 1}, \quad \mathrm{Var}[X] = \frac{r \beta (\alpha + r - 1)(\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)},$

where $$\mathrm{E}[X]$$ is defined for $$\alpha > 1$$ and $$\mathrm{Var}[X]$$ is defined for $$\alpha > 2$$.

## Proofs¶

Posterior predictive probability density function

$\begin{split}f(x; r, \alpha, \beta) &= \int_0^1 \binom{x + r - 1}{r - 1}p^r (1-p)^x \frac{p^{\alpha - 1} (1-p)^{\beta - 1}}{B(\alpha, \beta)} \mathop{dp}\\ &= \binom{x + r - 1}{r - 1}\frac{1}{B(\alpha, \beta)} \int_0^1 p^{\alpha + r - 1} (1-p)^{\beta + x - 1} \mathop{dt} = \binom{x + r - 1}{r - 1}\frac{B(\alpha + r, \beta + x)}{B(\alpha, \beta)}.\end{split}$

Posterior predictive expected value

$\mathrm{E}[X] = \mathrm{E}[\mathrm{E}[X | p]] = r \mathrm{E} \left[\frac{1 - p}{p}\right] = r\frac{\beta}{\alpha - 1}.$

Posterior predictive variance

$\begin{split}\mathrm{Var}[X] &= \mathrm{E}[\mathrm{Var}[X | p]] + \mathrm{Var}[\mathrm{E}[X | p]]\\ &= \mathrm{E}\left[r\frac{1 - p}{p^2}\right] + \mathrm{Var}\left[r\frac{1-p}{p}\right] = r \left(\mathrm{E}\left[\frac{1}{p^2}\right] - \mathrm{E}\left[\frac{1}{p}\right]\right) + r^2 \mathrm{Var}\left[\frac{1}{p}\right]\\ &= r \frac{\alpha + \beta - 1}{\alpha - 1}\left(\frac{\alpha + \beta - 2}{\alpha - 2} - 1\right) + r^2\frac{\beta^2 (\alpha + \beta - 1)}{(\alpha - 1)^2(\alpha - 2)} = \frac{r \beta (\alpha + r - 1)(\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)}.\end{split}$

Note

Recall the calculation of $$\mathrm{E}\left[\frac{1}{p}\right]$$, $$\mathrm{E}\left[\frac{1}{p^2}\right]$$ and $$\mathrm{Var}\left[\frac{1}{p}\right]$$ derived for the geometric-beta model.