Posterior predictive distribution
If \(X|p \sim \mathcal{G}(p)\) with \(p \sim \mathcal{B}(\alpha, \beta)\),
then the posterior predictive probability density function, the expected value
and variance of \(X\) are
\[f(x; \alpha, \beta) = \frac{B(\alpha + 1, \beta + x - 1)}{B(\alpha, \beta)},
\quad x = 0, 1, 2, \ldots.\]
\[\mathrm{E}[X] = \frac{\alpha + \beta - 1}{\alpha - 1}, \quad \mathrm{Var}[X] =
\frac{\beta (\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)},\]
where \(\mathrm{E}[X]\) is defined for \(\alpha > 1\) and
\(\mathrm{Var}[X]\) is defined for \(\alpha > 2\).
Proofs
Posterior predictive probability density function
\[\begin{split}f(x; \alpha, \beta) &= \int_0^1 (1 - p)^{x - 1} p \frac{p^{\alpha - 1} (1-p)^{\beta - 1}}{B(\alpha, \beta)} \mathop{dp}\\
&= \frac{1}{B(\alpha, \beta)} \int_0^1 p^{\alpha} (1-p)^{\beta + x - 2} \mathop{dp}
= \frac{B(\alpha + 1, \beta + x - 1)}{B(\alpha, \beta)}.\end{split}\]
Posterior predictive expected value
\[\mathrm{E}[X] = \mathrm{E}[\mathrm{E}[X | p]] = \mathrm{E}\left[\frac{1}{p}\right] = \frac{\alpha + \beta - 1}{\alpha - 1}.\]
Note that,
\[\mathrm{E}\left[\frac{1}{p}\right] = \int_0^1 \frac{1}{p} \frac{p^{\alpha - 1} (1-p)^{\beta - 1}}{B(\alpha, \beta)} \mathop{dp} = \int_0^1 \frac{p^{\alpha - 2} (1-p)^{\beta - 1}}{B(\alpha - 1, \beta)} \frac{\alpha + \beta - 1}{\alpha - 1} \mathop{dp} = \frac{\alpha + \beta - 1}{\alpha - 1},\]
where we use the property of the beta function: \(B(a -1, b) = \frac{a + b - 1}{a - 1} B(a, b)\).
Posterior predictive variance
\[\mathrm{Var}[X] = \mathrm{E}[X^2] - \mathrm{E}[X]^2 = \frac{\beta (\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)}.\]
Similarly, we have that
\[\mathrm{E}[X^2] = \frac{\alpha + \beta - 1}{\alpha - 1}\frac{\alpha + \beta - 2}{\alpha - 2}\]
and
\[\mathrm{Var}[X] = \frac{\alpha + \beta - 1}{\alpha - 1}\frac{\alpha + \beta - 2}{\alpha - 2} - \left(\frac{\alpha + \beta - 1}{\alpha - 1}\right)^2 = \frac{\beta (\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)}.\]
Note
The same can be proven applying properties of the beta and beta prime distribution. Given that if \(X \sim \mathcal{B}(a, b) \rightarrow \frac{X}{1 - X} \sim \beta'(a, b)\) and if \(Y \sim \beta'(a, b) \rightarrow \frac{1}{Y} \sim \beta'(b, a)\), we get that if \(X \sim \mathcal{B}(a, b)\) then \(\frac{1 - X}{X} \sim \beta'(b, a)\), thus
\[\mathrm{E}\left[\frac{1}{X}\right] = \frac{\beta}{\alpha - 1} + 1 = \frac{\alpha + \beta - 1}{\alpha - 1},\]
and
\[\mathrm{Var}\left[\frac{1}{X}\right] = \frac{\beta (\alpha + \beta - 1)}{(\alpha - 1)^2 (\alpha - 2)}.\]