Binomial-beta conjugate model¶
Posterior predictive distribution¶
If \(X|p \sim \mathcal{BI}(m, p)\) with \(p \sim \mathcal{B}(\alpha, \beta)\), then the posterior predictive probability density function, the expected value and variance of \(X\) are
\[f(x; m, \alpha, \beta) = \binom{m}{x}\frac{B(\alpha + x, m - x + \beta)}
{B(\alpha, \beta)}, \quad x = 0, 1, 2, \ldots.\]
\[\mathrm{E}[X] = m\frac{\alpha}{\alpha + \beta}, \quad \mathrm{Var}[X] =
\frac{m \alpha \beta (m + \alpha + \beta)}{(\alpha + \beta)^2
(\alpha + \beta + 1)}.\]
Proofs¶
Posterior predictive probability density function
\[\begin{split}f(x; m, \alpha, \beta) &= \int_0^1 \binom{m}{x}p^x (1-p)^{m - x}
\frac{p^{\alpha - 1} (1-p)^{\beta - 1}}{B(\alpha, \beta)} \mathop{dp}\\
&= \binom{m}{x}\frac{1}{B(\alpha, \beta)} \int_0^1 p^{\alpha + x - 1}
(1-p)^{\beta + m - x - 1} \mathop{dp} = \binom{m}{x}\frac{B(\alpha + x, m - x + \beta)}
{B(\alpha, \beta)},\end{split}\]
Note that this is the probability density function of the beta-binomial distribution, thus
\[X \sim \mathcal{BB}(m, \alpha, \beta),\]
see https://en.wikipedia.org/wiki/Beta-binomial_distribution.
Posterior predictive expected value
\[\mathrm{E}[X] = \mathrm{E}[\mathrm{E}[X | p]] = \mathrm{E}[mp] = m\frac{\alpha}{\alpha + \beta}.\]
Posterior predictive variance
Applying properties of the beta-binomial distribution, we obtain
\[\mathrm{Var}[X] = \frac{m \alpha \beta (m + \alpha + \beta)}{(\alpha + \beta)^2
(\alpha + \beta + 1)}.\]