# Bayesian applications

The *generalized Beta distribution* \(\beta_\tau(c, d, \kappa)\) is a continuous distribution on \((0,1)\) with density function proportional to \[
{u}^{c-1}{(1-u)}^{d-1}{\bigl(1+(\tau-1)u\bigr)}^\kappa, \quad u \in (0,1),
\] with parameters \(c>0\), \(d>0\), \(\kappa \in \mathbb{R}\) and \(\tau>0\).

The *(scaled) generalized Beta prime distribution* \(\beta'_\tau(c, d, \kappa, \sigma)\) is the distribution of the random variable \(\sigma \times \tfrac{U}{1-U}\) where \(U \sim \beta_\tau(c, d, \kappa)\).

## Application to the Bayesian binomial model

Assume a \(\beta_\tau(c, d, \kappa)\) prior distribution is assigned to the success probability parameter \(\theta\) of the binomial model with \(n\) trials. Then the posterior distribution of \(\theta\) after \(x\) successes have been observed is \((\theta \mid x) \sim \beta_\tau(c+x, d+n-x, \kappa)\).

## Application to the Bayesian ‘two Poisson samples’ model

Let the statistical model given by two independent observations \[
x \sim \mathcal{P}(\lambda T), \qquad y \sim \mathcal{P}(\mu S),
\] where \(S\) and \(T\) are known design parameters and \(\mu\) and \(\lambda\) are the unknown parameters.

Assign the following independent prior distributions on \(\mu\) and \(\phi := \tfrac{\lambda}{\mu}\) (the relative risk): \[
\mu \sim \mathcal{G}(a,b), \quad \phi \sim \beta'(c, d, \sigma),
\] where \(\mathcal{G}(a,b)\) is the Gamma distribution with shape parameter \(a\) and rate parameter \(b\), and \(\beta'(c, d, \sigma)\) is the scaled Beta prime distribution with shape parameters \(c\) and \(d\) and scale \(\sigma\), that is the distribution of the random variable \(\sigma \times \tfrac{U}{1-U}\) where \(U \sim \beta(c, d)\).

Then the posterior distribution of \(\phi\) is \[
(\phi \mid x, y) \sim \beta'_{\rho/\sigma}(c+x, a+d+y, c+d, \rho)
\] where \(\rho = \tfrac{b+T}{S}\).