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 $\rho = \tfrac{b+T}{S}$ .