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