---
title: "Bayesian applications"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Bayesian applications}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

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