The generalized Beta distribution βτ(c, d, κ) is a continuous distribution on (0, 1) with density function proportional to uc − 1(1 − u)d − 1(1 + (τ − 1)u)κ, u ∈ (0, 1), with parameters c > 0, d > 0, κ ∈ ℝ and τ > 0.
The (scaled) generalized Beta prime distribution β′τ(c, d, κ, σ) is the distribution of the random variable $\sigma \times \tfrac{U}{1-U}$ where U ∼ βτ(c, d, κ).
Assume a βτ(c, d, κ) prior distribution is assigned to the success probability parameter θ of the binomial model with n trials. Then the posterior distribution of θ after x successes have been observed is (θ ∣ x) ∼ βτ(c + x, d + n − x, κ).
Let the statistical model given by two independent observations x ∼ 𝒫(λT), y ∼ 𝒫(μS), where S and T are known design parameters and μ and λ are the unknown parameters.
Assign the following independent prior distributions on μ and $\phi := \tfrac{\lambda}{\mu}$ (the relative risk): μ ∼ 𝒢(a, b), ϕ ∼ β′(c, d, σ), where 𝒢(a, b) is the Gamma distribution with shape parameter a and rate parameter b, and β′(c, d, σ) is the scaled Beta prime distribution with shape parameters c and d and scale σ, that is the distribution of the random variable $\sigma \times \tfrac{U}{1-U}$ where U ∼ β(c, d).
Then the posterior distribution of ϕ is (ϕ ∣ x, y) ∼ β′ρ/σ(c + x, a + d + y, c + d, ρ) where $\rho = \tfrac{b+T}{S}$.