Quadrature rule for an expectation over a Beta-distributed power parameter

Integrates on the logit scale, gamma = plogis(t), where the integrand is the Beta density times gamma (1 - gamma). That is smooth and decays exponentially in both tails whatever the shapes, including the U-shaped priors whose density is unbounded at 0 and 1, so the trapezoidal rule converges geometrically. The mass beyond the end points is put on gamma = 0 and gamma = 1, where the integrands used here are flat to well below the rule's accuracy.

Usage

npp_gamma_rule(p, q, t_limit = 30, t_step = 0.05)

Arguments

p, q

Shape parameters of the Beta prior.

t_limit

Half-width of the logit range.

t_step

Spacing on the logit scale.

Value

A list with nodes in \([0, 1]\) and weights summing to one.