Expectation over a Beta-distributed control rate of a function of the treatment rate

Computes \(E_v[g(\theta + v)]\) for each effect \(\theta\), where \(v \sim Beta(a, b)\) is the control rate and \(g\) is the treatment arm's distribution function or density - the distribution function and density of a difference of independent Beta variables.

The control rate is integrated in rate space by Gauss-Legendre quadrature over the Beta's effective support, cut where \(\theta + v\) leaves \([0, 1]\). There \(g\) has a corner, or a jump for a density with a shape parameter of one, and a rule on a piece that straddled it would converge slowly; on each smooth piece a few dozen nodes reach an accuracy a midpoint rule in probability space needs hundreds of thousands of nodes for.

Usage

beta_difference_expectation(
  effect,
  shape1,
  shape2,
  treatment_function,
  n_nodes = 32L,
  tail = 1e-15
)

Arguments

effect: Treatment effects at which to evaluate the expectation.
shape1, shape2: Shape parameters of the control rate's Beta.
treatment_function: Function of the treatment rate, vectorised.
n_nodes: Nodes per smooth piece.
tail: Probability left outside the effective support on each side.

Value

One value per effect.