Density of w₁

Computes the probability density p(w₁ = x | a, b).

Usage

density_w1(x, a, b, log = FALSE)

Arguments

x

Numeric vector; evaluation points.

a

Numeric; shape parameter of the Gamma prior on α (a > 0).

b

Numeric; rate parameter of the Gamma prior on α (b > 0).

log

Logical; if TRUE, returns log-density. Default is FALSE.

Value

Numeric vector of density (or log-density if log = TRUE) values. Returns 0 (or -Inf on log scale) for x outside (0, 1).

Details

The marginal density of w₁ is: $$p(w_1 | a, b) = \frac{a \cdot b^a}{(1-w_1) \cdot [b - \log(1-w_1)]^{a+1}}$$

Important: For small values of a (a < 1), the density has significant mass concentrated very close to x = 1.

See also

cdf_w1, quantile_w1

Examples

# Density at several points
x <- seq(0.1, 0.9, by = 0.1)
density_w1(x, a = 2, b = 1)
#> [1] 1.6454157 1.3661794 1.1442068 0.9665754 0.8240923 0.7105355 0.6227160
#> [8] 0.5628065 0.5552236

# Log-density for numerical stability
density_w1(0.5, a = 2, b = 1, log = TRUE)
#> [1] -0.1934727