Score Function with Respect to Rate Parameter b

Computes the score function \(s_b(\alpha) = \partial/\partial b \log g_{a,b}(\alpha)\) for the Gamma(a, b) distribution.

Usage

score_b(alpha, a, b)

Arguments

alpha

Numeric vector; points at which to evaluate.

a

Numeric scalar; shape parameter of the Gamma distribution (> 0).

b

Numeric scalar; rate parameter of the Gamma distribution (> 0).

Value

Numeric vector of the same length as alpha.

Details

For the Gamma(shape = a, rate = b) distribution, the score function with respect to b is: $$s_b(\alpha) = \frac{a}{b} - \alpha.$$

A fundamental property of score functions is that their expectation is zero: $$E_{\alpha \sim g_{a,b}}[s_b(\alpha)] = 0.$$

Unlike score_a, this function is linear in \(\alpha\), so its expectation converges very quickly with quadrature.

References

RN-04, Section 4.2: Jacobian via score identities

See also

score_a for the score with respect to a

Examples

# Evaluate score at several points
alpha_vals <- c(0.5, 1.0, 2.0, 5.0)
score_b(alpha_vals, a = 2.0, b = 1.0)
#> [1]  1.5  1.0  0.0 -3.0