Score Function with Respect to Shape Parameter a

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

Usage

score_a(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 with density $$g_{a,b}(\alpha) = \frac{b^a}{\Gamma(a)} \alpha^{a-1} e^{-b\alpha},$$ the score function with respect to a is: $$s_a(\alpha) = \log b - \psi(a) + \log \alpha,$$ where $\psi$ is the digamma function.

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

Numerical Note: The log(alpha) term causes slower quadrature convergence for score-weighted integrands compared to the moments themselves. Use higher M (e.g., 120-200) for verification purposes.

References

RN-04, Section 4.2: Jacobian via score identities

Examples

# Evaluate score at several points
alpha_vals <- c(0.5, 1.0, 2.0, 5.0)
score_a(alpha_vals, a = 2.0, b = 1.0)
#> [1] -1.1159315 -0.4227843  0.2703628  1.1866536