tfp.substrates.jax.math.log_gamma_correction

Returns the error of the Stirling approximation to lgamma(x) for x >= 8.

This is useful for accurately evaluating ratios between Gamma functions, as happens when trying to compute Beta functions.

Specifically,

lgamma(x) approx (x - 0.5) * log(x) - x + 0.5 log (2 pi)

                 + log_gamma_correction(x)

for x >= 8.

This is the function called Delta in [1], eq (30). We implement it with the rational minimax approximation given in [1], eq (32).

References:

[1] DiDonato and Morris, "Significant Digit Computation of the Incomplete Beta Function Ratios", 1988. Technical report NSWC TR 88-365, Naval Surface Warfare Center (K33), Dahlgren, VA 22448-5000. Section IV, Auxiliary Functions. https://apps.dtic.mil/dtic/tr/fulltext/u2/a210118.pdf

x Floating-point Tensor at which to evaluate the log gamma correction elementwise. The approximation is accurate when x >= 8.
name Optional Python str naming the operation.

lgamma_corr Tensor of elementwise log gamma corrections.