Semi-analytical approximations to statistical moments of sigmoid and softmax mappings of normal variables

Understand

This note is concerned with accurate and computationally efficient approximations of moments of Gaussian random variables passed through sigmoid or softmax mappings.

• These approximations are semi-analytical (i.e.
• they involve the numerical adjustment of parametric forms) and highly accurate (they yield 5% error at most).
• We also highlight a few niche applications of these approximations, which arise in the context of, e.g., drift-diffusion models of decision making or non-parametric data clustering approaches.