Title: Semi-analytical approximations to statistical moments of sigmoid and softmax mappings of normal variables,
Abstract: 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. We provide these
as examples of efficient alternatives to more tedious derivations that would be
needed if one was to approach the underlying mathematical issues in a more
formal way. We hope that this technical note will be helpful to modellers
facing similar mathematical issues, although maybe stemming from different
academic prospects.