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