Semi-analytical approximations to statistical moments of sigmoid and softmax mappings of normal variables
This provides efficient alternatives for modellers in fields like decision-making models or data clustering, though it is incremental in nature.
The paper tackles the problem of approximating statistical moments of Gaussian random variables after sigmoid or softmax mappings, achieving highly accurate results with at most 5% error using semi-analytical methods.
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.