On the validity of using the delta method for calculating the uncertainty of the predictions from an overparameterized model

TL;DR

This paper demonstrates that the delta method provides an accurate or conservative estimate of prediction uncertainty in overparameterized models, validated through analytical proofs and simulation experiments.

Contribution

It analytically proves and empirically validates that the delta method does not underestimate uncertainty in overparameterized models.

Findings

Delta method uncertainty is always ≥ that of minimal models.

Equality occurs when extra parameters do not add flexibility.

Validation through simulation of a neural network model.

Abstract

The uncertainty in the prediction calculated using the delta method for an overparameterized (parametric) black-box model is shown to be larger or equal to the uncertainty in the prediction of a canonical (minimal) model. Equality holds if the additional parameters of the overparameterized model do not add flexibility to the model. As a conclusion, for an overparameterized black-box model, the calculated uncertainty in the prediction by the delta method is not underestimated. The results are shown analytically and are validated in a simulation experiment where the relationship between the normalized traction force and the wheel slip of a car is modelled using e.g., a neural network