Usefulness of structural identifiability when practical identifiability is absent

Ascertain whether ideal structural identifiability results—defined as injectivity of the parameter-to-data-distribution mapping under perfect data—are useful for inference and model development in scenarios where, at the observation grid level with finite noisy data, the parameters are practically non-identifiable.

Background

The paper distinguishes structural identifiability (uniqueness under perfect data) from practical identifiability (estimability with finite, noisy observations) and introduces an auxiliary mapping from parameters to data-distribution parameters. It notes that matching structural identifiability requires working at the solution grid level, while practical analyses operate at the observation grid level.

In ill-posed problems or models with parameter-dependent limiting behavior, the auxiliary mapping can exhibit arbitrarily small singular values in certain regions, blurring the distinction between practical and structural non-identifiability. This context raises a foundational question about the practical value of structural identifiability conclusions in settings where practical identifiability fails.