Identifiability

Gist

Cool you have a model! But can you uh tell me about your parameter estimates? You can’t? Well then what you have is an identifiability issue.

Definition

Identifiability is

A model is structurally identifiable when there is a single value of the parameters consistent with noise-free data observations. Problem: the data that is observed tends to be noisy.

A model is practically identifiable if we lose the ability to discern different parameters values baesd on noisy observations

SIR MODEL

If you only have incidence data, then your model is basically unidentifiable (Calpadi et al.)

General notes

As the magnitude of the noise is increased, there can be a lack of practical identifiability even when parameters are known to be structurally identifiable

How to know if your model is practically

1. The model is simulated for a set value of the parameters and noise is added and a fitting procedure is conducted on the noisy data - this is summarized using the average relative estimation error which is then ploited as a function of a noise intensity

In an Model - SIR:
It can be shown that, even when the trajectory 𝑝⁢Δ𝑡 is observed only at discrete time intervals and the peak of infections has not yet occurred, the maximum likelihood estimator (MLE) exists and is unique, and so the model is structurally identifiable (Hamelin et al. 2020)