Identifying unidentifiability

Leutenant Cauchy: Sir, we have an unidentified parameter entering regression airspace.

Admiral Gauss: Can we reparameterize and obtain their identity?

Leutenant Cauchy: Trying now, Sir. No success.

Admiral Gauss: Do we have any prior information?

Leutenant Cauchy: No, Sir.

Admiral Gauss: Shoot it down…

What is identifiability?

Formal definition

A statistical model \(\mathcal{P} = \{P_\theta: \theta \in \Theta\}\) is identifiable if the mapping of \(\theta \to P_\theta\) is one-to-one and invertible.