A non-identifiable model is one that has parameters that cannot be dis-entangled (ie estimated distinctly) given the likelihood alone. If we want to fit such a model, something extra is needed. That ‘extra’ might be a constraint (hard) or a prior distributions (soft).
Here’s a simple example of non-identifiability:
\[ y_1, ..., y_n \sim Normal(a+b, \sigma^2) \]
From the likelihood alone, the parameters \(a\) and \(b\) are non estimable, unless we introduce something extra (prior, constraints, etc.)
Examples of non-identifiable models abound in non-linear regression. In the compartmental model, the identifiability issue (ID) is dealt with by assuming that one parameter is greater than the other. I recommend Doug Bates’s book on non-linear regression for further reading. The nlme book, Mixed Effects Models in S and S-plus is also a treasure.
I think a lot of these type of models now because it happens that a non-identifiable model is the only way I can learn about certain parameters that I’m interested in. I take a Bayesian approach, using informative priors to parse out the effects, but I’m generally concerned about situations where some form of complete non-ID could prohibit the priors from being updated at all from the data. The problem could depend on the priors and the likelihood.
I’m not sure I can recommend these models to researchers that won’t take the ID problem seriously. In my case, my models can be fit with non-informative priors, but the results may be unbearably sensitive to prior assumptions. Therefore, a sensitivity analysis is a strict requirement when overcoming the ID problem Bayesianly.