Identifiability

Identifiability is a property of a mathematical model that concerns whether the parameters of the model can be uniquely determined from observed data under an assumed data-generating process. In practice, identifiability questions ask whether the model’s parameters can be recovered, in principle, from perfect measurements of the outputs produced by the model.

Structural identifiability (also called a priori identifiability) deals with the ideal case of noise-free data and complete observation of the model outputs. It asks whether, given the model equations and outputs, there exists a unique parameter value (or a unique set of parameter values) that could have produced those outputs. Structural identifiability can be global (a single unique value) or local (unique in a neighborhood but not globally).

Practical identifiability concerns real-world data, which are finite in length and corrupted by noise. Even if a model is structurally identifiable, finite data may make parameter estimates non-unique or highly imprecise. Practical identifiability is influenced by experimental design, data quality, and the amount of information contained in the observations. It is often assessed through methods such as profile likelihood, Fisher information matrix analysis, or other information-theoretic criteria.

Assessing identifiability informs model design and data collection. If parameters or combinations of parameters are not identifiable, one may reparameterize the model, fix certain parameters, or design experiments to elicit informative outputs. Identities such as certain parameter combinations being identifiable rather than individual parameters are also common in practice.

Applications appear across statistics, econometrics, systems biology, pharmacokinetics/pharmacodynamics, and engineering. Methods for identifiability analysis include symbolic approaches, differential algebra, and numerical techniques, used to determine whether a model’s parameters can be uniquely inferred from the available data.