  What does it mean to say that a quantity is identifiable from the data?
Statisticians seem to agree on a definition in the context of parametric
statistical models --- roughly, a parameter $\theta$ in a model $\mathcal{P} =
\{P_\theta: \theta \in \Theta\}$ is identifiable if the mapping $\theta \mapsto
P_\theta$ is injective. This definition raises important questions: Are
parameters the only quantities that can be identified? Is the concept of
identification meaningful outside of parametric statistics? Does it even
require the notion of a statistical model? Partial and idiosyncratic answers to
these questions have been discussed in econometrics, biological modeling, and
in some subfields of statistics like causal inference. This paper proposes a
unifying theory of identification that incorporates existing definitions for
parametric and nonparametric models and formalizes the process of
identification analysis. The applicability of this framework is illustrated
through a series of examples and two extended case studies.
