Weak and Finite Sample Identifiability

Thomas M. Menino Convention & Exhibition Center 

Identifiability is a foundational concept in statistics, formalizing what can and cannot be learned from data. It underlies classical results on minimax risk lower bounds and fundamental limits in testing and estimation. More recently, work on identifiability has motivated the study of partially identified models, leading to robust methods in econometrics, the social sciences, and genomics. However, the classical notion of identifiability is inherently asymptotic, characterizing learnability under the true data-generating model–which is only available in the infinite-sample limit. Yet there is no general framework for understanding what is identifiable in finite samples. We propose a definition of finite-sample identifiability that generalizes classical identifiability and is asymptotically consistent with it. Using this framework, we extend core theoretical results to finite-sample and partially identified settings, including bounds on minimax risk, testing power, and confidence set size. We illustrate the implications through examples involving zero-inflated Poisson model and nonparametric average treatment effect estimation under imperfect compliance.

Identifiability

Random measure

Finite-sample

Total variation distance

Minimax risk bound

Confidence set 

Section on Statistical Learning and Data Science