What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems
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Here we consider, in the context of causal inference, the general question: 'what can be estimated from data?'. We call this the question of estimability. We consider the usual definition adopted in the causal inference literature -- identifiability -- in a general mathematical setting and show why it is an inadequate formal translation of the concept of estimability. Despite showing that identifiability implies the existence of a Fisher-consistent estimator, we show that this estimator may be discontinuous, hence unstable, in general. The source of the difficulty is that the general form of the causal inference problem is an ill-posed inverse problem. Inverse problems have three conditions which must be satisfied in order to be considered well-posed: existence, uniqueness, and stability of solutions. We illustrate how identifiability corresponds to the question of uniqueness; in contrast, we take estimability to mean satisfaction of all three conditions, i.e. well-posedness. Well-known results from the inverse problems literature imply that mere identifiability does not guarantee well-posedness of the causal inference procedure, i.e. estimability, and apparent solutions to causal inference problems can be essentially useless with even the smallest amount of imperfection. Analogous issues with attempts to apply standard statistical procedures to very general settings were raised in the statistical literature as far back as the 60s and 70s. Thus, in addition to giving general characterisations of identifiability and estimability, we demonstrate how the issues raised in both the theory of inverse problems and in the theory of statistical inference lead to concerns over the stability of general nonparametric approaches to causal inference. These apply, in particular, to those that focus on identifiability while excluding the additional stability requirements required for estimability.
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