What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems

Here we consider, in the context of causal inference, the basic 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, and hence unstable, in general. The difficulty arises because the causal inference problem is in general 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. It follows that mere identifiability does not guarantee well-posedness of a causal inference procedure, i.e. estimability, and apparent solutions to causal inference problems can be essentially useless with even the smallest amount of imperfection. These concerns apply, in particular, to causal inference approaches that focus on identifiability while ignoring the additional stability requirements needed for estimability.