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

We consider basic conceptual questions concerning the relationship between\nstatistical estimation and causal inference. Firstly, we show how to translate\ncausal inference problems into an abstract statistical formalism without\nrequiring any structure beyond an arbitrarily-indexed family of probability\nmodels. The formalism is simple but can incorporate a variety of causal\nmodelling frameworks, including 'structural causal models', but also models\nexpressed in terms of, e.g., differential equations. We focus primarily on the\nstructural/graphical causal modelling literature, however. Secondly, we\nconsider the extent to which causal and statistical concerns can be cleanly\nseparated, examining the fundamental question: 'What can be estimated from\ndata?'. We call this the problem of estimability. We approach this by analysing\na standard formal definition of 'can be estimated' commonly adopted in the\ncausal inference literature -- identifiability -- in our abstract statistical\nformalism. We use elementary category theory to show that identifiability\nimplies the existence of a Fisher-consistent estimator, but also show that this\nestimator may be discontinuous, and thus unstable, in general. This difficulty\narises because the causal inference problem is, in general, an ill-posed\ninverse problem. Inverse problems have three conditions which must be satisfied\nto be considered well-posed: existence, uniqueness, and stability of solutions.\nHere identifiability corresponds to the question of uniqueness; in contrast, we\ntake estimability to mean satisfaction of all three conditions, i.e.\nwell-posedness. Lack of stability implies that naive translation of a causally\nidentifiable quantity into an achievable statistical estimation target may\nprove impossible. Our article is primarily expository and aimed at unifying\nideas from multiple fields, though we provide new constructions and proofs.\n