Zusammenfassung
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.
Nutzer