A Formal Treatment of Sequential Ignorability

TL;DR

This paper rigorously analyzes conditions under which sequential ignorability can be inferred in decision problems with observable and unobservable variables, highlighting the role of stability, randomization, irrelevance, and discreteness.

Contribution

It formalizes the conditions for identifying sequential ignorability, especially emphasizing the case where all variables are discrete, removing the need for positivity conditions.

Findings

Sequential randomization implies simple stability under certain conditions.

Sequential irrelevance can infer sequential ignorability with positivity when variables are discrete.

Positivity conditions are unnecessary for discrete variables in this context.

Abstract

Taking a rigorous formal approach, we consider sequential decision problems involving observable variables, unobservable variables, and action variables. We can typically assume the property of extended stability, which allows identification (by means of G-computation) of the consequence of a specified treatment strategy if the unobserved variables are, in fact, observed - but not generally otherwise. However, under certain additional special conditions we can infer simple stability (or sequential ignorability), which supports G-computation based on the observed variables alone. One such additional condition is sequential randomization, where the unobserved variables essentially behave as random noise in their effects on the actions. Another is sequential irrelevance, where the unobserved variables do not influence future observed variables. In the latter case, to deduce sequential ignorability in full generality requires additional positivity conditions. We show here that these positivity conditions are not required when all variables are discrete.