A Formal Treatment of Sequential Ignorability

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 "[Formula: see text]-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 [Formula: see text]-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.