Entropy property testing with finitely many errors

Let P be a class of distributions over natural numbers, and A be a subset of ℝ
⁺
. We study the problem of deciding, using i.i.d. samples X
₁
,X
₂
,... from an unknown p ∈ P, whether the entropy H(p) is in A or not. The decision is updated based on every new observation X
_n
-we are interested in decision rules that make only finitely many errors no matter what the underlying source is. We give necessary and sufficient conditions on the class P and A that can be decided with only finitely many errors. We show for example that such rules exist for testing the rationality of entropy within a given interval, for testing if the entropy falls in an interval of form (a,b], but no such decision rule exists to determine if the entropy is finite or if the entropy falls in an interval of form [a,b]. In the process, we also highlight the conceptual foundation this framework shares with regularization.