Binary Hypothesis Testing Game With Training Data

We introduce a game-theoretic framework to study the hypothesis testing problem in the presence of an adversary aiming to prevent a correct decision. Specifically, this paper considers a scenario in which an analyst has to accept or reject the null hypothesis
\(H_0\)
characterized by a probability mass function (pmf)
\(P_X\)
based on the evidence provided by a test sequence. In turn, the goal of the adversary is to take a sequence generated according to a different pmf and modify it in such a way to induce a decision error.
\(P_X\)
is known only through one or more training sequences. We derive the asymptotic equilibrium of the game under the assumption that the analyst relies only on first order statistics of the test and training sequences, and compute the asymptotic payoff of the game when the length of the sequences tends to infinity. We introduce the concept of indistinguishability region, defined as the set of pmfs that can not be distinguished reliably from
\(P_X\)
in the presence of attacks. Two different scenarios are considered: in the first one the analyst and the adversary share the same training sequence, in the second scenario, they rely on independent sequences. The obtained results are compared with a version of the game in which the pmf
\(P_X\)
is perfectly known to both the analyst and the adversary.