  Private information retrieval (PIR) is a protocol that guarantees the privacy
of a user who is in communication with databases. The user wants to download
one of the messages stored in the databases while hiding the identity of the
desired message. Recently, the benefits that can be obtained by weakening the
privacy requirement have been studied, but the definition of weak privacy needs
to be elaborated upon. In this paper, we attempt to quantify the weak privacy
(i.e., information leakage) in PIR problems by using the R\'enyi divergence
that generalizes the Kullback-Leibler divergence. By introducing R\'enyi
divergence into the existing PIR problem, the tradeoff relationship between
privacy (information leakage) and PIR performance (download cost) is
characterized via convex optimization. Furthermore, we propose an alternative
PIR scheme with smaller message sizes than the Tian-Sun-Chen (TSC) scheme. The
proposed scheme cannot achieve the PIR capacity of perfect privacy since the
message size of the TSC scheme is the minimum to achieve the PIR capacity.
However, we show that the proposed scheme can be better than the TSC scheme in
the weakly PIR setting, especially under a low download cost regime.
