We consider the problem of noisy private information retrieval (NPIR) from $N$ non-communicating databases, each storing the same set of $M$ messages. In this model, the answer strings are not returned through noiseless bit pipes, but rather through \emph{noisy} memoryless channels. We aim at characterizing the PIR capacity for this model as a function of the statistical information measures of the noisy channels such as entropy and mutual information. We derive a general upper bound for the retrieval rate in the form of a max-min optimization. We use the achievable schemes for the PIR problem under asymmetric traffic constraints and random coding arguments to derive a general lower bound for the retrieval rate. The upper and lower bounds match for $M=2$ and $M=3$, for any $N$, and any noisy channel. The results imply that separation between channel coding and retrieval is optimal except for adapting the traffic ratio from the databases. We refer to this as \emph{almost separation}. Ne 查看全文>>