  Private information retrieval (PIR) is the problem of retrieving as
efficiently as possible, one out of $K$ messages from $N$ non-communicating
replicated databases (each holds all $K$ messages) while keeping the identity
of the desired message index a secret from each individual database. The
information theoretic capacity of PIR (equivalently, the reciprocal of minimum
download cost) is the maximum number of bits of desired information that can be
privately retrieved per bit of downloaded information. $T$-private PIR is a
generalization of PIR to include the requirement that even if any $T$ of the
$N$ databases collude, the identity of the retrieved message remains completely
unknown to them. Robust PIR is another generalization that refers to the
scenario where we have $M \geq N$ databases, out of which any $M - N$ may fail
to respond. For $K$ messages and $M\geq N$ databases out of which at least some
$N$ must respond, we show that the capacity of $T$-private and Robust PIR is
$\left(1+T/N+T^2/N^2+\cdots+T^{K-1}/N^{K-1}\right)^{-1}$. The result includes
as special cases the capacity of PIR without robustness ($M=N$) or $T$-privacy
constraints ($T=1$).
