Actually, perfect secrecy refers to the ability to determine any information about the plaintext at all, given arbitrary decoding power. It's quite simple -- it means that given a standard distribution of keys and an a priori distribution over the plain texts the best estimate of the plaintext given the ciphertext is simply the a priori distribution (no additional information). For the binary case, Y(any distribution)+X(uniform)=Z(uniform) (mod 2), so that this is satisfied for any prior.