  Privacy-constrained source coding problems have become increasingly important
recently, and the utility-privacy trade-off has been investigated for various
systems. As pioneering work, Yamamoto (1983) found theoretical limits of the
coding rate, privacy and utility in two cases; (i) both public and private
information is encoded and (ii) only public information is encoded. However,
the theoretical limit has not been characterized in a more general case; (iii)
encoded messages consist of public information and a part of private
information. Then in this paper, we characterize the trade-off relation in case
(iii) using rate-distortion theory. The obtained expression of the achievable
region is a "unified expression" because it includes the ones in cases (i) and
(ii) as special cases. The numerical results also demonstrate that neither case
(i) nor (ii) are the best cases, and it is important to select the encoded
information adequately in case (iii).
