  A rate-distortion problem motivated by the consideration of semantic
information is formulated and solved. The starting point is to model an
information source as a pair consisting of an intrinsic state which is not
observable, corresponding to the semantic aspect of the source, and an
extrinsic observation which is subject to lossy source coding. The proposed
rate-distortion problem seeks a description of the information source, via
encoding the extrinsic observation, under two distortion constraints, one for
the intrinsic state and the other for the extrinsic observation. The
corresponding state-observation rate-distortion function is obtained, and a few
case studies of Gaussian intrinsic state estimation and binary intrinsic state
classification are studied.
