  A new source model, which consists of an intrinsic state part and an
extrinsic observation part, is proposed and its information-theoretic
characterization, namely its rate-distortion function, is defined and analyzed.
Such a source model is motivated by the recent surge of interest in the
semantic aspect of information: the intrinsic state corresponds to the semantic
feature of the source, which in general is not observable but can only be
inferred from the extrinsic observation. There are two distortion measures, one
between the intrinsic state and its reproduction, and the other between the
extrinsic observation and its reproduction. Under a given code rate, the
tradeoff between these two distortion measures is characterized by the
rate-distortion function, which is solved via the indirect rate-distortion
theory and is termed as the semantic rate-distortion function of the source. As
an application of the general model and its analysis, the case of Gaussian
extrinsic observation is studied, assuming a linear relationship between the
intrinsic state and the extrinsic observation, under a quadratic distortion
structure. The semantic rate-distortion function is shown to be the solution of
a convex programming programming problem with respect to an error covariance
matrix, and a reverse water-filling type of solution is provided when the model
further satisfies a diagonalizability condition.
