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A new source model, which consists of an intrinsic state part and anextrinsic observation part, is proposed and its information-theoreticcharacterization, namely its rate-distortion function, is defined and analyzed.Such a source model is motivated by the recent surge of interest in thesemantic aspect of information: the intrinsic state corresponds to the semanticfeature of the source, which in general is not observable but can only beinferred from the extrinsic observation. There are two distortion measures, onebetween the intrinsic state and its reproduction, and the other between theextrinsic observation and its reproduction. Under a given code rate, thetradeoff between these two distortion measures is characterized by therate-distortion function, which is solved via the indirect rate-distortiontheory and is termed as the semantic rate-distortion function of the source. Asan application of the general model and its analysis, the case of Gaussianextrinsic observation is studied, assuming a linear relationship between theintrinsic state and the extrinsic observation, under a quadratic distortionstructure. The semantic rate-distortion function is shown to be the solution ofa convex programming programming problem with respect to an error covariancematrix, and a reverse water-filling type of solution is provided when the modelfurther satisfies a diagonalizability condition.

An Indirect Rate-Distortion Characterization for Semantic Sources: General Model and the Case of Gaussian Observation