Log-Likelihood Loss for Semantic Compression

We study lossy source coding under a distortion measure defined by the negative log-likelihood induced by a prescribed conditional distribution PX|U. This log-likelihood distortion models compression settings in which the reconstruction is a semantic representation from which the source can be probabilistically generated, rather than a pointwise approximation. We formulate the corresponding rate–distortion problem and characterize fundamental properties of the resulting rate–distortion function, including its connections to lossy compression under log-loss, classical rate–distortion problems with arbitrary distortion measures, and rate–distortion with perfect perception.