Single-Shot Lossy Compression for Joint Inference and Reconstruction

In the classical source coding problem, the compressed source is reconstructed at the decoder with respect to some distortion metric. Motivated by settings in which we are interested in more than simply reconstructing the compressed source, we investigate a single-shot compression problem where the decoder is tasked with reconstructing the original data as well as making inferences from it. Quality of inference and reconstruction is determined by a distortion criteria for each task. Given allowable distortion levels, we are interested in characterizing the probability of excess distortion. Modeling the joint inference and reconstruction problem as direct-indirect source coding one, we obtain lower and upper bounds for excess distortion probability. We specialize the converse bound and present a new easily computable achievability bound for the case where the distortion metric for reconstruction is logarithmic loss.