  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.
