A connection between good rate-distortion codes and backward DMCs

Let Xⁿ ϵ Xⁿ be a sequence drawn from a discrete memoryless source, and let Yⁿ ϵ yⁿ be the corresponding reconstruction sequence that is output by a good rate-distortion code. This paper establishes a property of the joint distribution of (Xⁿ, Yⁿ). It is shown that for D > 0, the input-output statistics of a R(D)-achieving rate-distortion code converge (in normalized relative entropy) to the output-input statistics of a discrete memoryless channel (dmc). The dmc is “backward” in that it is a channel from the reconstruction space yⁿ to source space Xⁿ. It is also shown that the property does not necessarily hold when normalized relative entropy is replaced by variational distance.