A connection between good rate-distortion codes and backward DMCs

Let X
ⁿ
ϵ X
ⁿ
be a sequence drawn from a discrete memoryless source, and let Y
ⁿ
ϵ y
ⁿ
be the corresponding reconstruction sequence that is output by a good rate-distortion code. This paper establishes a property of the joint distribution of (X
ⁿ
, Y
ⁿ
). 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 y
ⁿ
to source space X
ⁿ
. It is also shown that the property does not necessarily hold when normalized relative entropy is replaced by variational distance.