  We derive a simple general parametric representation of the rate-distortion
function of a memoryless source, where both the rate and the distortion are
given by integrals whose integrands include the minimum mean square error
(MMSE) of the distortion $\Delta=d(X,Y)$ based on the source symbol $X$, with
respect to a certain joint distribution of these two random variables. At first
glance, these relations may seem somewhat similar to the I-MMSE relations due
to Guo, Shamai and Verd\'u, but they are, in fact, quite different. The new
relations among rate, distortion, and MMSE are discussed from several aspects,
and more importantly, it is demonstrated that they can sometimes be rather
useful for obtaining non-trivial upper and lower bounds on the rate-distortion
function, as well as for determining the exact asymptotic behavior for very low
and for very large distortion. Analogous MMSE relations hold for channel
capacity as well.
