  This paper shows new general nonasymptotic achievability and converse bounds
and performs their dispersion analysis for the lossy compression problem in
which the compressor observes the source through a noisy channel. While this
problem is asymptotically equivalent to a noiseless lossy source coding problem
with a modified distortion function, nonasymptotically there is a noticeable
gap in how fast their minimum achievable coding rates approach the common
rate-distortion function, as evidenced both by the refined asymptotic analysis
(dispersion) and the numerical results. The size of the gap between the
dispersions of the noisy problem and the asymptotically equivalent noiseless
problem depends on the stochastic variability of the channel through which the
compressor observes the source.
