  Consider the problem of estimating a latent signal from a lossy compressed
version of the data when the compressor is agnostic to the relation between the
signal and the data. This situation arises in a host of modern applications
when data is transmitted or stored prior to determining the downstream
inference task. Given a bitrate constraint and a distortion measure between the
data and its compressed version, let us consider the joint distribution
achieving Shannon's rate-distortion (RD) function. Given an estimator and a
loss function associated with the downstream inference task, define the
rate-distortion risk as the expected loss under the RD-achieving distribution.
We provide general conditions under which the operational risk in estimating
from the compressed data is asymptotically equivalent to the RD risk. The main
theoretical tools to prove this equivalence are transportation-cost
inequalities in conjunction with properties of compression codes achieving
Shannon's RD function. Whenever such equivalence holds, a recipe for designing
estimators from datasets undergoing lossy compression without specifying the
actual compression technique emerges: design the estimator to minimize the RD
risk. Our conditions simplified in the special cases of discrete memoryless or
multivariate normal data. For these scenarios, we derive explicit expressions
for the RD risk of several estimators and compare them to the optimal source
coding performance associated with full knowledge of the relation between the
latent signal and the data.
