For several styles of fidelity constraints -- guaranteed distortion, conditional excess distortion, excess distortion -- we show mutual information upper bounds on the minimum expected description length needed to represent a random variable. Coupled with the corresponding converses, these results attest that as long as the information content in the data is not too low, minimizing the mutual information under an appropriate fidelity constraint serves as a reasonable proxy for the minimum description length of the data. We provide alternative characterizations of all three convex proxies, shedding light on the structure of their solutions.