Loss-Shift Transfer via Bayes Quotients

Transfer learning is usually studied as a consequence of distribution shift. This paper identifies an orthogonal failure mode in which the data distribution is fixed and the loss changes. This setting is called loss shift. A loss determines which information in (X) is Bayes-relevant, and two losses may therefore require different representations even under the same joint law (P(X,Y)). The idea is formalized using Bayes quotients, which allow losses to be ordered by refinement. In the Bayes-quotient formulation, strict refinement gives an immediate qualitative obstruction. A source-minimal representation for a coarser loss is insufficient for a strictly finer target loss. For finite-output log loss, this obstruction becomes an exact quantitative identity. The excess risk is the conditional information about (Y) discarded by the representation. Experiments in controlled, learned, synthetic-image, and real-image settings show the predicted effect, i.e., classification-equivalent representations can have different optimal log-loss performance under a fixed data distribution.