Supplementary Material AProof of Proposition 2

Proposition 2. (From main text) The Bayes error of flow models is monotonically increasing in . That is, for 0 < 0, we have that EBayes(ˆp) EBayes(ˆp 0). B.1 Hardness of Classes In addition to measuring the difficulty of classification tasks relative to one another, it also may be of interest to evaluate the relative difficulty of individual classes within a particular task. A natural way to do this is by looking at the error of one-vs-all classification tasks. The optimal Bayes classifier in this task is CBayes(x)= 0 if logpj(x) logp j(x), 1 otherwise .