All you need is log

Abstract

Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the R\'enyi divergences of order $\alpha\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the R\'enyi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_{\alpha}(\pi_1,\dots,\pi_W) := -\log\int \pi_1^{\alpha_1}\cdots\pi_W^{\alpha_W}$ (with $\sum_k \alpha_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of R\'enyi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with R\'enyi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution R\'enyi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard R\'enyi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.