AN INVERSE VARIATIONAL CHARACTERIZATION OF THE KULLBACK–LEIBLER DIVERGENCE

抄録

We pose an inverse problem in information geometry: we identify minimal variational assumptions under which the Kullback–Leibler form is uniquely forced (up to scale), rather than postulated. Working in one dimension on the positive half-line R>0, we consider a C2 generating potential F ∶ R>0 → R in a dimensionless ratio coordinate ψ, normalized at the equilibrium ψ = 1. Our key postulate is imposed not on a statistical manifold or a divergence class but on the marginal field: we assume F'(ψ) ∈ span{1, ψ−1}, i.e., F' is a linear combination of the canonical 0- and (−1)-homogeneous types on R>0. Under strict convexity and equilibrium normalization, this determines the potential uniquely up to a positive constant, F (ψ) = c(ψ − 1 − ln ψ), c > 0, so the excess ψ − 1 − ln ψ is forced rather than selected from a broad class of divergences. We then record standard interpretations of this excess as the scalar Itakura–Saito divergence (a scale-invariant loss on R>0) in ratio form and as a concrete Kullback–Leibler divergence within a one-parameter exponential family, and briefly summarize the induced one-dimensional dually flat structure in the sense of information geometry.