Similarity-Sensitive Entropy: Induced Kernels and Data-Processing Inequalities

We study an entropy functional $H_K$ that is sensitive to a prescribed similarity structure on a state space. For finite spaces, $H_K$ coincides with the order-1 similarity-sensitive entropy of Leinster and Cobbold. We work in the general measure-theoretic setting of kernelled probability spaces $(Ω,μ,K)$ introduced by Leinster and Roff, and develop basic structural properties of $H_K$. Our main results concern the behavior of $H_K$ under coarse-graining. For a measurable map $f:Ω\to Y$ and input law $μ$, we define a law-induced kernel on $Y$ whose pullback minimally dominates $K$, and show that it yields a coarse-graining inequality and a data-processing inequality for $H_K$, for both deterministic maps and general Markov kernels. We also introduce conditional similarity-sensitive entropy and an associated mutual information, and compare their behavior to the classical Shannon case.