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.

翻译：我们研究一种对状态空间上指定相似度结构敏感的熵泛函 $H_K$。对于有限空间，$H_K$ 与 Leinster 和 Cobbold 提出的一阶相似度敏感熵一致。我们在 Leinster 和 Roff 引入的核化概率空间 $(Ω,μ,K)$ 的一般测度论框架下展开工作，并发展了 $H_K$ 的基本结构性质。我们的主要结果关注 $H_K$ 在粗粒化下的行为。对于可测映射 $f:Ω\to Y$ 和输入分布 $μ$，我们定义了 $Y$ 上的一个由分布诱导的核，其回拉最小支配 $K$，并证明该核为 $H_K$ 导出了粗粒化不等式和数据处理不等式，这些结果对确定性映射和一般马尔可夫核均成立。我们还引入了条件相似度敏感熵及其对应的互信息，并将其行为与经典香农情形进行了比较。