Along the way we prove several properties of Ramsey degrees that suggest that Ramsey degrees actually behave as a measure of diversity for combinatorial objects. Motivated by combinatorial phenomena (structural Ramsey degrees) we build the necessary infrastructure and prove the fundamental properties using only special partitions imposed on homsets. We require no additional categorical machinery to introduce and prove the properties of this entropy. Our entropy function, the Ramsey entropy, is a real invariant of an object in an arbitrary small category. In this paper we add to the body of arguments in favor of the categorical treatment of entropy. Close connections between various notions of entropy and the apparatus of category theory have been observed already in the 1980's and more vigorously developed in the past ten years. In this paper we propose a way to compute entropy of a finite structure not as a measure of statistical, but as a measure of its combinatorial complexity. Our proof is independent of all earlierД¬haracterizations, but inspired by the work of Petz. Monoid $$ which vanishes when $s$ is optimal must be a scalar We show thatДЄny convex linear, lower semicontinuous functor from FinStat to the additive We say that $s$ is 'optimal' if these distributions agree. 'prior' given by pushing the probability distribution on $Y$ forwards along Given this data we canДefine the entropy of the probability distribution on $X$ relative to the Measured system given the result of a measurement.

Measurement process, while s provides a hypothesis about the state of the Morphism is a measure-preserving function $f: X o Y$ together with a Object is a finite set equipped with a probability distribution, while a In particular, we consider a category FinStat where an We use a number of interesting categories related We give a new characterization of relative entropy, also known as the

Relative entropy