Tom describes operads, and gives simple examples (constant, monoidal, polynomials), and a probabilistic one. He then describes operad algebras, interconnecting operads. For probabilistic operads it leads to Shannon entropy. This is a feature of operad algebras in general, a fact Tom apologizes or it to be not very intersting! The main result can be brought down to an axiomatic description of information-losing processes modelled as functions. Linearity.
Tom Leinster: "The categorical origins of entropy" youtube