Probabilistic Monads, Domains and Classical Information

Shannon’s classical information theory [18] uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results from [14] for binary channels to show how some more modern tools — probabilistic monads and domain theory in particular — can be used to model classical channels. As initiated in [14], the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the n× n-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.