# Modelling channels without specifying input alphabets

The standard mathematical model of a communication channel is that of a stochastic matrix $$(C(x|a))_{a \in A, x \in X}$$, where $$A$$ is the input alphabet and $$X$$ the output alphabet.

This definition works very well for the purposes of Shannon theory in general and the noisy channel coding theorem in particular. However, it seems to me that this model of a channel still contains some redundancy, since all that matters for Shannon theory is presumably which probability distributions over output symbols are achievable by the sender. In other words, it should be enough to work with the following definition:

A communication channel is a pair $$(X,C)$$ consisting of an output alphabet $$X$$ and a closed convex subset $$C \subseteq PX$$, where $$PX$$ denotes the space of probability distributions.

Here, we no longer keep track of how the distributions over outputs are indexed by the input symbols, but only of which such distributions are achievable by the sender. The standard case of finite input alphabet corresponds to the case where $$C$$ is polyhedral and given by the convex hull of the output distributions for each input symbol. For example, the perfect channel with output alphabet $$X$$ simply has $$C = PX$$. (I suspect that the polyhedrality/finiteness is not such an important condition after all, which is why it's not part of the definition itself.)

Here are my questions:

1. Am I right to think that this simplified channel model is enough to develop all (or most?) of the standard theory based on it?

2. Where has this simplified model of channel been considered before?

I have not seen this observation in the literature that I'm familiar with. I find it hard to imagine that it could be new, but I'm not enough of an expert on Shannon theory to assess novelty myself.