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Stream: theory: categorical probability

Topic: Entropy as a Functor

view this post on Zulip Gurkenglas (Oct 18 2022 at 20:33):

In https://youtu.be/5phJVSWdWg4?t=2230 , @John Baez says that the only role of the continuity condition is to rule out nonobvious solutions to ϕ(mn)=ϕ(m)+ϕ(n)\phi(mn)=\phi(m)+\phi(n). Equivalently we can define ψ=ϕ∘exp⁡ \psi = \phi \circ \exp and rule out nonobvious solutions to ψ(m+n)=ψ(m)+ψ(n)\psi(m+n)=\psi(m)+\psi(n) and ψ(m)>0\psi(m)>0. Surely we should get that by requiring ψ(am)=aψ(m)\psi(am)=a\psi(m) for all a>0a>0. Instead of "convex" we would say "conical", instead of probability measures we would have measures. Right?

view this post on Zulip John Baez (Oct 19 2022 at 06:57):

Yes, I guess people find that starting with a function ψ:R→R\psi : \mathbb{R} \to \mathbb{R} and using the axiom ψ(am)=aψ(m)\psi(am) = a \psi(m) to deduce that ψ(m)=cm\psi(m) = c m for some constant cc is "too easy to be interesting": you just take c=ψ(1)c = \psi(1) and it's obvious.

view this post on Zulip John Baez (Oct 19 2022 at 06:57):

I'm not sure why you say "instead of probability measures we have measures", though.

view this post on Zulip Gurkenglas (Oct 19 2022 at 21:32):

Because when you allow not just convex combinations but conical ones, aka remove the constraint that the coefficients of a combination sum to 1, then you get all the positive multiples of each probability distribution, which are all the measures. Uh, all the finite measures, once we go beyond FinSet.