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Stream: theory: applied category theory

Topic: A simple adjunction in probability theory

Joshua Meyers (Feb 21 2021 at 22:26):

Let $F:\mathbb{R}^*\to [0,1]$ be a weakly monotonic, right-continuous function from the extended reals to the unit interval, satisfying $F(-\infty)=0$. In other words, $F$ is a limit-preserving functor. By the adjoint functor theorem for preorders, $F$ has a right adjoint $X:[0,1]\to\mathbb{R}^*$. We can define $X$ explicitly by $X(\omega)\coloneqq \text{inf}\{x:\omega\leq F(x)\}$. Then $X$ is a random variable with cdf $F$.

John Baez (Feb 21 2021 at 23:02):

:tada:

It's already nice how "weakly monotonic, right-continuous map" collapses to "limit-preserving functor".

Joshua Meyers (Feb 21 2021 at 23:07):

And we also get $F(-\infty)=0$ for free!

Jade Master (Feb 22 2021 at 17:03):

This is nice.

Evan Patterson (Feb 22 2021 at 23:13):

Fun fact: I once tried to say that "the quantile function is the left adjoint of the CDF" in a stats paper, but one of my coauthors made me take out the word "adjoint." At least I got the defining inequalities in there!