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Stream: theory: categorical probability

Topic: Probability monad on Euc

Jules Hedges (Sep 07 2021 at 14:57):

I don't remember whether I asked this question here before! $\mathbf{Euc}$ is the (cartesian) prop whose morphisms $m \to n$ are smooth functions $\mathbb R^m \to \mathbb R^n$. I feel intuitively like there ought to be a monad on $\mathbf{Euc}$ describing normal distributions, which is given on objects by $G(n) = n + n^2$, ie. to a space with enough dimensions to describe a mean vector and a covariance matrix. Has anyone worked this out, or does anyone know good reasons why it should or shouldn't work? The obvious barrier is that not every point in $\mathbb R^{n^2}$ is a valid covariance matrix, but my gut feeling is that there should be some simple trick to avoid that.

Tobias Fritz (Sep 07 2021 at 16:15):

That's an interesting question. A more severe problem is going to be the functoriality of $G$: usually we'd expect the action of $G$ on morphisms to be given by taking the pushforward of measures, right? Now the problem is that a pushfoward of a Gaussian along a smooth map is typically not a Gaussian.

Intuitively I'd say that there probably isn't a probability monad for Gaussian measures for any reasonable choice of morphisms. But Gaussian kernels nevertheless form a Markov category.

Jules Hedges (Sep 07 2021 at 16:24):

Didn't think of that! Good point...

Jules Hedges (Sep 07 2021 at 16:25):

What maps can you push forward Gaussian distributions along? Just affine? But then the monad multiplication probably isn't an affine map...

Tobias Fritz (Sep 07 2021 at 16:29):

Exactly! And even then it'll be pretty difficult to find a family of bijections $G(n) \cong \mathbb{R}^{n+n^2}$ such that conjugating with them will turn the pushfoward maps $G(n) \to G(m)$ into affine maps.

Jules Hedges (Sep 07 2021 at 16:31):

Interesting. Thanks! That makes this an extra use-case for Markov categories, vs the monad centric way of thinking about categorical probability