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Stream: theory: applied category theory

Topic: Crámer transform of Markov kernels

Peva Blanchard (Aug 29 2026 at 21:52):

Consider the 2-category $\mathbf{Stoch}$

• objects: measurable spaces
• arrows: Markov kernel $K : X \to Y$
• 2-cells: $\kappa : K \Rightarrow K'$ is a nonnegative real $\kappa \ge 0$ such that

$$e^{-\kappa} K(B \mid x) \le K'(B \mid x)$$

for all measurable subset $B$ of $Y$ and element $x$ of $X$.

Similarly, we can also define the 2-category $\mathbf{CostMat}$

• objects: measurable spaces
• arrows: $L : X \to Y$ is a matrix $M : Y \times X \to [0,+\infty]$ valued in the tropical rig $\mathbf{Cost} = ([0,\infty], \inf, \infty, +, 0)$
• 2-cells: $\kappa : L \Rightarrow L'$ is a nonnegative real $\kappa \ge 0$ such that

$$L(y,x) + \kappa \ge L'(y,x)$$

Every stochastic kernel $K : X \to Y$ yields a Cost-matrix $L : X \to Y$, defined as the Crámer transform of $K(\cdot \mid x)$, i.e. the Legendre-Fenchel transform of the cumulant-generating function of $K$

$$\begin{align*} H(z,x) &= \log \mathbb{E}_{y\sim K(\cdot\mid x)} \big[e^{\langle z, y \rangle}\big] \\ L(y,x) &= \sup_z \big(\langle z, y \rangle - H(z,x)\big) \end{align*}$$

I am wondering if the Crámer transform is part of a (lax?) functor.

Peva Blanchard (Aug 29 2026 at 23:20):

Oops, it seems that Crámer transform does not behave well wrt arrow composition.

Vincent Moreau (Aug 30 2026 at 19:43):

I did not hear about these 2-cells on Stoch before, they look interesting. Maybe I am wrong, but I have the impression that they come from the fact that the Giry monad yields the object part of a category internal to measurable spaces. Are there some references where this 2-category is defined, and do the 2-cells correspond to a some (standard?) notion from probability theory?

Tobias Fritz (Aug 31 2026 at 06:30):

Those 2-cells are relative Rényi entropy at parameter $\infty$. This is usually considered for probability measures, where $D_\infty(P,Q) = \inf \{ \kappa \ge 0 \mid e^{-\kappa} P \le Q\}$.

Tobias Fritz (Aug 31 2026 at 06:30):

In the categorical setting, this makes a brief appearance in Section 2.3.2 of @Paolo Perrone's Markov Categories and Entropy. Perhaps the $\alpha \in [0,1]$ there in Proposition 2.18 should be $\alpha \in [0,\infty]$? At least the proof seems to do $\alpha > 1$ as well. If so, then this is one place where $D_\infty$ has come up in categorical probability, but in Paolo's setting it defines an enrichment rather than 2-cells.

Vincent Moreau (Aug 31 2026 at 07:44):

Thanks!

Peva Blanchard (Aug 31 2026 at 15:04):

Thanks @Tobias Fritz @Vincent Moreau

Actually, I started from the Cost-matrices where $\kappa$ is interpreted as a robustness level (see my previous post). In trying to relate to $\mathbf{Stoch}$, I tried to lift the definition.

I'm glad this lands on something well-known.