nLab zero-one kernel

Idea

Zero-one kernels are Markov kernels whose only values are zero and one.

In probability theory, they model situations which “are not really random”, where we are almost surely certain of which transitions take place and which do not.

They are the kernel version of a zero-one measure.

Zero-one kernels form a category, which is used in categorical probability to model situations of determinism in a point-free way.

Definition

A Markov kernel $k:X\to Y$ is said to be zero-one if and only if for every $x\in X$ and every measurable subset $B$ of $Y$,

$$k(B|x) \;=\; 0 \qquad or \qquad k(B|x) \;=\; 1 .$$

Examples

• Every Dirac delta measure is a zero-one kernel from the one-point spae:

  $$\delta_x(A) \;=\; 1_A(x) \;=\; \begin{cases} 1 & x\in A ; \\ 0 & x\notin A . \end{cases}$$

  If $X$ is standard Borel, or more more generally if it has enough points, every zero-one measure on $X$ is a Dirac delta.

• More generally, every kernel induced by a function is zero-one:

  $$\delta_f(A|x) \;=\; 1_A(f(x)) \;=\; \begin{cases} 1 & f(x)\in A ; \\ 0 & f(x)\notin A . \end{cases}$$

  Once again, if $Y$ is sober, every zero-one Markov kernel $X\to Y$ is in this form.

(See also at zero-one measure.)

The category of zero-one kernels

Zero-one Markov kernels are closed under composition, and hence they form a subcategory of Stoch, sometimes denoted by $Stoch_det$.

This category is useful in categorical probability since it provides a point-free point of view on some probabilistic concepts. Given measurable spaces $X$ and $Y$, denote their sigma-algebras by $\Sigma_X$ and $\Sigma_Y$. A zero-one kernel $k:X\to Y$ induces an assignment $k^*:\Sigma_Y\to\Sigma_X$ via

$$k^*B \;\coloneqq\; \{x\in X \,:\, k(B|x) = 1 \} .$$

The map $k^*:\Sigma_Y\to\Sigma_X$ is a morphism of sigma-algebras (i.e. it preserves countable unions and complements), and every morphism of sigma-algebras $\Sigma_Y\to\Sigma_X$ is in this form for some kernel $k$. In other words, zero-one kernels are analogous to morphisms of locales in point-free topology.

In particular, similar to the case of sober topological spaces, $Stoch_det$ is equivalent to the category of sober measurable spaces. Equivalently, it can also be seen as the Kleisli category of the zero-one measure monad (equivalently, the sobrification monad of measurable spaces).

Almost surely zero-one kernels

Given probability spaces $(X,p)$ and $(Y,q)$, a measure-preserving kernel $k:(X,p)\to(Y,q)$ is almost surely zero-one if for every measurable subset $B\subseteq Y$,

$$k(B|x) \;=\; 0 \qquad or \qquad k(B|x) \;=\; 1$$

for $p$-almost sure all $X$. Almost surely zero-one kernels are closed under composition, and so are their almost sure equivalence classes.

Further properties

• Zero-one kernels are exactly the deterministic morphisms of Stoch, in the sense of Markov categories.

• Zero-one kernels are exactly the thunkable morphisms of Stoch, seen as the Kleisli category of the Giry monad with its canonical thunk-force structure.

• Almost surely zero-one kernels are exactly the dagger epimorphisms (or coisometries) of Krn.

• The invariant sigma-algebra is a colimit over an action in the category of zero-one kernels (and also in the one of all kernels).

Related entries

• zero-one measure
• Markov kernel, Markov category, Stoch, Krn
• thunk-force category
• sober measurable space
• point-free topology, point of a locale

References

• Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370:107239, 2020. arXiv:1908.07021.

• Sean Moss, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

Introductory material

• Paolo Perrone, When measurable spaces don’t have enough points, introductory talk. (YouTube)

category: probability