nLab zero-one kernel

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Measure and probability theory

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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:XYk:X\to Y is said to be zero-one if and only if for every xXx\in X and every measurable subset BB of YY,

k(B|x)=0ork(B|x)=1. 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:

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

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

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

    δ f(A|x)=1 A(f(x))={1 f(x)A; 0 f(x)A. \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 YY is sober, every zero-one Markov kernel XYX\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 detStoch_det.

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

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

The map k *:Σ YΣ Xk^*:\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 Σ YΣ X\Sigma_Y\to\Sigma_X is in this form for some kernel kk. 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 detStoch_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)(X,p) and (Y,q)(Y,q), a measure-preserving kernel k:(X,p)(Y,q)k:(X,p)\to(Y,q) is almost surely zero-one if for every measurable subset BYB\subseteq Y,

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

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

Further properties

References

Introductory material

category: probability

Last revised on July 13, 2024 at 21:09:00. See the history of this page for a list of all contributions to it.