nLab Stoch

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Idea

The category whose objects are measurable spaces and whose morphisms are stochastic kernels (or Markov kernels) is often denotes Stoch, or similar.

It is the motivating example of a Markov category, and together with its subcategory BorelStoch, one of the most important categories of categorical probability.

Definition

Stoch is the category whose

  • Objects are measurable spaces, i.e. pairs (X,π’œ)(X,\mathcal{A}) where XX is a set and π’œ\mathcal{A} is a sigma-algebra on XX;
  • Morphisms(X,π’œ)β†’(Y,ℬ)(X,\mathcal{A})\to(Y,\mathcal{B}) are Markov kernels of entries k(B|x)k(B|x), for x∈Xx\in X and Bβˆˆβ„¬B\in\mathcal{B};
  • The identities (X,π’œ)β†’(X,π’œ)(X,\mathcal{A})\to(X,\mathcal{A}) are given by the Dirac delta kernels
    Ξ΄(A|x)=1 A(x)={1 x∈A; 0 xβˆ‰A; \delta(A|x) = 1_A(x) = \begin{cases} 1 & x\in A ; \\ 0 & x\notin A ; \end{cases}
  • The composition of kernels k:(X,π’œ)β†’(Y,ℬ)k:(X,\mathcal{A})\to(Y,\mathcal{B}) and h:(Y,ℬ)β†’(Z,π’ž)h:(Y,\mathcal{B})\to(Z,\mathcal{C}) is given by the Lebesgue integral
    (h∘k)(C|x)=∫ Yh(C|y)k(dy|x) (h\circ k) (C|x) = \int_Y h(C|y)\,k(dy|x)

    for all x∈Xx\in X and Cβˆˆπ’žC\in\mathcal{C}. This is sometimes called Chapman-Kolmogorov formula. (Compare with the composition formula for stochastic matrices.)

Properties

As a Kleisli category

Stoch can be equivalently described as the Kleisli category of the Giry monad on Meas.

It can also be described as the Kleisli category of the Giry monad restricted to the category SoberMeasSoberMeas of sober measurable spaces. Indeed,

In other words, this triangle commutes up to natural isomorphism, where SS denotes the sobrification, and L PL_P denotes the left adjoint of the Kleisli adjunction given by the Giry monad on Meas, as well as its restriction to sober measurable spaces.

As a Markov category

As a Markov category, Stoch is causal and hence positive. It does not have all conditionals, but its subcategory BorelStoch does.

Its deterministic morphisms are exactly the zero-one kernels, forming the subcategory Stoch detStoch_det. Note that, outside of the sober case, these are more general than measurable functions.

Representability

Stoch is a representable Markov category:

Consider now the following diagram, where

By the considerations above, Rβ€²R' is right adjoint to the inclusion functor Stoch detβ†ͺStochStoch_det\hookrightarrow Stoch, making StochStoch a representable Markov category.

Explicitly, the sampling map samp:PX→Xsamp:P X\to X is the usual measure-theoretic one:

samp(A|p)=p(A). samp(A|p) \;=\; p(A) .

The resulting monad on SubStochSubStoch, chasing the diagram above, can be seen as induced the Giry monad on SoberMeasSoberMeas through the equivalence SoberMeas≃Stoch detSoberMeas \simeq Stoch_det.

One should always keep in mind that the probability monad making Stoch representable is on Stoch det Stoch_det , or equivalently on SoberMeas, but not on Meas, even though StochStoch is also the Kleisli category of the Giry monad on Meas. (The reason is that measurable functions are not the deterministic morphisms of StochStoch, zero-one kernels are.)

Particular limits and colimits

Notable subcategories

BorelStoch

BorelStoch is the full subcategory of Stoch whose objects are standard Borel measurable spaces. This is particularly important as a Markov category because it has conditionals and countable Kolmogorov products. It is the Kleisli category of the Giry monad restricted to standard Borel spaces.

As proven here, in BorelStoch all idempotents split, making it a Cauchy-complete category.

FinStoch

FinStoch is the full subcategory of Stoch whose objects are discrete finite sets. Equivalently, it is the category of finite stochastic matrices.

It is closely related to the Kleisli category of the distribution monad.

References

Last revised on July 14, 2024 at 13:50:23. See the history of this page for a list of all contributions to it.