Markov kernel: Difference between revisions

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Given measurable spaces <math>(X, \mathcal A)</math>, <math>(Y, \mathcal B) </math> we consider a Markov kernel <math>\kappa: \mathcal B \times X \to [0,1]</math> as a morphism <math>\kappa: X \to Y</math>. Intuitively, rather than assigning to each <math>x \in X</math> a sharply defined point <math> y \in Y</math> the kernel assigns a "fuzzy" point in <math>Y</math> which is only known with some level of uncertainty, much like actual physical measurements. If we have a third measurable space <math>(Z, \mathcal C)</math>, and probability kernels <math>\kappa: X \to Y</math> and <math>\lambda: Y \to Z</math>, we can define a composition <math>\lambda \circ \kappa : X \to Z</math> by

:<math>(\lambda \circ \kappa) (dz|x) = \int_Y \lambda(dz | y)\kappa(dy|x)</math>.

The composition is associative by [[Fubini'sthe theorem#Tonelli's_theorem_for_non-negative_measurable_function|Tonelli'sMonotone theorem]]Convergence Theorem and the identity function considered as a Markov kernel (i.e. the delta measure <math> \kappa_{1}(dx'|x) = \delta_x(dx')</math>) is the unit for this composition.

This composition defines the structure of a [[category (mathematics)|category]] on the measurable spaces with Markov kernels as morphisms first defined by Lawvere.<ref>{{cite web|author = F. W. Lawvere|title = The Category of Probabilistic Mappings|date = 1962|url = https://ncatlab.org/nlab/files/lawvereprobability1962.pdf}}</ref> The category has the empty set as initial object and the one point set <math>*</math> as the terminal object. From this point of view a probability space <math>(\Omega, \mathcal A, \mathbb P)</math> is the same thing as a pointed space <math>* \to \Omega</math> in the [[Markov category]].