Electron rest mass

In probability theory, a Markov kernel (or stochastic kernel) is a map that plays the role, in the general theory of Markov processes, that the transition matrix does in the theory of Markov processes with a finite state space.

Formal definition

Let $(X,{\mathcal {A}})$, $(Y,{\mathcal {B}})$ be measurable spaces. A Markov kernel with source $(X,{\mathcal {A}})$ and target $(Y,{\mathcal {B}})$ is a map $K$ that associates to each point $x\in X$ a probability measure $K(x)$ on $(Y,{\mathcal {B}})$ such that, for every measurable set $B\in {\mathcal {B}}$, the map $x\mapsto K(x)(B)$ is measurable with respect to the $\sigma$-algebra ${\mathcal {A}}$.

Let ${\mathcal {P}}(X,{\mathcal {A}})$ denote the set of all probability measures on the measurable space $(X,{\mathcal {A}})$. If $K$ is a Markov kernel with source $(X,{\mathcal {A}})$ and target $(Y,{\mathcal {B}})$ then we can naturally associate to $K$ a map ${\widehat {K}}:{\mathcal {P}}(X,{\mathcal {A}})\to {\mathcal {P}}(Y,{\mathcal {B}})$ defined as follows: given $P$ in ${\mathcal {P}}(X,{\mathcal {A}})$, we set ${\widehat {K}}(P)(B)=\int _{X}K(x)(B)\,\mathrm {d} P(x)$, for all $B$ in ${\mathcal {B}}$.

Estimation

The kernel can be estimated using kernel density estimation.

References

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§36. Kernels and semigroups of kernels