Markov kernel

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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.^[1]^[2]

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.^[3]

References

↑ Epstein, P.; Howlett, P.; Schulze, M. S. (2003). "Distribution dynamics: Stratification, polarization, and convergence among OECD economies, 1870–1992". Explorations in Economic History 40: 78. doi:10.1016/S0014-4983(02)00023-2.
↑ Reiss, R. D. (1993). A Course on Point Processes. Springer Series in Statistics. doi:10.1007/978-1-4613-9308-5. ISBN 978-1-4613-9310-8.
↑ Poletti Laurini, M. R.; Valls Pereira, P. L. (2009). "Conditional stochastic kernel estimation by nonparametric methods". Economics Letters 105 (3): 234. doi:10.1016/j.econlet.2009.08.012.

Bauer, Heinz (1996), Probability Theory, de Gruyter, ISBN 3-11-013935-9

§36. Kernels and semigroups of kernels

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