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[edit]

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 \kappa \colon X \times \mathcal B \to [0,1]
with the following properties:

  1. The map  x \mapsto \kappa(x,B) is \mathcal A - measureable for every B \in \mathcal B .
  2. The map  B \mapsto \kappa(x,B) is a probability measure on (Y, \mathcal B) for every  x \in X.

(i.e. It associates to each point x \in X a probability measure \kappa(x,.)
on (Y,\mathcal B) such that, for every measurable set B\in\mathcal B, the map x\mapsto \kappa(x,B) is measurable with respect to the \sigma-algebra \mathcal A.)


\kappa(x,B)=\frac{1}{2}\mathbf{1}_{x-1}(B)+\frac{1}{2}\mathbf{1}_{x+1}(B), \quad \forall x \in \Z, \quad \forall B \in \mathcal P(\Z),

describes the transition rule for the random walk on \Z. Where \mathbf{1} is the indicator function.

\mathbf{1}_0(B)  & \quad x=0,\\
P[\xi_1 + \dots + \xi_x \in B] & \quad \text{else,}\\

with i.i.d. random variables \xi_i.

  • General Markov processes with finite state space: Take X=Y, \mathcal A = \mathcal B = \mathcal P(X) = \mathcal P(Y) and |X|=|Y|=n, then the transition rule can be represented as a stochastic matrix (K_{ij})_{1 \leq i,j \leq n} with \Sigma_{j \in Y}K_{ij}=1 for every i \in X. In the convention of Markov kernels we write
\kappa(i,B)=\Sigma_{j \in B}K_{ij}, \quad \forall i \in X, \quad \forall B \in \mathcal B.


Semidirect product[edit]

Let (X, \mathcal A, P) be a probability space and \kappa a Markov kernel from (X, \mathcal A) to some (Y, \mathcal B). Then there exists a unique measure Q on (X \times Y, \mathcal A \otimes \mathcal B), s.t.

Q(A \times B) = \int_A \kappa(x,B)dP(x), \quad \forall A \in \mathcal A, \quad \forall B \in
\mathcal B.

Regular conditional distribution[edit]

Let (S,Y) be a Borel space, X a (S,Y) - valued random variable on the measure space (\Omega, \mathcal F,P) and \mathcal G \subseteq \mathcal F a sub-\sigma-algebra. Then there exists a Markov kernel \kappa from (\Omega, \mathcal G) to (S,Y), s.t. \kappa(.,B) is a version of the conditional expectation E[\mathbf 1_{\{X \in B\}}| \mathcal G] for every B \in Y, i.e.

P[X \in B|\mathcal G]=E[\mathbf 1_{\{X \in B\}}|\mathcal G]=\kappa(\omega,B), \quad P-a.s. \forall B \in \mathcal G.

It is called regular conditional distribution of X given \mathcal G and is not uniquely defined.


  1. ^ 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. 
  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. 
§36. Kernels and semigroups of kernels