Definition of a probability kernel

I don't quite understand the definition of a probability kernel (or Markov kernel).

Is this correct, the reason to introduce this a transition kernel is, if we have a source $$(X,\mathcal{A})$$ and a target $$(Y,\mathcal{B})$$, both measurable spaces, we want to have a new measurable space $$(X,\mathcal{B})$$? There is this example on Wikipedia for a random walk:

Take $$X=Y=\mathbb{Z}$$ and $$\mathcal A = \mathcal B = \mathcal P(\mathbb{Z})$$, then the Markov kernel $$\kappa$$ with $$\kappa(x,B)=\frac{1}{2}\mathbf{1}_{B}(x-1)+\frac{1}{2}\mathbf{1}_{B}(x+1), \quad \forall x \in \mathbb{Z}, \quad \forall B \in \mathcal P(\mathbb{Z})$$ describes the transition rule

But I don't understand it. Why are we using here $$Y$$ and $$\mathcal{A}$$? Is the measurable space $$(Y,\mathcal{B})$$ the next position on the random walk with new event from $$\mathcal{B}$$?

For two measurable space $(E,\mathcal{E})$ and $(F,\mathcal{F})$, we call a mapping $\kappa:E\times\mathcal{F}\rightarrow \mathbb{R}$ a kernel, if $\kappa(x,.):\mathcal{F}\rightarrow\mathbb{R}$ is a measure for all $x\in E$ and $\kappa(.,B):E\rightarrow\mathbb{R}$ is $\mathcal{E}-\mathcal{B}(\mathbb{R})$-measurable for all $B\in\mathcal{F}$. It is called a Markov kernel, if in addition $\kappa(x,.):\mathcal{F}\rightarrow[0,1]$ is a probability measure.

Your example just says that if you are in state $x\in\mathbb{Z}$, then you can jump to state $x+1$ or $x-1$ and you do this with probability $1/2$ resp (since $\kappa(x,\{x+i\})=0$ for $i\in\mathbb{Z}\backslash \{1;-1\}$).

• Hi, thank you for the answer, but this doesn't help me a lot. I know the definition but I have no intuition for it, it seems quite abstract. Is my intuition that we make a new measurable space $(E,\mathcal{F})$ wrong? Mar 20, 2017 at 11:21
• What confuses me is that it is a map from $E \times \mathcal{F}$ but the transition says $(x,x+i)$ which seems to me like a map from $E\times F$ Mar 20, 2017 at 11:34
• @MarcE Sorry, i forgot to add the brackets in my post. But in general, as your second argument you want to have an event, that is an element of the sigma-algebra $\mathcal{F}$. Note that in general, $(X,\mathcal{B})$ is not a measurable space, since $\mathcal{B}\subseteq \mathcal{P}(X)$ is not true in general, so $\mathcal{B}$ is not a sigma-algebra on $X$.
– peer
Mar 20, 2017 at 11:53
• @MarcE Answering your last question in your post, i would say that the values in Y are the possible states of your random walk in the next step.
– peer
Mar 20, 2017 at 12:10
• @MarcE Yes, you want for example write $\kappa(X_s,B)$ which is a mapping from $\Omega$ into $[0,1]$ for fixed $B\in\mathcal{F}$ defined by $\omega\mapsto\kappa(X_s(\omega),B)$ and $X_s(\omega)$ is in $E$ (or in your case in $\mathbb{Z}$) (here $X=(X_n)_{n\in\mathbb{N}}$ is a stochastic process; for example your random walk). Does that help you?
– peer
Mar 20, 2017 at 16:39

To explain the wikipedia definition:

Let $$(X, \mathcal{A})$$ and $$(Y,\mathcal{B})$$ be two measurable spaces. A markov kernel with source $$(X, \mathcal{A})$$ and target $$(Y,\mathcal{B})$$ is a map $$\kappa: \mathcal{B} \times X \rightarrow [0,1]$$ such that

1. for any fixed $$B \in \mathcal{B}$$, the map $$x\mapsto \kappa(B,x)$$ is $$\mathcal{A}$$-measurable.
2. for any fixed $$x\in X$$, the map $$B\mapsto \kappa(B,x)$$ is a probability measure on $$(Y,\mathcal{B})$$.

Point 2. gives some intuition on what exactly the probability kernel is doing:

1. tells us that $$\kappa$$ sends any element $$x\in X$$ to a probability measure on $$(Y,\mathcal{B})$$. Another way to put this is "for any $$x \in X$$, $$\kappa_x$$ is a measure on $$(Y,\mathcal{B})$$.

So let's use the notation $$\kappa_x(\cdot) = \kappa(\cdot,x)$$. Point 1. requires some sense of injectivity:

1. tells us that for any fixed $$B\in\mathcal{B}$$, every measure $$\kappa_x(\cdot)$$ corresponds to some element of $$\mathcal{A}$$ (see definition of $$\mathcal{A}$$-measurable).

$$\kappa(x,B)=\frac{1}{2}\mathbf{1}_{B}(x-1)+\frac{1}{2}\mathbf{1}_{B}(x+1), \quad \forall x \in \mathbb{Z}, \quad \forall B \in \mathcal P(\mathbb{Z})$$
This means, for any state $$x \in \mathbb{Z}$$ there is an associated probability measure $$\kappa(x, \cdot)$$ (i.e. $$\kappa_x$$) on the state space $$\mathbb{Z}$$, and that measure represents the probability of where the particle will go. It places $$\frac12$$ mass at the $$x\pm 1$$ and $$0$$ everywhere else.