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I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.

If $(X,\Sigma)$ is a measurable space, then the function $\mu : \Sigma\times\Sigma \to [0,1]$ is called a Conditional Probability measure if it satisfies (For any $A,B,C \in \Sigma$):

  1. $\mu(A,B) \geq 0$; and $\mu(B,B)= 1$,

  2. For any $B\neq \emptyset$, $\mu(\cdot,B)$ is $\sigma$-additive

  3. $\mu(A\cap B,C) = \mu(A,B\cap C)\cdot \mu(B,C)$

It is a standard definition for the product for (non-conditional) probability function:

If $X$ and $Y$ are the support of measurable spaces and $\mu,\eta$ are probability measures over them, then the product measure over $X\times Y$ is given by:

$\lambda(U) = \int\mu(U_y)d\eta(y)= \int\eta(U_x)d\mu(x)$

where $U_x = \{y \in Y : (x,y)\in U\}$

On a similar way we could (try to) define the product of conditional probability measures as follows(Since I am more interested on the case where one of the spaces is finite and discrete I will make this assumption):

Let $X$ be a measurable space and $\mu$ a conditional probability measure over $X$. Let $Y$ be a finite discrete conditional probability space and $\eta$ a conditional probability measure over $Y$.

Define the following function: \begin{equation} \lambda(A,B) = \sum_{y\in Y}\mu(A_y,B_y)\cdot\eta(y,\pi_2(B))\quad (*)\end{equation}

where $\pi_2(B) = \{y \in Y:$for some $x\in X, (x,y)\in B\}$.

With this definition we have that if $U$ and $V$ are rectangles in $X\times Y$, i.e., $U = A\times B$ and $V = C\times D$, then $\lambda(U,V)= \mu(A,C)\cdot\eta(B,D)$.

Also the properties 1. and 2. from the definition of a conditional probability function hold.

However we have the following concerning property 3.:

On one hand:

$\lambda(A\cap B,C) = \sum_{y\in Y} \mu(A_y\cap B_y,C_y)\cdot \eta(y,\pi_2(C))$

On the other hand:

$\lambda(A,B\cap C)\cdot \lambda(B,C) = \sum_{y\in Y} \mu(A_y, B_y\cap C_y)\cdot \eta(y,\pi_2(B\cap C)\cdot \sum_{x\in Y}\mu(B_x,C_x)\cdot \eta(x,\pi_2(C))$.

Which I do not know if one can prove that an equality holds between both equations.

QUESTIONS:

  1. If the definition $(*)$ makes sense, how one could prove that the product is a conditional probability measure?

  2. Does anyone know a reference for the product of conditional probability spaces?

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