Let $X, Y, Z$ be three random discrete variables. Consider the below random variables:
$A = X\vert Y\vert Z$ ,$B= X\vert Y,Z$
Question: Can I conclude that $A$ and $B$ are the same random variable? And if yes, how do I construct a formal proof?
Clarification of the notation:
We can define the random variable $X \vert Y$ as the random discrete variable which can take the value $X=x \vert Y = y$ with the probability of this outcome denoted by $P(X=x \vert Y = y)$.
We let $T = X \vert Y$. Now we introduce a new variable $Z$. From $Z, T$ we define two new variables:
The first one we denote by $A$ and define it as:
$A = T \vert Z$
The second one we denote by $B$ and define it as the multivariable
$B = (T, Z)$
My attempt
I have tried to show that $A$ and $B$ is the same variable. I have first deduce that $A$ is the same variable as $B$ if the following statement is true:
$P(X\vert Y \vert Z) = P(X \vert Y , Z)$
If $X_{1}, X_{2}$ are discrete random variables we have the below result:
$P( X_{1} \vert X_{2} ) = \frac{P(X_{1}, X_{2})}{P(X_{2})} \Leftrightarrow P(X_{1},X_{2}) = P(X_{2})P(X_{1}\vert X_{2})$
In this case we can obtain the following identity:
$P(X\vert Y \vert Z ) = \frac{P(X,Y\vert Z)}{P(Y\vert Z)} = \frac{P(X,Y\vert Z)P(Z)}{P(Y,Z)} = \frac{P(X,Y,Z)}{P(Y,Z)} = P(X\vert Y, Z)$
And hence we are done.