Rewriting channel transition probability as equations.

Question

I read from this paper (Equations 2 & 3) that for a given channel $p_{Y|Z}(y|z)$, we can write $Y$ as a function of $Z$ and another random variable $W$. For example, for additive white Gaussian noise, we have $$ p_{Y|Z}(y|z)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\Big(-\frac{(y-z)^2}{2\sigma^2}\Big)}, $$ which we can rewrite as $$ Y=f(Z,W)=Z+W, $$ where $W\sim N(0,\sigma)$. Is this true that we can do this for any channel $p_{Y|Z}(y|z)$? Specifically, I saw another channel from this paper (Equation 2): $$ p_{Y|Z}(y=0|z)=(1-p_2)(p_1)^z, $$ and $$ p_{Y|Z}(y=1|z)=1-p_{Y|Z}(y=0|z)=1-(1-p_2)(p_1)^z, $$ where $p_1$ and $p_2$ are fixed constants to be chosen. I was wondering if it is possible to write $Y$ as a function of $Z$ and another random variable $W$ (which we can choose), but wasn't able to come up with one. Is this possible? Thanks.

Answer 1

Here's a partial answer, which I'm not entirely satisfied with. Nevertheless, I'm confident that in almost all practical cases the answer to your question is "yes", with some qualifications.

If $\ Y\ $ and $\ Z\ $ are any random quantities taking values in the sets $\ \mathscr{Y}\ $ and $\ \mathscr{Z}\ $ respectively, and the cardinality of $\ \mathscr{Y}\ $ is no greater than than that of the continuum, then for each $\ z\in\mathscr{Z}\ $ let $\ \phi_z:\mathscr{Y}\rightarrow A\subseteq\mathbb{R}\ $ be an arbitrary bijective function, and $$ W=\phi_Z(Y) $$ If $\ y,z\mapsto\phi_z(y)\ $ is measurable with respect to appropriate $\ \sigma$-algebras, then $\ W\ $ will be a random variable, and $$ Y=\phi_Z^{-1}(W)\ . $$ At this level of generality, some qualifications are:

• There are obviously many (typically an infinite number) of possible families $\ \phi_z\ $ of bijective functions to choose from, and the vast majority of these (or perhaps even all of them in some cases) won't give anything very useful.
• While I believe it will nearly always be possible to choose $\ \phi_z\ $ so that the measurability criteria are satisfied, there may be pathological examples where this isn't possible.

In your second example, $\ y\ $ is $\ 0\ $ or $\ 1\ $ and $\ z\ $ is a real number (in equation $(2)$ of your second cited paper, where it appears as $\ w_m\ $, it's actually a natural number). The same is true for equation $(4)$ of your first cited paper. Since $\ y+z\ $ is therefore a well-defined quantity in both cases, you can still take $\ \phi_z(y)=y+z\ $ and have $\ Y=Z+W\ $ , where $\ W=Y-Z\ $.