Is it abuse of notation to use tilde operator in this context?

Question

The following is a way to use tilde (∼) in context of random variables or random vectors.

In statistics, the tilde is frequently used to mean "has the distribution (of)," for instance, $X∼N(0,1)$ means "the stochastic (random) variable $X$ has the distribution $N(0,1)$ (the standard normal distribution). If X and Y are stochastic variables then $X∼Y$ means "$X$ has the same distribution as $Y$.

Consider the following usage of tilde in the paper titled Generative Adversarial Nets

$$x ∼ p_{data}(x)$$ $$z ∼ p_z(z)$$

I am thinking that the following is the standard (and possibly correct) notation

$$x ∼ p_{data}$$ $$z ∼ p_z$$

$p_{data}$ is a probability distribution and $p_{data}(x)$ is not a probability distribution and it is a value in $[0, 1]$. It is same in case of noise probability distribution.

Is it an abuse of notation to use in such a way or is it also a standard and allowed notation to use?

Answer 1

The notation $p(x)$ is widely used in machine learning (e.g. here) and even statistics (e.g. here). People often use $p(x)$ to refer to a probability distribution (either pmf, pdf, or cdf) rather than just $p$. There is also the notation $p_x$ (or things like $p_{x \mid y}$ for conditional p.d.s), which you will find in some statistics books.

Of course, if you interpret the notation $p(x)$ as the evaluation of e.g. the pdf at $x$, it's a density value, so not a p.d. However, I think, in your case, $p(x)$ is used as $p_x$, i.e. to emphasize that the pd $p$ is associated with the r.v. $x$ or that it's a function of one variable (in this case denoted by $x$ to remind the reader that this pd is associated with the r.v. $x$). That's how I would interpret that notation in this case.

This can be useful if you use the letter $p$ to denote multiple p.d.s associated with different r.v.s (which is often the case); so, by using $p_x$ or $p(z)$, you make it clear which r.v. $p$ is associated with.

I looked at the GAN paper, and sometimes they use $z \sim p_z(z)$ and other times $z \sim p_z$, so there are some inconsistencies in the paper, which is not so uncommon in machine learning papers (sometimes people do that just to avoid verbosity).

One that knows the mathematical definition of an r.v. will also know that an r.v. is a function. So, if $p$ is e.g. a pdf and $x$ in $p(x)$ is an r.v., then $p(x)$ would be a composition of functions. I don't think that people use the notation $p(x)$ for that case, as this can become more complicated: we can easily start to talk about measure theory or pushforward measures (which I only barely heard of). I think that many people in the machine learning community do not have a formal statistics or mathematical background, so they probably just use $p(x)$ because everyone else does.

In this answer that I wrote a while ago, I also say that $p(x)$ could be a shorthand for $p(x = x_i)$, i.e. the probability that $x = x_i$ (some event). This is also common, but that's not how I would interpret $p(x)$ in your case.