Answers.Body: <p>The notation <span class="math-container">$p(x)$</span> is widely used in machine learning (e.g. <a href="http://proceedings.mlr.press/v37/blundell15.pdf" rel="nofollow noreferrer">here</a>) and even statistics (e.g. <a href="https://arxiv.org/pdf/1601.00670.pdf" rel="nofollow noreferrer">here</a>). People often use <span class="math-container">$p(x)$</span> to refer to a probability distribution (either pmf, pdf, or cdf) rather than just <span class="math-container">$p$</span>. There is also the notation <span class="math-container">$p_x$</span> (or things like <span class="math-container">$p_{x \mid y}$</span> for conditional p.d.s), which you will find in some statistics books.</p>
  <p>Of course, if you interpret the notation <span class="math-container">$p(x)$</span> as the evaluation of e.g. the pdf at <span class="math-container">$x$</span>, it's a density value, so not a p.d. However, I think, in your case, <span class="math-container">$p(x)$</span> is used as <span class="math-container">$p_x$</span>, i.e. <strong>to emphasize</strong> that the pd <span class="math-container">$p$</span> is associated with the r.v. <span class="math-container">$x$</span> or that it's a function of one variable (in this case denoted by <span class="math-container">$x$</span> to remind the reader that this pd is associated with the r.v. <span class="math-container">$x$</span>). That's how I would interpret that notation in this case.</p>
  <p>This can be useful if you use the letter <span class="math-container">$p$</span> to denote multiple p.d.s associated with different r.v.s (which is often the case); so, by using <span class="math-container">$p_x$</span> or <span class="math-container">$p(z)$</span>, you make it clear which r.v. <span class="math-container">$p$</span> is associated with.</p>
  <p>I looked at the GAN paper, and sometimes they use <span class="math-container">$z \sim p_z(z)$</span> and other times <span class="math-container">$z \sim p_z$</span>, 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).</p>
  <p>One that knows the mathematical definition of an r.v. will also know that an r.v. is a function. So, if <span class="math-container">$p$</span> is e.g. a pdf and <span class="math-container">$x$</span> in <span class="math-container">$p(x)$</span> is an r.v., then <span class="math-container">$p(x)$</span> would be a composition of functions. I don't think that people use the notation <span class="math-container">$p(x)$</span> for that case, as this can become more complicated: we can easily start to talk about measure theory or <a href="https://en.wikipedia.org/wiki/Pushforward_measure" rel="nofollow noreferrer">pushforward measures</a> (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 <span class="math-container">$p(x)$</span> because everyone else does.</p>
  <p>In <a href="https://ai.stackexchange.com/a/16842/2444">this answer</a> that I wrote a while ago, I also say that <span class="math-container">$p(x)$</span> could be a shorthand for <span class="math-container">$p(x = x_i)$</span>, i.e. the probability that <span class="math-container">$x = x_i$</span> (some event). This is also common, but that's not how I would interpret <span class="math-container">$p(x)$</span> in your case.</p>
  
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Body: <p>The <a href="https://mathworld.wolfram.com/Tilde.html" rel="nofollow noreferrer">following is a way</a> to use tilde (∼) in context of random variables or random vectors.</p>
  <blockquote>
  <p>In statistics, the tilde is frequently used to mean &quot;has the
  distribution (of),&quot; for instance, <span class="math-container">$X∼N(0,1)$</span> means &quot;the stochastic
  (random) variable <span class="math-container">$X$</span> has the distribution <span class="math-container">$N(0,1)$</span> (the standard
  normal distribution). If X and Y are stochastic variables then <span class="math-container">$X∼Y$</span>
  means &quot;<span class="math-container">$X$</span> has the same distribution as <span class="math-container">$Y$</span>.</p>
  </blockquote>
  <p>Consider the following usage of tilde in the paper titled <a href="https://papers.nips.cc/paper/2014/file/5ca3e9b122f61f8f06494c97b1afccf3-Paper.pdf" rel="nofollow noreferrer">Generative Adversarial Nets</a></p>
  <p><span class="math-container">$$x ∼ p_{data}(x)$$</span>
  <span class="math-container">$$z ∼ p_z(z)$$</span></p>
  <p>I am thinking that the following is the standard (and possibly correct) notation</p>
  <p><span class="math-container">$$x ∼ p_{data}$$</span>
  <span class="math-container">$$z ∼ p_z$$</span></p>
  <p><span class="math-container">$p_{data}$</span> is a probability distribution and <span class="math-container">$p_{data}(x)$</span> is not a probability distribution and it is a value in <span class="math-container">$[0, 1]$</span>. It is same in case of noise probability distribution.</p>
  <p>Is it an abuse of notation to use in such a way or is it also a standard and allowed notation to use?</p>
  
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Comments.Text: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/128104/discussion-on-question-by-hanugm-is-it-abuse-of-notation-to-use-tilde-operator-i).
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Comments.Text: Since the question is now closed as off-topic by the community, I honestly hope that it will not be re-opened unilaterally by the mod that has an obvious conflict of interest (i.e. has provided an answer below), at least not without a discussion in Meta. As it seems, the community has real concerns about if such a question about *general* mathematical notation, and tagged as `probability` and `random-variable`, has really anything *specific* to do with AI, just because it happened to be stumbled upon in a ML paper.
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Comments.Text: Most of the AI textbooks has initial units dedicated to **Mathematics useful to AI**. So, I think these are on-topic. If some maths question is asked that is not directly or indirectly related to AI, then it can be off-topic.
Comments.UserDisplayName: hanugm
Tags: <probability-distribution><notation><random-variable>
Title: Is it abuse of notation to use tilde operator in this context?
UserDisplayName: hanugm
ThreadText: # Is it abuse of notation to use tilde operator in this context?
  
  (Asked by: username_0 on 2021-08-01)
  
  The [following is a way](https://mathworld.wolfram.com/Tilde.html) 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](https://papers.nips.cc/paper/2014/file/5ca3e9b122f61f8f06494c97b1afccf3-Paper.pdf)
  
  $$x ∼ p\_{data}(x)$$
  $$z ∼ p\_z(z)$$
  
  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?
  
  **Tags:** probability-distribution, notation, random-variable
  
  > **Comments:**
  > **username_1**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/128104/discussion-on-question-by-username_0-is-it-abuse-of-notation-to-use-tilde-operator-i).  
  > **username_2**: Since the question is now closed as off-topic by the community, I honestly hope that it will not be re-opened unilaterally by the mod that has an obvious conflict of interest (i.e. has provided an answer below), at least not without a discussion in Meta. As it seems, the community has real concerns about if such a question about *general* mathematical notation, and tagged as `probability` and `random-variable`, has really anything *specific* to do with AI, just because it happened to be stumbled upon in a ML paper.  
  > **username_0**: Most of the AI textbooks has initial units dedicated to **Mathematics useful to AI**. So, I think these are on-topic. If some maths question is asked that is not directly or indirectly related to AI, then it can be off-topic.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_3 on 2021-08-01. Score: 2 | Accepted answer
  
  The notation $p(x)$ is widely used in machine learning (e.g. [here](http://proceedings.mlr.press/v37/blundell15.pdf)) and even statistics (e.g. [here](https://arxiv.org/pdf/1601.00670.pdf)). 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](https://en.wikipedia.org/wiki/Pushforward_measure) (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](https://ai.stackexchange.com/a/16842/2444) 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.
  
  
  ---
  
  