Answers.Body: <p>Machine learning papers are often somewhat confused about the distinction between a distribution and its probability density. I would rewrite this</p>
  <blockquote>
  <p>The process consists of two steps: (1) a value <span class="math-container">$\mathbf{z}^{(i)}$</span> is generated from some prior distribution <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span>; (2) a value <span class="math-container">$\mathbf{x}^{(i)}$</span> is generated from some conditional distribution <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{x} \mid \mathbf{z})$</span>.</p>
  </blockquote>
  <p>as follows</p>
  <blockquote>
  <p>The process consists of two steps: (1) a value <span class="math-container">$\mathbf{z}^{(i)}$</span> is generated from some prior distribution. The probability density of selecting <span class="math-container">$\mathbf{z}^{(i)}$</span> is known and denoted as <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z}^{(i)})$</span>. (2) a value <span class="math-container">$\mathbf{x}^{(i)}$</span> is generated from some conditional distribution. The probability density of selecting <span class="math-container">$\mathbf{x}^{(i)}$</span> given <span class="math-container">$\mathbf{z}^{(i)}$</span> is known and denoted as <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{x}^{(i)} \mid \mathbf{z}^{(i)})$</span>.</p>
  </blockquote>
  <p>As for the uppercase/lower case notation, this notation is not used in machine learning. <span class="math-container">$z, \, x$</span> are both random variables. In this paper, the authors use <span class="math-container">$z^{(i)}$</span>, <span class="math-container">$x^{(i)}$</span> to indicate specific realizations of the random variables <span class="math-container">$z, \, x$</span>. Probably a notation like <span class="math-container">$x_i, z_i$</span> is more common in general.</p>
  <p>The notation/explanation is quite bad, because they should not say <span class="math-container">$p_{\theta}(z)$</span> is the distribution. <span class="math-container">$p_{\theta}(z)$</span>, or if you wanted to be more precise, <span class="math-container">$p_{\theta}(z^{(i)})$</span>, refers to the probability density.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-02T18:06:31.093Z
Answers.Text: I noticed Kingma added notation guide in his other paper on VAEs, in appendix A. 
  https://arxiv.org/pdf/1906.02691.pdf  Seems like he uses the same notation for random vectors and their instances here too?
Answers.UserDisplayName: a12345
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-02T06:21:51.660Z
Answers.LastActivityDate: 2021-12-02T06:21:51.660Z
Answers.UserDisplayName: Taw
Answers.Body: <p>You can read <span class="math-container">$X=\{x^{(i)}\}_{i=1}^N$</span> as <span class="math-container">$X$</span> represents the sequence of all values of <span class="math-container">$x$</span> from <span class="math-container">$x_i$</span> to <span class="math-container">$x_N$</span> where <span class="math-container">$i$</span> is all values from 1 to <span class="math-container">$N$</span>.</p>
  <p>To me, the notation is confusing since my experience tells me that curly braces are used for sets, but this seems to be the best interpretation.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-03T01:10:10.930Z
Answers.Text: I guess its more like a sequence than a set?
Answers.UserDisplayName: a12345
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-03T11:31:50.593Z
Answers.Text: Ah yes, that would be a good interpretation.  Let me update the answer.
Answers.UserDisplayName: David Hoelzer
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-02T19:23:54.877Z
Answers.LastActivityDate: 2021-12-03T11:32:29.883Z
Answers.LastEditDate: 2021-12-03T11:32:29.883Z
Answers.UserDisplayName: David Hoelzer
Answers.Body: <p>When it comes to notation/terminology, often, people in machine learning are (a bit?) sloppy, which causes a lot of confusion, especially for newcomers to the field or people not very math-savvy. I was also confused about this notation at some point (see my last questions <a href="https://math.stackexchange.com/users/168764/nbro?tab=questions">here</a>, which are all about this confusing topic). See also <a href="https://ai.stackexchange.com/a/16842/2444">this answer</a>.</p>
  <p>In the VAE paper, <span class="math-container">$\mathbf{X}$</span> is a dataset, as the authors write.</p>
  <p>Your confusion also arises because the authors vaguely use the term &quot;probability distribution&quot;, rather than pdf or pmf, to refer, for example, to <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span>, which thus does not refer to a pdf or pmf. In fact, the authors also write</p>
  <blockquote>
  <p><strong>their PDFs</strong> are differentiable almost everywhere w.r.t. both <span class="math-container">$\boldsymbol{\theta}$</span> and <span class="math-container">$\mathbf{z}$</span></p>
  </blockquote>
  <p>The <span class="math-container">$\mathbf{z}$</span> can refer to</p>
  <ol>
  <li>a random variable, <strong>or</strong></li>
  <li>an input to the function <span class="math-container">$p_{\boldsymbol{\theta}^{*}}$</span></li>
  </ol>
  <p>If it's the first case, then <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> is the composition of 2 functions (because a rv is also a function).</p>
  <p>If it's the second case, then <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> is the evaluation of <span class="math-container">$p_{\boldsymbol{\theta}^{*}}$</span> at <span class="math-container">$\mathbf{z}$</span>.</p>
  <p>I think the 2nd case is the most likely. In addition, people are being sloppy here and use the notation <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> (rather than just <span class="math-container">$p_{\boldsymbol{\theta}^{*}}$</span>) to <strong>emphasize</strong> <span class="math-container">$p_{\boldsymbol{\theta}^{*}}$</span> is a function of some <strong>input variable</strong> (not random variable!), which we denote with the letter <span class="math-container">$\mathbf{z}$</span> to remind ourselves that <span class="math-container">$\mathbf{z}$</span> is associated with a random variable denoted with the same letter (and maybe also in bold and lowercase).</p>
  <p>So, in this case, let's say we denote the random variable associated with <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> with <span class="math-container">$\mathbf{z}$</span>, then we could refer to this associated prior more explicitly as follows <span class="math-container">$p_\mathbf{z}(\mathbf{z})$</span> (but that would even be more confusing). It would have been a better idea to use <span class="math-container">$\mathbf{Z}$</span>, but then we may use the upper case letters to denote matrices or sets (like the VAE paper), so we end up with this mess (which is one of the 2 mythical difficult problems well-known in Computer Science, i.e. naming things), which we need to learn to deal with or just ignore.</p>
  <p><strong>Conclusion</strong>: when I look at <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span>, which has been referred to as a probability distribution, I think there's also some associated random variable, which people, in that same context, will probably denote as <span class="math-container">$\mathbf{z}$</span> or <span class="math-container">$\mathbf{Z}$</span>. There may also be some input variable (not a random variable), which we denote by <span class="math-container">$\mathbf{z}$</span> or <span class="math-container">$z$</span>. If they are not mentioned, then I just ignore that. I never think that <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> is the composition of 2 functions (even if that's the case), because that case was never useful in my readings.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-02T17:58:21.953Z
Answers.Text: I just noticed in Appendix A of his other paper on VAEs, he goes over the notation.  Seems like he's using the same notation for random vectors and instances of random vectors. https://arxiv.org/pdf/1906.02691.pdf
Answers.UserDisplayName: a12345
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-02T18:03:27.533Z
Answers.Text: and yeah, figuring out which notation to use is a mess. I was thinking italics, using mathcal in latex , and something like this:https://tex.stackexchange.com/questions/520412/is-there-a-latex-font-correspond-to-the-symbol-an-x-with-bars-at-the-top-and-bo  gives more variety there.
Answers.UserDisplayName: a12345
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-12-01T16:49:44.127Z
Answers.LastActivityDate: 2021-12-02T08:14:47.683Z
Answers.LastEditDate: 2021-12-02T08:14:47.683Z
Answers.UserDisplayName: nbro
Body: <p>Usually, I see the conventions:</p>
  <ul>
  <li>discrete random variable is denoted as <span class="math-container">$X$</span>,</li>
  <li>the pmf is written as <span class="math-container">$P(X=x)$</span> or <span class="math-container">$p(X=x)$</span> or <span class="math-container">$p_{X}(x)$</span> or <span class="math-container">$p(x)$</span>, where <span class="math-container">$x$</span> is an instance of <span class="math-container">$X$</span></li>
  <li>a continuous random variable is denoted as <span class="math-container">$X$</span>,</li>
  <li>the pdf is denoted as <span class="math-container">$f_{X}(x)$</span> or <span class="math-container">$f(x)$</span>, where <span class="math-container">$x$</span> is an instance of <span class="math-container">$X$</span>; sometimes <span class="math-container">$p$</span> is used here too instead of <span class="math-container">$f$</span>.</li>
  </ul>
  <p>However, <a href="https://arxiv.org/pdf/1312.6114.pdf" rel="nofollow noreferrer">the VAE paper</a> uses slightly different notation that I'm trying to understand</p>
  <blockquote>
  <p>Let us consider some dataset <span class="math-container">$\mathbf{X}=\left\{\mathbf{x}^{(i)}\right\}_{i=1}^{N}$</span> consisting of <span class="math-container">$N$</span> i.i.d. samples of some continuous or discrete variable <span class="math-container">$\mathrm{x}$</span>. We assume that the data are generated by some random process, involving an unobserved continuous random variable <span class="math-container">$\mathbf{z}$</span>. The process consists of two steps: (1) a value <span class="math-container">$\mathbf{z}^{(i)}$</span> is generated from some prior distribution <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z}) ;(2)$</span> a value <span class="math-container">$\mathbf{x}^{(i)}$</span> is generated from some conditional distribution <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{x} \mid \mathbf{z})$</span>. We assume that the prior <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span> and likelihood <span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{x} \mid \mathbf{z})$</span> come from parametric families of distributions <span class="math-container">$p_{\boldsymbol{\theta}}(\mathbf{z})$</span> and <span class="math-container">$p_{\boldsymbol{\theta}}(\mathbf{x} \mid \mathbf{z})$</span>, and that their PDFs are differentiable almost everywhere w.r.t. both <span class="math-container">$\boldsymbol{\theta}$</span> and <span class="math-container">$\mathbf{z}$</span>. Unfortunately, a lot of this process is hidden from our view: the true parameters <span class="math-container">$\theta^{*}$</span> as well as the values of the latent variables <span class="math-container">$\mathrm{z}^{(i)}$</span> are unknown to us.</p>
  </blockquote>
  <p>So I am looking at these:</p>
  <ul>
  <li><span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{z})$</span></li>
  <li><span class="math-container">$p_{\boldsymbol{\theta}^{*}}(\mathbf{x} \mid \mathbf{z})$</span></li>
  <li>dataset <span class="math-container">$\mathbf{X}=\left\{\mathbf{x}^{(i)}\right\}_{i=1}^{N}$</span></li>
  </ul>
  <p>So I know the subscript for <span class="math-container">$\theta$</span> denotes those are the parameters for the pdf. It says &quot;discrete variable <span class="math-container">$\mathrm{x}$</span>&quot;, &quot;unobserved continuous random variable <span class="math-container">$\mathbf{z}$</span>&quot;, and  &quot;latent variables <span class="math-container">$\mathrm{z}^{(i)}$</span>&quot;.  In the top, where I wrote &quot; discrete random variable <span class="math-container">$X$</span>&quot;, seems like that's the equivalent of &quot;discrete variable <span class="math-container">$\mathrm{x}$</span>&quot; in this paper.</p>
  <p>So, it looks like they're writing the PDFs as a function of the random variables. Is my assumption correct? Because it is different than the typical conventions I see.</p>
  <p>edit: looks like his other paper has a notation guide, in the appendix, though it seems like he's conflating both random vector and instances of vector in the notation?
  <a href="https://arxiv.org/pdf/1906.02691.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1906.02691.pdf</a></p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-11-30T23:04:24.483Z
Comments.Text: no they’re not writing the pdf/pmf as a function of a random variable. The capital X here is just denoting your dataset of realisations of random variables
Comments.UserDisplayName: David
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-11-30T23:17:59.333Z
Comments.Text: Sorry I rewrote my comment, that particular part about p(X) might have made things more confusing.
Comments.UserDisplayName: a12345
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-11-30T23:28:28.377Z
Comments.Text: I would say that here they’re saying ‘discrete variable’ to denote a realisation of the random variable, so $p(x)$ in this setting is still in line with standard notation. $p(\cdot)$ is a function of $x$ (discrete in this case), it’s just that $x$ are some random realisations of a random variable $X$.
Comments.UserDisplayName: David
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-12-01T00:39:37.763Z
Comments.Text: One other possibility I'm considering is that they could just be using it for both the instance and random variable?
Comments.UserDisplayName: a12345
Tags: <papers><variational-autoencoder><probability-distribution><notation>
Title: Are the authors of the VAE paper writing the PDFs as a function of the random variables?
UserDisplayName: a12345
ThreadText: # Are the authors of the VAE paper writing the PDFs as a function of the random variables?
  
  (Asked by: username_0 on 2021-11-30)
  
  Usually, I see the conventions:
  
  * discrete random variable is denoted as $X$,
  * the pmf is written as $P(X=x)$ or $p(X=x)$ or $p\_{X}(x)$ or $p(x)$, where $x$ is an instance of $X$
  * a continuous random variable is denoted as $X$,
  * the pdf is denoted as $f\_{X}(x)$ or $f(x)$, where $x$ is an instance of $X$; sometimes $p$ is used here too instead of $f$.
  
  However, [the VAE paper](https://arxiv.org/pdf/1312.6114.pdf) uses slightly different notation that I'm trying to understand
  
  > Let us consider some dataset $\mathbf{X}=\left\{\mathbf{x}^{(i)}\right\}\_{i=1}^{N}$ consisting of $N$ i.i.d. samples of some continuous or discrete variable $\mathrm{x}$. We assume that the data are generated by some random process, involving an unobserved continuous random variable $\mathbf{z}$. The process consists of two steps: (1) a value $\mathbf{z}^{(i)}$ is generated from some prior distribution $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z}) ;(2)$ a value $\mathbf{x}^{(i)}$ is generated from some conditional distribution $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{x} \mid \mathbf{z})$. We assume that the prior $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ and likelihood $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{x} \mid \mathbf{z})$ come from parametric families of distributions $p\_{\boldsymbol{\theta}}(\mathbf{z})$ and $p\_{\boldsymbol{\theta}}(\mathbf{x} \mid \mathbf{z})$, and that their PDFs are differentiable almost everywhere w.r.t. both $\boldsymbol{\theta}$ and $\mathbf{z}$. Unfortunately, a lot of this process is hidden from our view: the true parameters $\theta^{\*}$ as well as the values of the latent variables $\mathrm{z}^{(i)}$ are unknown to us.
  
  So I am looking at these:
  
  * $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$
  * $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{x} \mid \mathbf{z})$
  * dataset $\mathbf{X}=\left\{\mathbf{x}^{(i)}\right\}\_{i=1}^{N}$
  
  So I know the subscript for $\theta$ denotes those are the parameters for the pdf. It says "discrete variable $\mathrm{x}$", "unobserved continuous random variable $\mathbf{z}$", and "latent variables $\mathrm{z}^{(i)}$". In the top, where I wrote " discrete random variable $X$", seems like that's the equivalent of "discrete variable $\mathrm{x}$" in this paper.
  
  So, it looks like they're writing the PDFs as a function of the random variables. Is my assumption correct? Because it is different than the typical conventions I see.
  
  edit: looks like his other paper has a notation guide, in the appendix, though it seems like he's conflating both random vector and instances of vector in the notation?
  <https://arxiv.org/pdf/1906.02691.pdf>
  
  **Tags:** papers, variational-autoencoder, probability-distribution, notation
  
  > **Comments:**
  > **username_1**: no they’re not writing the pdf/pmf as a function of a random variable. The capital X here is just denoting your dataset of realisations of random variables  
  > **username_0**: Sorry I rewrote my comment, that particular part about p(X) might have made things more confusing.  
  > **username_1**: I would say that here they’re saying ‘discrete variable’ to denote a realisation of the random variable, so $p(x)$ in this setting is still in line with standard notation. $p(\cdot)$ is a function of $x$ (discrete in this case), it’s just that $x$ are some random realisations of a random variable $X$.  
  > **username_0**: One other possibility I'm considering is that they could just be using it for both the instance and random variable?  
  
  ---
  
  ## 3 answers
  
  ### Answer by username_4 on 2021-12-01. Score: 4 | Accepted answer
  
  When it comes to notation/terminology, often, people in machine learning are (a bit?) sloppy, which causes a lot of confusion, especially for newcomers to the field or people not very math-savvy. I was also confused about this notation at some point (see my last questions [here](https://math.stackexchange.com/users/168764/username_4?tab=questions), which are all about this confusing topic). See also [this answer](https://ai.stackexchange.com/a/16842/2444).
  
  In the VAE paper, $\mathbf{X}$ is a dataset, as the authors write.
  
  Your confusion also arises because the authors vaguely use the term "probability distribution", rather than pdf or pmf, to refer, for example, to $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$, which thus does not refer to a pdf or pmf. In fact, the authors also write
  
  > **their PDFs** are differentiable almost everywhere w.r.t. both $\boldsymbol{\theta}$ and $\mathbf{z}$
  
  The $\mathbf{z}$ can refer to
  
  1. a random variable, **or**
  2. an input to the function $p\_{\boldsymbol{\theta}^{\*}}$
  
  If it's the first case, then $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ is the composition of 2 functions (because a rv is also a function).
  
  If it's the second case, then $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ is the evaluation of $p\_{\boldsymbol{\theta}^{\*}}$ at $\mathbf{z}$.
  
  I think the 2nd case is the most likely. In addition, people are being sloppy here and use the notation $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ (rather than just $p\_{\boldsymbol{\theta}^{\*}}$) to **emphasize** $p\_{\boldsymbol{\theta}^{\*}}$ is a function of some **input variable** (not random variable!), which we denote with the letter $\mathbf{z}$ to remind ourselves that $\mathbf{z}$ is associated with a random variable denoted with the same letter (and maybe also in bold and lowercase).
  
  So, in this case, let's say we denote the random variable associated with $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ with $\mathbf{z}$, then we could refer to this associated prior more explicitly as follows $p\_\mathbf{z}(\mathbf{z})$ (but that would even be more confusing). It would have been a better idea to use $\mathbf{Z}$, but then we may use the upper case letters to denote matrices or sets (like the VAE paper), so we end up with this mess (which is one of the 2 mythical difficult problems well-known in Computer Science, i.e. naming things), which we need to learn to deal with or just ignore.
  
  **Conclusion**: when I look at $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$, which has been referred to as a probability distribution, I think there's also some associated random variable, which people, in that same context, will probably denote as $\mathbf{z}$ or $\mathbf{Z}$. There may also be some input variable (not a random variable), which we denote by $\mathbf{z}$ or $z$. If they are not mentioned, then I just ignore that. I never think that $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$ is the composition of 2 functions (even if that's the case), because that case was never useful in my readings.
  
  > **Comments:**
  > **username_0**: I just noticed in Appendix A of his other paper on VAEs, he goes over the notation.  Seems like he's using the same notation for random vectors and instances of random vectors. https://arxiv.org/pdf/1906.02691.pdf  
  > **username_0**: and yeah, figuring out which notation to use is a mess. I was thinking italics, using mathcal in latex , and something like this:https://tex.stackexchange.com/questions/520412/is-there-a-latex-font-correspond-to-the-symbol-an-x-with-bars-at-the-top-and-bo  gives more variety there.  
  
  ---
  
  ### Answer by username_2 on 2021-12-02. Score: 2
  
  Machine learning papers are often somewhat confused about the distinction between a distribution and its probability density. I would rewrite this
  
  > The process consists of two steps: (1) a value $\mathbf{z}^{(i)}$ is generated from some prior distribution $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z})$; (2) a value $\mathbf{x}^{(i)}$ is generated from some conditional distribution $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{x} \mid \mathbf{z})$.
  
  as follows
  
  > The process consists of two steps: (1) a value $\mathbf{z}^{(i)}$ is generated from some prior distribution. The probability density of selecting $\mathbf{z}^{(i)}$ is known and denoted as $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{z}^{(i)})$. (2) a value $\mathbf{x}^{(i)}$ is generated from some conditional distribution. The probability density of selecting $\mathbf{x}^{(i)}$ given $\mathbf{z}^{(i)}$ is known and denoted as $p\_{\boldsymbol{\theta}^{\*}}(\mathbf{x}^{(i)} \mid \mathbf{z}^{(i)})$.
  
  As for the uppercase/lower case notation, this notation is not used in machine learning. $z, \, x$ are both random variables. In this paper, the authors use $z^{(i)}$, $x^{(i)}$ to indicate specific realizations of the random variables $z, \, x$. Probably a notation like $x\_i, z\_i$ is more common in general.
  
  The notation/explanation is quite bad, because they should not say $p\_{\theta}(z)$ is the distribution. $p\_{\theta}(z)$, or if you wanted to be more precise, $p\_{\theta}(z^{(i)})$, refers to the probability density.
  
  > **Comments:**
  > **username_0**: I noticed Kingma added notation guide in his other paper on VAEs, in appendix A.  https://arxiv.org/pdf/1906.02691.pdf  Seems like he uses the same notation for random vectors and their instances here too?  
  
  ---
  
  ### Answer by username_3 on 2021-12-02. Score: 1
  
  You can read $X=\{x^{(i)}\}\_{i=1}^N$ as $X$ represents the sequence of all values of $x$ from $x\_i$ to $x\_N$ where $i$ is all values from 1 to $N$.
  
  To me, the notation is confusing since my experience tells me that curly braces are used for sets, but this seems to be the best interpretation.
  
  > **Comments:**
  > **username_0**: I guess its more like a sequence than a set?  
  > **username_3**: Ah yes, that would be a good interpretation.  Let me update the answer.  
  
  ---
  
  