Answers.Body: <p>The <span class="math-container">$\sim$</span> symbol means that a random variable is drawn from the given distribution, i.e. if I were to say <span class="math-container">$X$</span> has a Standard Normal distribution I would write <span class="math-container">$X \sim \text{Normal}(0,1)$</span>. They write two explicit expectations here because <span class="math-container">$a$</span> is a random variable with distribution <span class="math-container">$\mu_x$</span> but <span class="math-container">$X$</span> is also a random variable with distribution <span class="math-container">$V$</span>. I believe you are right that <span class="math-container">$V$</span> would be analogous to a set of features. So we are saying that <span class="math-container">$X$</span> is a random variable over the distribution over the features, or individuals in this context.</p>
  <p>As for the <span class="math-container">$\Delta(A)$</span>, I have never seen this notation used before -- I am not sure if it is standard notation. I Googled to see if it was something that I just hadn't seen before, but there was no such answer. However, from the context of the paper they define <span class="math-container">$M: V \rightarrow \Delta (A)$</span> to be mappings from individuals to probability distributions over <span class="math-container">$A$</span>, so I guess that <span class="math-container">$\Delta (A) $</span> is probability distributions over <span class="math-container">$A$</span>.</p>
  
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Body: <p>In this paper <a href="https://arxiv.org/pdf/1104.3913.pdf" rel="nofollow noreferrer">Fairness Through Awareness</a>, the notation <span class="math-container">$\mathbb{E}_{x \sim V} \mathbb{E}_{a \sim \mu_x} L(x,a)$</span> is being used (page 5 top line), where <span class="math-container">$V$</span> denotes the set of individuals (so I guess set of feature vectors?) and the meaning of the other variables can be found in the paragraph above the mentioned notation. What does the <span class="math-container">$\sim$</span> in the expectation stand for?</p>
  <p>Another notation that I do not know is <span class="math-container">$\Delta (A) $</span>, where <span class="math-container">$A$</span> is the set of outcomes, for instance, <span class="math-container">$A = \{ 0,1\}$</span>. What does it stand for?</p>
  
Tags: <machine-learning><papers><notation><explainable-ai>
Title: What do the notations $\sim$ and $\Delta (A) $ mean in the paper "Fairness Through Awareness"?
UserDisplayName: MMM
ThreadText: # What do the notations $\sim$ and $\Delta (A) $ mean in the paper "Fairness Through Awareness"?
  
  (Asked by: username_0 on 2020-06-19)
  
  In this paper [Fairness Through Awareness](https://arxiv.org/pdf/1104.3913.pdf), the notation $\mathbb{E}\_{x \sim V} \mathbb{E}\_{a \sim \mu\_x} L(x,a)$ is being used (page 5 top line), where $V$ denotes the set of individuals (so I guess set of feature vectors?) and the meaning of the other variables can be found in the paragraph above the mentioned notation. What does the $\sim$ in the expectation stand for?
  
  Another notation that I do not know is $\Delta (A) $, where $A$ is the set of outcomes, for instance, $A = \{ 0,1\}$. What does it stand for?
  
  **Tags:** machine-learning, papers, notation, explainable-ai
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2020-06-19. Score: 2 | Accepted answer
  
  The $\sim$ symbol means that a random variable is drawn from the given distribution, i.e. if I were to say $X$ has a Standard Normal distribution I would write $X \sim \text{Normal}(0,1)$. They write two explicit expectations here because $a$ is a random variable with distribution $\mu\_x$ but $X$ is also a random variable with distribution $V$. I believe you are right that $V$ would be analogous to a set of features. So we are saying that $X$ is a random variable over the distribution over the features, or individuals in this context.
  
  As for the $\Delta(A)$, I have never seen this notation used before -- I am not sure if it is standard notation. I Googled to see if it was something that I just hadn't seen before, but there was no such answer. However, from the context of the paper they define $M: V \rightarrow \Delta (A)$ to be mappings from individuals to probability distributions over $A$, so I guess that $\Delta (A) $ is probability distributions over $A$.
  
  
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