Answers.Body: <p>This is a commonly used notation in theoretical computer science.</p>
  
  <p><span class="math-container">$[m]$</span> is <em>not</em> the variable <span class="math-container">$m$</span>, but is instead the set of integers from <span class="math-container">$1$</span> to <span class="math-container">$m$</span> inclusive. The empirical error equation thus reads in English:</p>
  
  <p>The cardinality of a set consisting of the elements <span class="math-container">$i$</span> of the set of integers <span class="math-container">$[m]$</span> such that the hypothesis given input <span class="math-container">$x_i$</span> disagrees with label <span class="math-container">$y_i$</span>, normalized by <span class="math-container">$m$</span>. </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2019-10-11T04:25:48.507Z
Answers.LastActivityDate: 2019-10-11T04:25:48.507Z
Answers.UserDisplayName: John Doucette
Body: <p>The empirical error equation given in the book <a href="https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf" rel="nofollow noreferrer">Understanding Machine Learning: From Theory to Algorithms</a> is </p>
  
  <p><a href="https://i.sstatic.net/XPFmE.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XPFmE.png" alt="enter image description here"></a></p>
  
  <p>My intuition for this equation is: total wrong predictions divided by the total number of samples <span class="math-container">$m$</span> in the given sample set <span class="math-container">$S$</span> (Correct me if I'm wrong). But, in this equation, the <span class="math-container">$m$</span> takes <span class="math-container">$\{ 1, \dots, m \}$</span>. How is this actually calculated, as I thought it should be one number (the size of the sample)?</p>
  
Tags: <machine-learning><computational-learning-theory><notation><books>
Title: What does the notation $[m]=\{1, \ldots, m\}$ mean in the equation of the empirical error?
UserDisplayName: user30381
ThreadText: # What does the notation $[m]=\{1, \ldots, m\}$ mean in the equation of the empirical error?
  
  (Asked by: username_0 on 2019-10-11)
  
  The empirical error equation given in the book [Understanding Machine Learning: From Theory to Algorithms](https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf) is
  
  [![enter image description here](https://i.sstatic.net/XPFmE.png)](https://i.sstatic.net/XPFmE.png)
  
  My intuition for this equation is: total wrong predictions divided by the total number of samples $m$ in the given sample set $S$ (Correct me if I'm wrong). But, in this equation, the $m$ takes $\{ 1, \dots, m \}$. How is this actually calculated, as I thought it should be one number (the size of the sample)?
  
  **Tags:** machine-learning, computational-learning-theory, notation, books
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2019-10-11. Score: 2 | Accepted answer
  
  This is a commonly used notation in theoretical computer science.
  
  $[m]$ is *not* the variable $m$, but is instead the set of integers from $1$ to $m$ inclusive. The empirical error equation thus reads in English:
  
  The cardinality of a set consisting of the elements $i$ of the set of integers $[m]$ such that the hypothesis given input $x\_i$ disagrees with label $y\_i$, normalized by $m$.
  
  
  ---
  
  