Answers.Body: <p>If by &quot;effective&quot; hypotheses you mean distinct hypotheses, this sentence is false. VC dimension of a hypothesis set <span class="math-container">$H$</span> is simply the cardinality of a largest set of data points which can be shattered by <span class="math-container">$H$</span>. Therefore <span class="math-container">$H$</span> containing a large number or even infinite number of distinct hypotheses could still have a low VC dimension, if <span class="math-container">$H$</span> cannot shatter many distinct arrangements of a small set of data points. For a simple example of <span class="math-container">$H$</span> which is the set of all 2D linear classifiers, its VC dimension is obviously 3 though <span class="math-container">$H$</span> has infinitely many distinct hypotheses as distinct lines in a 2D plane. You need to know <span class="math-container">$H$</span>'s shattering capacity to decide its VC dimension in any proportional sense.</p>
  <p>However, if by &quot;effective&quot; hypotheses you mean only those representative hypotheses from their own equivalence class of a hypothesis set <span class="math-container">$H$</span> which can shatter all arrangements of a set of data points, this sentence is (trivially) true, since per PAC theory, VC dimension <span class="math-container">$n$</span> is just defined as the finite &quot;effective&quot; size or complexity of an arbitrary hypothesis set <span class="math-container">$H$</span> to shatter a largest set of <span class="math-container">$n$</span> data points.</p>
  
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Body: <p>True or False. The VC dimension of a hypothesis set H is proportional to the number of effective hypotheses in H. Explain the reason if it is true.</p>
  <p>This is what ChatGPT answers:</p>
  <blockquote>
  <p>True. The VC (Vapnik-Chervonenkis) dimension of a hypothesis set
  <span class="math-container">$$</span> is proportional to the number of effective hypotheses in
  <span class="math-container">$$</span> because the VC dimension measures the capacity of
  <span class="math-container">$H$</span> to shatter different arrangements of data points. A hypothesis set with a larger number of distinct, effective hypotheses will be capable of shattering more combinations of points, leading to a higher VC dimension.</p>
  </blockquote>
  
Tags: <machine-learning><vc-dimension>
Title: Is VC dimension of a hypothesis set H proportional to the number of effective hypotheses in H?
UserDisplayName: DSPinfinity
ThreadText: # Is VC dimension of a hypothesis set H proportional to the number of effective hypotheses in H?
  
  (Asked by: username_0 on 2024-11-07)
  
  True or False. The VC dimension of a hypothesis set H is proportional to the number of effective hypotheses in H. Explain the reason if it is true.
  
  This is what ChatGPT answers:
  
  > True. The VC (Vapnik-Chervonenkis) dimension of a hypothesis set
  > $$ is proportional to the number of effective hypotheses in
  > $$ because the VC dimension measures the capacity of
  > $H$ to shatter different arrangements of data points. A hypothesis set with a larger number of distinct, effective hypotheses will be capable of shattering more combinations of points, leading to a higher VC dimension.
  
  **Tags:** machine-learning, vc-dimension
  
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  ## 1 answers
  
  ### Answer by username_1 on 2024-11-07. Score: 2 | Accepted answer
  
  If by "effective" hypotheses you mean distinct hypotheses, this sentence is false. VC dimension of a hypothesis set $H$ is simply the cardinality of a largest set of data points which can be shattered by $H$. Therefore $H$ containing a large number or even infinite number of distinct hypotheses could still have a low VC dimension, if $H$ cannot shatter many distinct arrangements of a small set of data points. For a simple example of $H$ which is the set of all 2D linear classifiers, its VC dimension is obviously 3 though $H$ has infinitely many distinct hypotheses as distinct lines in a 2D plane. You need to know $H$'s shattering capacity to decide its VC dimension in any proportional sense.
  
  However, if by "effective" hypotheses you mean only those representative hypotheses from their own equivalence class of a hypothesis set $H$ which can shatter all arrangements of a set of data points, this sentence is (trivially) true, since per PAC theory, VC dimension $n$ is just defined as the finite "effective" size or complexity of an arbitrary hypothesis set $H$ to shatter a largest set of $n$ data points.
  
  
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