Answers.Body: <p>You're pretty close:</p>
  <ul>
  <li><span class="math-container">$x,y$</span> correspond to a single data point</li>
  <li><span class="math-container">$\mathcal{D}$</span> is the whole dataset</li>
  </ul>
  <p>Given this, you can read the posterior <span class="math-container">$p(y|x,D)$</span> as <em>&quot; what's the distribution of <span class="math-container">$y$</span> given that I observe <span class="math-container">$x$</span> and my dataset is <span class="math-container">$D$</span>&quot;</em></p>
  <p>At this point, they say that reasoning over a whole dataset is not feasible, so they introduce a new random variable <span class="math-container">$\theta$</span>, from which you can reason over how to solve this</p>
  
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Answers.CreationDate: 2023-09-19T17:30:15.353Z
Answers.Text: @Seo you’re welcome
Answers.UserDisplayName: Alberto
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2023-09-15T10:52:19.133Z
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Answers.UserDisplayName: Alberto
Body: <p>I'm new to the Bayesian perspective and would appreciate clarity on this.</p>
  <p>In a few resources concerning Bayesian deep learning (such as <a href="https://uvadlc-notebooks.readthedocs.io/en/latest/tutorial_notebooks/DL2/Bayesian_Neural_Networks/dl2_bnn_tut1_students_with_answers.html" rel="nofollow noreferrer">this one</a>), I see this notation:</p>
  <p><span class="math-container">$p(y|x, D) = \int p(y|x, \theta)p(\theta|D)d\theta$</span></p>
  <p>for input input <span class="math-container">$x$</span>, corresponding output <span class="math-container">$y$</span>, network weights and biases <span class="math-container">$\theta$</span>, likelihood function <span class="math-container">$p(y|x, \theta)$</span>, and &quot;posterior distribution over the weights and biases given the observed data <span class="math-container">$D$</span>&quot; <span class="math-container">$p(\theta|D)$</span>.</p>
  <p>I would have thought that the observed data would be the input, but apparently <span class="math-container">$x$</span> and <span class="math-container">$D$</span> are not the same thing. My question is what is <span class="math-container">$D$</span> in this context, and how is it different than <span class="math-container">$x$</span>?</p>
  
Tags: <notation><bayesian-networks><bayesian-deep-learning><bayesian-neural-networks>
Title: What is the difference between an input and observed data in a Bayesian neural network?
UserDisplayName: Seo
ThreadText: # What is the difference between an input and observed data in a Bayesian neural network?
  
  (Asked by: username_0 on 2023-09-15)
  
  I'm new to the Bayesian perspective and would appreciate clarity on this.
  
  In a few resources concerning Bayesian deep learning (such as [this one](https://uvadlc-notebooks.readthedocs.io/en/latest/tutorial_notebooks/DL2/Bayesian_Neural_Networks/dl2_bnn_tut1_students_with_answers.html)), I see this notation:
  
  $p(y|x, D) = \int p(y|x, \theta)p(\theta|D)d\theta$
  
  for input input $x$, corresponding output $y$, network weights and biases $\theta$, likelihood function $p(y|x, \theta)$, and "posterior distribution over the weights and biases given the observed data $D$" $p(\theta|D)$.
  
  I would have thought that the observed data would be the input, but apparently $x$ and $D$ are not the same thing. My question is what is $D$ in this context, and how is it different than $x$?
  
  **Tags:** notation, bayesian-networks, bayesian-deep-learning, bayesian-neural-networks
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2023-09-15. Score: 1 | Accepted answer
  
  You're pretty close:
  
  * $x,y$ correspond to a single data point
  * $\mathcal{D}$ is the whole dataset
  
  Given this, you can read the posterior $p(y|x,D)$ as *" what's the distribution of $y$ given that I observe $x$ and my dataset is $D$"*
  
  At this point, they say that reasoning over a whole dataset is not feasible, so they introduce a new random variable $\theta$, from which you can reason over how to solve this
  
  > **Comments:**
  > **username_1**: @username_0 you’re welcome  
  
  ---
  
  