Answers.Body: <p>Formally, the functions of your conditional normalizing flow are <span class="math-container">$f(y, x) = z$</span> and <span class="math-container">$f^{-1}(z, x) = y$</span>, where we model the likelihood of <span class="math-container">$y$</span> conditional on <span class="math-container">$x$</span>.</p>
  <p>The probability density can be expressed as:</p>
  <p><span class="math-container">$$
  p_{Y \vert X}(y \vert x) = p_{Z \vert X}(z \vert x) \vert J \vert
  $$</span></p>
  <p>In optimizing the <span class="math-container">$p_{Y \vert X}$</span> term of the objective, the model learns <span class="math-container">$z$</span> to represent how &quot;unpredictable&quot; <span class="math-container">$y$</span> is, being incentivised to learn the highest probability that doesn't lose any information.</p>
  <blockquote>
  <p>Is <span class="math-container">$f^{−1}$</span> able to &quot;cheat&quot; by leaking information about <span class="math-container">$y$</span> directly into <span class="math-container">$z$</span>, or do the theoretical guarantees of normalizing flows side-step this?</p>
  </blockquote>
  <p>Now, when passed through the inverse, only the information about <span class="math-container">$y$</span> that it &quot;couldn't have known&quot; is preserved through <span class="math-container">$z$</span>, which is what <span class="math-container">$f^{-1}$</span> receives.</p>
  <p>Conditional normalizing flows might not be what you're looking for in adapting BYOL-Hindsight because the likelihood consists of both what it doesn't know and what it knows it can't know. Having the actor look for places with lower probability would likely reproduce the noisy tv problem.</p>
  <p>Perhaps you could instead use it to task the agent to find states with high probability, that the model still doesn't predict well.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-03-03T01:26:07.053Z
Answers.LastActivityDate: 2024-03-03T01:26:07.053Z
Answers.UserDisplayName: Richie Bendall
Body: <p>Consider the Conditional Normalizing Flow (informally): <span class="math-container">$f(x, z) = y$</span> with its inverse <span class="math-container">$f^{-1}(x, y) = z$</span> where <span class="math-container">$x$</span> is the prior and <span class="math-container">$z$</span> is the posterior.</p>
  <p>Is <span class="math-container">$f^{-1}$</span> able to &quot;cheat&quot; by leaking information about <span class="math-container">$y$</span> directly into <span class="math-container">$z$</span>, or do the theoretical guarantees of normalizing flows side-step this?</p>
  <p>I want to use <span class="math-container">$z$</span> as a way to represent anything about <span class="math-container">$y$</span> that can't be deduced from only <span class="math-container">$x$</span> (similar to <a href="https://openreview.net/pdf?id=fIH2G4fnSy" rel="nofollow noreferrer">BYOL-Hindsight</a>) so that I can randomly sample from <span class="math-container">$z$</span> for inference.</p>
  
Tags: <bayesian-deep-learning>
Title: Can Conditional Normalizing Flows Cheat?
UserDisplayName: Richie Bendall
ThreadText: # Can Conditional Normalizing Flows Cheat?
  
  (Asked by: username_0 on 2023-10-02)
  
  Consider the Conditional Normalizing Flow (informally): $f(x, z) = y$ with its inverse $f^{-1}(x, y) = z$ where $x$ is the prior and $z$ is the posterior.
  
  Is $f^{-1}$ able to "cheat" by leaking information about $y$ directly into $z$, or do the theoretical guarantees of normalizing flows side-step this?
  
  I want to use $z$ as a way to represent anything about $y$ that can't be deduced from only $x$ (similar to [BYOL-Hindsight](https://openreview.net/pdf?id=fIH2G4fnSy)) so that I can randomly sample from $z$ for inference.
  
  **Tags:** bayesian-deep-learning
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_0 on 2024-03-03. Score: 0 | Accepted answer
  
  Formally, the functions of your conditional normalizing flow are $f(y, x) = z$ and $f^{-1}(z, x) = y$, where we model the likelihood of $y$ conditional on $x$.
  
  The probability density can be expressed as:
  
  $$
  p\_{Y \vert X}(y \vert x) = p\_{Z \vert X}(z \vert x) \vert J \vert
  $$
  
  In optimizing the $p\_{Y \vert X}$ term of the objective, the model learns $z$ to represent how "unpredictable" $y$ is, being incentivised to learn the highest probability that doesn't lose any information.
  
  > Is $f^{−1}$ able to "cheat" by leaking information about $y$ directly into $z$, or do the theoretical guarantees of normalizing flows side-step this?
  
  Now, when passed through the inverse, only the information about $y$ that it "couldn't have known" is preserved through $z$, which is what $f^{-1}$ receives.
  
  Conditional normalizing flows might not be what you're looking for in adapting BYOL-Hindsight because the likelihood consists of both what it doesn't know and what it knows it can't know. Having the actor look for places with lower probability would likely reproduce the noisy tv problem.
  
  Perhaps you could instead use it to task the agent to find states with high probability, that the model still doesn't predict well.
  
  
  ---
  
  