Answers.Body: <p>The equation you are referring to is called <a href="https://en.wikipedia.org/wiki/Mean_squared_error" rel="nofollow noreferrer"><em>Mean Squared Error</em></a> (or <span class="math-container">$L_2$</span> loss) and it is used for <a href="https://en.wikipedia.org/wiki/Regression_analysis" rel="nofollow noreferrer"><em>regression</em></a> tasks, where the goal is to predict a real value given some input.</p>
  <p>In your case, the inputs are measurements of temperature <span class="math-container">$y$</span>, either at a certain point in time or point in space or both or none, this is not clear from the image. Now, the goal would be to predict the temperature at a new point in space, time, or both, where we don't have access to a measurement. That is we would like to find a function <span class="math-container">$f$</span> (e.g. a simple linear function) which we can use for prediction. But how can we measure which function is &quot;best&quot;? We introduce a <em>loss function</em> <span class="math-container">$L(f,y)$</span>, another function which tells us how good our proposed function is.</p>
  <p>Visually it looks like this (<a href="https://medium.com/@jorgesleonel/linear-regression-307937441a8b" rel="nofollow noreferrer">image source</a>):</p>
  <p><a href="https://i.sstatic.net/im1oe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/im1oe.png" alt="enter image description here" /></a></p>
  <p>Red crossed are measurements, the black line is our function we use for prediction and the green dotted lines are the errors (the distance from our prediction to the real measurement). In this example salary depends on experience.</p>
  <p>Now, the paper introduces the constant mean of all measurements as <span class="math-container">$y$</span>, <span class="math-container">$z = \mathbb{E}_y\{y\} = \frac{1}{N}\sum_i^N y_i$</span>, as our function <span class="math-container">$f$</span>, which is known to be the minimizer for the <span class="math-container">$L_2$</span> loss in the case where there is no dependence on other variables (e.g. time or space).</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-11-21T14:22:47.180Z
Answers.Text: Note that the expectation is not exactly an average, unless you have access to all possible $y_i$ values. Moreover, you should probably comment on the symbol $\mathbb{E}$.
Answers.UserDisplayName: nbro
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Answers.CreationDate: 2020-11-25T10:54:59.990Z
Answers.Text: $\mathbb{E}_y\{y\}$ is for expected value of y?
Answers.UserDisplayName: Markov
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-11-25T12:26:49.250Z
Answers.Text: Yes, correct. $\mathbb{E}_y\{y\}$ is the expected value of the variable y.
Answers.UserDisplayName: Tinu
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Answers.CreationDate: 2020-11-21T11:38:55.193Z
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Answers.LastEditDate: 2020-11-21T14:18:27.513Z
Answers.UserDisplayName: Tinu
Body: <p>I am trying to understand what is meant by following equations in the <a href="https://arxiv.org/abs/1803.04189" rel="nofollow noreferrer">Noise2Noise paper</a> by Nvidia.</p>
  <p><a href="https://i.sstatic.net/gSKGh.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gSKGh.jpg" alt="enter image description here" /></a></p>
  <p>What is meant by the equation in this image? What is <span class="math-container">$\mathbb{E}_y\{y\}$</span>? And how should I try to visualize these equations?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-11-21T14:15:18.423Z
Comments.Text: Which equations are you referring to? There are at least 3 equations there and other math formulas.
Comments.UserDisplayName: nbro
Tags: <papers><supervised-learning><notation><expectation><mean-squared-error>
Title: What is the meaning of these equations in Noise2Noise paper?
UserDisplayName: Markov
ThreadText: # What is the meaning of these equations in Noise2Noise paper?
  
  (Asked by: username_0 on 2020-11-21)
  
  I am trying to understand what is meant by following equations in the [Noise2Noise paper](https://arxiv.org/abs/1803.04189) by Nvidia.
  
  [![enter image description here](https://i.sstatic.net/gSKGh.jpg)](https://i.sstatic.net/gSKGh.jpg)
  
  What is meant by the equation in this image? What is $\mathbb{E}\_y\{y\}$? And how should I try to visualize these equations?
  
  **Tags:** papers, supervised-learning, notation, expectation, mean-squared-error
  
  > **Comments:**
  > **username_1**: Which equations are you referring to? There are at least 3 equations there and other math formulas.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2020-11-21. Score: 1 | Accepted answer
  
  The equation you are referring to is called [*Mean Squared Error*](https://en.wikipedia.org/wiki/Mean_squared_error) (or $L\_2$ loss) and it is used for [*regression*](https://en.wikipedia.org/wiki/Regression_analysis) tasks, where the goal is to predict a real value given some input.
  
  In your case, the inputs are measurements of temperature $y$, either at a certain point in time or point in space or both or none, this is not clear from the image. Now, the goal would be to predict the temperature at a new point in space, time, or both, where we don't have access to a measurement. That is we would like to find a function $f$ (e.g. a simple linear function) which we can use for prediction. But how can we measure which function is "best"? We introduce a *loss function* $L(f,y)$, another function which tells us how good our proposed function is.
  
  Visually it looks like this ([image source](https://medium.com/@jorgesleonel/linear-regression-307937441a8b)):
  
  [![enter image description here](https://i.sstatic.net/im1oe.png)](https://i.sstatic.net/im1oe.png)
  
  Red crossed are measurements, the black line is our function we use for prediction and the green dotted lines are the errors (the distance from our prediction to the real measurement). In this example salary depends on experience.
  
  Now, the paper introduces the constant mean of all measurements as $y$, $z = \mathbb{E}\_y\{y\} = \frac{1}{N}\sum\_i^N y\_i$, as our function $f$, which is known to be the minimizer for the $L\_2$ loss in the case where there is no dependence on other variables (e.g. time or space).
  
  > **Comments:**
  > **username_1**: Note that the expectation is not exactly an average, unless you have access to all possible $y_i$ values. Moreover, you should probably comment on the symbol $\mathbb{E}$.  
  > **username_0**: $\mathbb{E}_y\{y\}$ is for expected value of y?  
  > **username_2**: Yes, correct. $\mathbb{E}_y\{y\}$ is the expected value of the variable y.  
  
  ---
  
  