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<!DOCTYPE html>
<html lang="en"><head>
<script src="index_files/libs/clipboard/clipboard.min.js"></script>
<script src="index_files/libs/quarto-html/tabby.min.js"></script>
<script src="index_files/libs/quarto-html/popper.min.js"></script>
<script src="index_files/libs/quarto-html/tippy.umd.min.js"></script>
<link href="index_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/light-border.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="index_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.3.361">
<title>index</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reset.css">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
</style>
<link rel="stylesheet" href="index_files/libs/revealjs/dist/theme/quarto.css">
<link href="index_files/libs/revealjs/plugin/quarto-line-highlight/line-highlight.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/quarto-menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
.callout.callout-style-simple {
padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
.callout.callout-style-default {
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
.callout .callout-body-container {
flex-grow: 1;
}
.callout.callout-style-simple .callout-body {
font-size: 1rem;
font-weight: 400;
}
.callout.callout-style-default .callout-body {
font-size: 0.9rem;
font-weight: 400;
}
.callout.callout-titled.callout-style-simple .callout-body {
margin-top: 0.2em;
}
.callout:not(.callout-titled) .callout-body {
display: flex;
}
.callout:not(.no-icon).callout-titled.callout-style-simple .callout-content {
padding-left: 1.6em;
}
.callout.callout-titled .callout-header {
padding-top: 0.2em;
margin-bottom: -0.2em;
}
.callout.callout-titled .callout-title p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.callout.callout-titled.callout-style-simple .callout-content p {
margin-top: 0;
}
.callout.callout-titled.callout-style-default .callout-content p {
margin-top: 0.7em;
}
.callout.callout-style-simple div.callout-title {
border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
.callout.callout-style-default div.callout-title {
border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-default div.callout-content {
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-simple .callout-icon::before {
height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
}
.callout.callout-style-default .callout-icon::before {
height: 0.9rem;
width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
}
.callout-title {
display: flex
}
.callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
.callout.no-icon::before {
display: none !important;
}
.callout.callout-titled .callout-body > .callout-content > :last-child {
margin-bottom: 0.5rem;
}
.callout.callout-titled .callout-icon::before {
margin-top: .5rem;
padding-right: .5rem;
}
.callout:not(.callout-titled) .callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
/* Callout Types */
div.callout-note {
border-left-color: #4582ec !important;
}
div.callout-note .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
}
div.callout-important .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-important.callout-style-default .callout-title {
background-color: #f7dddc
}
div.callout-warning {
border-left-color: #f0ad4e !important;
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<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section class="slide level2">
<h1>
Outrageously efficient<br>exploratory data analysis
</h1>
<h2>
with Apache Arrow and <code>dplyr</code>
</h2>
<hr>
<h3>
Tom Mock, Customer Enablement Lead at
</h3>
<h3>
2022-06-03
</h3>
<p><br></p>
<h3>
<p><code>r fontawesome::fa("github", "black")</code> <a href="https://github.com/jthomasmock/arrow-dplyr">github.com/jthomasmock/arrow-dplyr</a></p>
<p><img data-src="https://arrow.apache.org/img/offbrand_hex_2.png" class="absolute" style="top: 425px; left: 1100px; width: 300px; "> <img data-src="https://raw.githubusercontent.com/rstudio/hex-stickers/master/PNG/dplyr.png" class="absolute" style="top: 680px; left: 1250px; width: 300px; "></p>
<p><img data-src="https://www.rstudio.com/wp-content/uploads/2018/10/RStudio-Logo-Flat.png" class="absolute" style="top: 360px; left: 965px; height: 65px; "></p>
</h3></section>
<section id="introduction-to-gaussian-processes-1" class="slide level2">
<h2>Introduction to Gaussian Processes 1</h2>
<ul>
<li>Gaussian processes provide a mechanism for directly reasoning about the high-level properties of functions that could fit our data.</li>
</ul>
<div class="fragment">
<ul>
<li>For example, we may have a sense of whether these functions are quickly varying, periodic, involve conditional independencies, or translation invariance.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>Gaussian processes enable us to easily incorporate these properties into our model, by directly specifying a Gaussian distribution over the function values that could fit our data.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-2" class="slide level2">
<h2>Introduction to Gaussian Processes 2</h2>
<ul>
<li><p>Suppose we observe the following dataset, of regression targets (outputs), <span class="math inline">\(y\)</span>, indexed by inputs, <span class="math inline">\(x\)</span>. <img data-src="../img/gp-observed-data.svg" style="width:80.0%" alt="Observed data."> . . .</p></li>
<li><p>As an example, the targets could be changes in carbon dioxide concentrations, and the inputs could be the times at which these targets have been recorded.</p></li>
</ul>
<div class="fragment">
<ul>
<li>What are some features of the data? How quickly does it seem to varying? Do we have data points collected at regular intervals, or are there missing inputs? How would you imagine filling in the missing regions, or forecasting up until <span class="math inline">\(x=25\)</span>?</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-3" class="slide level2">
<h2>Introduction to Gaussian Processes 3</h2>
<ul>
<li>start by specifying a prior distribution over what types of functions we might believe to be reasonable.</li>
</ul>
<div class="fragment">
<ul>
<li>Here we show several sample functions from a Gaussian process. Does this prior look reasonable? Note here we are not looking for functions that fit our dataset, but instead for specifying reasonable high-level properties of the solutions, such as how quickly they vary with inputs. Note that we will see code for reproducing all of the plots in this notebook, in the next notebooks on priors and inference.</li>
</ul>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="../img/gp-sample-prior-functions.svg" style="width:70.0%"></p>
<figcaption>Sample prior functions that we may want to represent with our model.</figcaption>
</figure>
</div>
</div>
</section>
<section id="introduction-to-gaussian-processes-4" class="slide level2">
<h2>Introduction to Gaussian Processes 4</h2>
<p>Once we condition on data, we can use this prior to infer a posterior distribution over functions that could fit the data. Here we show sample posterior functions.</p>
<img data-src="../img/gp-sample-posterior-functions.svg" class="r-stretch quarto-figure-center"><p class="caption">Sample posterior functions, once we have observed the data.</p><div class="fragment">
<ul>
<li><p>each of these functions are entirely consistent with our data, perfectly running through each observation. . . .</p></li>
<li><p>In order to use these posterior samples to make predictions, we can average the values of every possible sample function from the posterior, to create the curve below, in thick blue. . . .</p></li>
<li><p>Note that we do not actually have to take an infinite number of samples to compute this expectation; as we will see later, we can compute the expectation in closed form.</p></li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-5" class="slide level2">
<h2>Introduction to Gaussian Processes 5</h2>
<p><img data-src="../img/gp-posterior-samples.svg" style="width:50.0%" alt="Posterior samples, alongside posterior mean, which can be used for point predictions, in blue."> . . .</p>
<ul>
<li><p>may also want a representation of uncertainty, so we know how confident we should be in our predictions. Intuitively, we should have more uncertainty where there is more variability in the sample posterior functions, as this tells us there are many more possible values the true function could take. . . .</p></li>
<li><p>This type of uncertainty is called <em>epistemic uncertainty</em>, which is the <em>reducible uncertainty</em> associated with lack of information. As we acquire more data, this type of uncertainty disappears, as there will be increasingly fewer solutions consistent with what we observe. Like with the posterior mean, we can compute the posterior variance (the variability of these functions in the posterior) in closed form.</p></li>
</ul>
<div class="fragment">
<ul>
<li>With shade, we show two times the posterior standard deviation on either side of the mean, creating a <em>credible interval</em> that has a 95% probability of containing the true value of the function for any input <span class="math inline">\(x\)</span>.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-6" class="slide level2">
<h2>Introduction to Gaussian Processes 6</h2>
<img data-src="../img/gp-posterior-samples-95.svg" style="width:50.0%" class="r-stretch quarto-figure-center"><p class="caption">Posterior samples, including 95% credible set.</p><p>The plot looks somewhat cleaner if we remove the posterior samples, simply visualizing the data, posterior mean, and 95% credible set. Notice how the uncertainty grows away from the data, a property of epistemic uncertainty.</p>
</section>
<section id="introduction-to-gaussian-processes-7" class="slide level2">
<h2>Introduction to Gaussian Processes 7</h2>
<img data-src="../img/gp-point-predictions.svg" class="r-stretch quarto-figure-center"><p class="caption">Point predictions, and credible set.</p><p>The properties of the Gaussian process that we used to fit the data are strongly controlled by what’s called a <em>covariance function</em>, also known as a <em>kernel</em>. The covariance function we used is called the <em>RBF (Radial Basis Function) kernel</em>, which has the form <span class="math display">\[ k_{\text{RBF}}(x,x') = \mathrm{Cov}(f(x),f(x')) = a^2 \exp\left(-\frac{1}{2\ell^2}||x-x'||^2\right) \]</span></p>
</section>
<section id="introduction-to-gaussian-processes-8" class="slide level2">
<h2>Introduction to Gaussian Processes 8</h2>
<p><span class="math display">\[ k_{\text{RBF}}(x,x') = \mathrm{Cov}(f(x),f(x')) = a^2 \exp\left(-\frac{1}{2\ell^2}||x-x'||^2\right) \]</span> . . .</p>
<ul>
<li><p>The <em>hyperparameters</em> of this kernel are interpretable. The <em>amplitude</em> parameter <span class="math inline">\(a\)</span> controls the vertical scale over which the function is varying, and the <em>length-scale</em> parameter <span class="math inline">\(\ell\)</span> controls the rate of variation (the wiggliness) of the function. . . .</p></li>
<li><p>Larger <span class="math inline">\(a\)</span> means larger function values, and larger <span class="math inline">\(\ell\)</span> means more slowly varying functions. Let’s see what happens to our sample prior and posterior functions as we vary <span class="math inline">\(a\)</span> and <span class="math inline">\(\ell\)</span>.</p></li>
</ul>
<div class="fragment">
<ul>
<li>The <em>length-scale</em> has a particularly pronounced effect on the predictions and uncertainty of a GP. At <span class="math inline">\(||x-x'|| = \ell\)</span> , the covariance between a pair of function values is <span class="math inline">\(a^2\exp(-0.5)\)</span>. At larger distances than <span class="math inline">\(\ell\)</span> , the values of the function values becomes nearly uncorrelated. This means that if we want to make a prediction at a point <span class="math inline">\(x_*\)</span>, then function values with inputs <span class="math inline">\(x\)</span> such that <span class="math inline">\(||x-x'||>\ell\)</span> will not have a strong effect on our predictions.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-9" class="slide level2">
<h2>Introduction to Gaussian Processes 9</h2>
<p>Let’s see how changing the lengthscale affects sample prior and posterior functions, and credible sets. The above fits use a length-scale of <span class="math inline">\(2\)</span>. Let’s now consider <span class="math inline">\(\ell = 0.1, 0.5, 2, 5, 10\)</span> .</p>
<div class="fragment">
<ul>
<li>A length-scale of <span class="math inline">\(0.1\)</span> is very small relative to the range of the input domain we are considering, <span class="math inline">\(25\)</span>. For example, the values of the function at <span class="math inline">\(x=5\)</span> and <span class="math inline">\(x=10\)</span> will have essentially no correlation at such a length-scale. On the other hand, for a length-scale of <span class="math inline">\(10\)</span>, the function values at these inputs will be highly correlated. Note that the vertical scale changes in the following figures.</li>
</ul>
<p><img data-src="../img/gp-priorpoint1.svg" style="width:30.0%" alt="priorpoint1"> <img data-src="../img/gp-postpoint1.svg" style="width:30.0%" alt="postpoint1"></p>
<p><img data-src="../img/gp-priorpoint5.svg" style="width:30.0%" alt="priorpoint5"> <img data-src="../img/gp-postpoint5.svg" style="width:30.0%" alt="postpoint5"></p>
</div>
</section>
<section id="introduction-to-gaussian-processes-10" class="slide level2">
<h2>Introduction to Gaussian Processes 10</h2>
<ul>
<li>A length-scale of <span class="math inline">\(0.1\)</span> is very small relative to the range of the input domain we are considering, <span class="math inline">\(25\)</span>. For example, the values of the function at <span class="math inline">\(x=5\)</span> and <span class="math inline">\(x=10\)</span> will have essentially no correlation at such a length-scale. On the other hand, for a length-scale of <span class="math inline">\(10\)</span>, the function values at these inputs will be highly correlated. Note that the vertical scale changes in the following figures.</li>
</ul>
<p><img data-src="../img/gp-prior2.svg" style="width:30.0%" alt="prior2"> <img data-src="../img/gp-post2.svg" style="width:30.0%" alt="post2"></p>
<p><img data-src="../img/gp-prior5.svg" style="width:30.0%" alt="prior5"> <img data-src="../img/gp-post5.svg" style="width:30.0%" alt="post5"></p>
<div class="fragment">
<ul>
<li>as the length-scale increases the ‘wiggliness’ of the functions decrease, and our uncertainty decreases.</li>
<li>If the length-scale is small, the uncertainty will quickly increase as we move away from the data, as the datapoints become less informative about the function values.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-11" class="slide level2">
<h2>Introduction to Gaussian Processes 11</h2>
<ul>
<li>now vary the amplitude parameter, holding the length-scale fixed at <span class="math inline">\(2\)</span>. Note the vertical scale is held fixed for the prior samples, and varies for the posterior samples, so you can clearly see both the increasing scale of the function, and the fits to the data.</li>
</ul>
<p><img data-src="../img/gp-priorap1.svg" style="width:30.0%" alt="priorap1"> <img data-src="../img/gp-postapoint1.svg" style="width:30.0%" alt="postapoint1"></p>
<p><img data-src="../img/gp-priora2.svg" style="width:30.0%" alt="priora2"> <img data-src="../img/gp-posta2.svg" style="width:30.0%" alt="posta2"></p>
<p><img data-src="../img/gp-priora8.svg" style="width:30.0%" alt="priora8"> <img data-src="../img/gp-posta8.svg" style="width:30.0%" alt="posta8"></p>
</section>
<section id="introduction-to-gaussian-processes-12" class="slide level2">
<h2>Introduction to Gaussian Processes 12</h2>
<p><span class="math display">\[ k_{\text{RBF}}(x,x') = \mathrm{Cov}(f(x),f(x')) = a^2 \exp\left(-\frac{1}{2\ell^2}||x-x'||^2\right) \]</span></p>
<div class="fragment">
<ul>
<li><p>amplitude parameter affects the scale of the function, but not the rate of variation. . . .</p></li>
<li><p>generalization performance of our procedure will depend on having reasonable values for these hyperparameters. . . .</p></li>
<li><p>Values of <span class="math inline">\(\ell=2\)</span> and <span class="math inline">\(a=1\)</span> appeared to provide reasonable fits, while some of the other values did not.</p></li>
</ul>
</div>
<div class="fragment">
<ul>
<li>Fortunately, there is a robust and automatic way to specify these hyperparameters, using what is called the <em>marginal likelihood</em>, which we will return to in the notebook on inference.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-13" class="slide level2">
<h2>Introduction to Gaussian Processes 13</h2>
<p>So what is a GP, really? As we started, a GP simply says that any collection of function values <span class="math inline">\(f(x_1),\dots,f(x_n)\)</span>, indexed by any collection of inputs <span class="math inline">\(x_1,\dots,x_n\)</span> has a joint multivariate Gaussian distribution. . . .</p>
<ul>
<li>The mean vector <span class="math inline">\(\mu\)</span> of this distribution is given by a <em>mean function</em>, which is typically taken to be a constant or zero. The covariance matrix of this distribution is given by the <em>kernel</em> evaluated at all pairs of the inputs <span class="math inline">\(x\)</span>.</li>
</ul>
<div class="fragment">
<ul>
<li><span class="math display">\[\begin{bmatrix}f(x) \\f(x_1) \\ \vdots \\ f(x_n) \end{bmatrix}\sim \mathcal{N}\left(\mu, \begin{bmatrix}k(x,x) & k(x, x_1) & \dots & k(x,x_n) \\ k(x_1,x) & k(x_1,x_1) & \dots & k(x_1,x_n) \\ \vdots & \vdots & \ddots & \vdots \\ k(x_n, x) & k(x_n, x_1) & \dots & k(x_n,x_n) \end{bmatrix}\right)\]</span> :eqlabel:<code>eq_gp_prior</code></li>
</ul>
</div>
<div class="fragment">
<ul>
<li>Equation :eqref:<code>eq_gp_prior</code> specifies a GP prior. We can compute the conditional distribution of <span class="math inline">\(f(x)\)</span> for any <span class="math inline">\(x\)</span> given <span class="math inline">\(f(x_1), \dots, f(x_n)\)</span>, the function values we have observed. This conditional distribution is called the <em>posterior</em>, and it is what we use to make predictions.</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-14" class="slide level2">
<h2>Introduction to Gaussian Processes 14</h2>
<p>In particular,</p>
<p><span class="math display">\[f(x) | f(x_1), \dots, f(x_n) \sim \mathcal{N}(m,s^2)\]</span></p>
<p>where</p>
<p><span class="math display">\[m = k(x,x_{1:n}) k(x_{1:n},x_{1:n})^{-1} f(x_{1:n})\]</span></p>
<p><span class="math display">\[s^2 = k(x,x) - k(x,x_{1:n})k(x_{1:n},x_{1:n})^{-1}k(x,x_{1:n})\]</span></p>
<p>where <span class="math inline">\(k(x,x_{1:n})\)</span> is a <span class="math inline">\(1 \times n\)</span> vector formed by evaluating <span class="math inline">\(k(x,x_{i})\)</span> for <span class="math inline">\(i=1,\dots,n\)</span> and <span class="math inline">\(k(x_{1:n},x_{1:n})\)</span> is an <span class="math inline">\(n \times n\)</span> matrix formed by evaluating <span class="math inline">\(k(x_i,x_j)\)</span> for <span class="math inline">\(i,j = 1,\dots,n\)</span>. <span class="math inline">\(m\)</span> is what we can use as a point predictor for any <span class="math inline">\(x\)</span>, and <span class="math inline">\(s^2\)</span> is what we use for uncertainty: if we want to create an interval with a 95% probability that <span class="math inline">\(f(x)\)</span> is in the interval, we would use <span class="math inline">\(m \pm 2s\)</span>. The predictive means and uncertainties for all the above figures were created using these equations. The observed data points were given by <span class="math inline">\(f(x_1), \dots, f(x_n)\)</span> and chose a fine grained set of <span class="math inline">\(x\)</span> points to make predictions.</p>
</section>
<section id="introduction-to-gaussian-processes-15" class="slide level2">
<h2>Introduction to Gaussian Processes 15</h2>
<p>Let’s suppose we observe a single datapoint, <span class="math inline">\(f(x_1)\)</span>, and we want to determine the value of <span class="math inline">\(f(x)\)</span> at some <span class="math inline">\(x\)</span>. Because <span class="math inline">\(f(x)\)</span> is described by a Gaussian process, we know the joint distribution over <span class="math inline">\((f(x), f(x_1))\)</span> is Gaussian:</p>
<p><span class="math display">\[
\begin{bmatrix}
f(x) \\
f(x_1) \\
\end{bmatrix}
\sim
\mathcal{N}\left(\mu,
\begin{bmatrix}
k(x,x) & k(x, x_1) \\
k(x_1,x) & k(x_1,x_1)
\end{bmatrix}
\right)
\]</span></p>
<p>The off-diagonal expression <span class="math inline">\(k(x,x_1) = k(x_1,x)\)</span> tells us how correlated the function values will be — how strongly determined <span class="math inline">\(f(x)\)</span> will be from <span class="math inline">\(f(x_1)\)</span>. We have seen already that if we use a large length-scale, relative to the distance between <span class="math inline">\(x\)</span> and <span class="math inline">\(x_1\)</span>, <span class="math inline">\(||x-x_1||\)</span>, then the function values will be highly correlated. We can visualize the process of determining <span class="math inline">\(f(x)\)</span> from <span class="math inline">\(f(x_1)\)</span> both in the space of functions, and in the joint distribution over <span class="math inline">\(f(x_1), f(x)\)</span>. Let’s initially consider an <span class="math inline">\(x\)</span> such that <span class="math inline">\(k(x,x_1) = 0.9\)</span>, and <span class="math inline">\(k(x,x)=1\)</span>, meaning that the value of <span class="math inline">\(f(x)\)</span> is moderately correlated with the value of <span class="math inline">\(f(x_1)\)</span>. In the joint distribution, the contours of constant probability will be relatively narrow ellipses.</p>
</section>
<section id="introduction-to-gaussian-processes-16" class="slide level2">
<h2>Introduction to Gaussian Processes 16</h2>
<p>Suppose we observe <span class="math inline">\(f(x_1) = 1.2\)</span>. To condition on this value of <span class="math inline">\(f(x_1)\)</span>, we can draw a horizontal line at <span class="math inline">\(1.2\)</span> on our plot of the density, and see that the value of <span class="math inline">\(f(x)\)</span> is mostly constrained to <span class="math inline">\([0.64,1.52]\)</span>. We have also drawn this plot in function space, showing the observed point <span class="math inline">\(f(x_1)\)</span> in orange, and 1 standard deviation of the Gaussian process predictive distribution for <span class="math inline">\(f(x)\)</span> in blue, about the mean value of <span class="math inline">\(1.08\)</span>.</p>
<p><img data-src="https://user-images.githubusercontent.com/6753639/206867364-b4707db5-0c2e-4ae4-a412-8292bca4d08d.svg" style="width:45.0%" alt="Contours of constant probability of a bivariate Gaussian density over f(x_1) and f(x) with k(x,x_1) = 0.9."> <img data-src="https://user-images.githubusercontent.com/6753639/206867367-3815720c-93c8-4b4b-80e7-296db1d3553b.svg" style="width:45.0%" alt="Gaussian process predictive distribution in function space at f(x), with k(x,x_1) = 0.9."></p>
</section>
<section id="introduction-to-gaussian-processes-17" class="slide level2">
<h2>Introduction to Gaussian Processes 17</h2>
<p>Now suppose we have a stronger correlation, <span class="math inline">\(k(x,x_1) = 0.95\)</span>. Now the ellipses have narrowed further, and the value of <span class="math inline">\(f(x)\)</span> is even more strongly determined by <span class="math inline">\(f(x_1)\)</span>. Drawing a horizontal line at <span class="math inline">\(1.2\)</span>, we see the contours for <span class="math inline">\(f(x)\)</span> support values mostly within <span class="math inline">\([0.83, 1.45]\)</span>. Again, we also show the plot in function space, with one standard deviation about the mean predictive value of <span class="math inline">\(1.14\)</span>.</p>
<p><img data-src="https://user-images.githubusercontent.com/6753639/206867797-20e42783-31de-4c50-8103-e9441ba6d0a9.svg" style="width:45.0%" alt="Contours of constant probability of a bivariate Gaussian density over f(x_1) and f(x) with k(x,x_1) = 0.95."> <img data-src="https://user-images.githubusercontent.com/6753639/206867800-d9fc7add-649d-492c-8848-cab07c8fb83e.svg" style="width:45.0%" alt="Gaussian process predictive distribution in function space at f(x), with k(x,x_1) = 0.95."></p>
<div class="fragment">
<ul>
<li>posterior mean predictor of our Gaussian process is closer to <span class="math inline">\(1.2\)</span>, because there is now a stronger correlation.</li>
<li>also uncertainty (the error bars) have somewhat decreased.</li>
<li>Despite the strong correlation between these function values, our uncertainty is still righly quite large, because we have only observed a single data point!</li>
</ul>
</div>
</section>
<section id="introduction-to-gaussian-processes-18" class="slide level2">
<h2>Introduction to Gaussian Processes 18</h2>
<p>This procedure can give us a posterior on <span class="math inline">\(f(x)\)</span> for any <span class="math inline">\(x\)</span>, for any number of points we have observed. Suppose we observe <span class="math inline">\(f(x_1), f(x_2)\)</span>.</p>
<div class="fragment">
<ul>
<li>We now visualize the posterior for <span class="math inline">\(f(x)\)</span> at a particular <span class="math inline">\(x=x'\)</span> in function space. The exact distribution for <span class="math inline">\(f(x)\)</span> is given by the above equations. <span class="math inline">\(f(x)\)</span> is Gaussian distributed, with mean</li>
</ul>
<p><span class="math display">\[m = k(x,x_{1:3}) k(x_{1:3},x_{1:3})^{-1} f(x_{1:3})\]</span></p>
<p>and variance</p>
<p><span class="math display">\[s^2 = k(x,x) - k(x,x_{1:3})k(x_{1:3},x_{1:3})^{-1}k(x,x_{1:3})\]</span></p>
</div>
<div class="fragment">
<ul>
<li><p>we have been considering <em>noise free</em> observations.</p></li>
<li><p>easy to include observation noise. If we assume that the data are generated from a latent noise free function <span class="math inline">\(f(x)\)</span> plus iid Gaussian noise <span class="math inline">\(\epsilon(x) \sim \mathcal{N}(0,\sigma^2)\)</span> with variance <span class="math inline">\(\sigma^2\)</span>, then our covariance function simply becomes <span class="math inline">\(k(x_i,x_j) \to k(x_i,x_j) + \delta_{ij}\sigma^2\)</span>, where <span class="math inline">\(\delta_{ij} = 1\)</span> if <span class="math inline">\(i=j\)</span> and <span class="math inline">\(0\)</span> otherwise.</p></li>
</ul>
</div>
</section>
<section id="summary-1" class="slide level2">
<h2>Summary 1</h2>
<ul>
<li>typical machine learning: we specify a function with some free parameters (such as a neural network and its weights), and we focus on estimating those parameters, which may not be interpretable.</li>
</ul>
<div class="fragment">
<ul>
<li>With a Gaussian process, we instead reason about distributions over functions directly, which enables us to reason about the high-level properties of the solutions.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>These properties are controlled by a covariance function (kernel), which often has a few highly interpretable hyperparameters. These hyperparameters include the <em>length-scale</em>, which controls how rapidly (how wiggily) the functions are. Another hyperparameter is the amplitude, which controls the vertical scale over which our functions are varying.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>Representing many different functions that can fit the data, and combining them all together into a predictive distribution, is a distinctive feature of Bayesian methods.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>greater amount of variability between possible solutions far away from the data –> uncertainty intuitively grows as we move from the data.</li>
</ul>
</div>
</section>
<section id="summary-2" class="slide level2">
<h2>Summary 2</h2>
<ul>
<li>Gaussian process represents a distribution over functions by specifying a multivariate normal (Gaussian) distribution over all possible function values.</li>
</ul>
<div class="fragment">
<ul>
<li>It is possible to easily manipulate Gaussian distributions to find the distribution of one function value based on the values of any set of other values.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>In other words, if we observe a set of points, then we can condition on these points and infer a distribution over what the value of the function might look like at any other input.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>How we model the correlations between these points is determined by the covariance function and is what defines the generalization properties of the Gaussian process.</li>
</ul>
</div>
<div class="fragment">
<ul>
<li>While it takes time to get used to Gaussian processes, they are easy to work with, have many applications, and help us understand and develop other model classes, like neural networks.</li>
</ul>
</div>
</section>
<section id="exercises" class="slide level2">
<h2>Exercises</h2>
<ol type="1">
<li>What is the difference between epistemic uncertainty versus observation uncertainty?</li>
<li>Besides rate of variation and amplitude, what other properties of functions might we want to consider, and what would be real-world examples of functions that have those properties?</li>
<li>The RBF covariance function we considered says that covariances (and correlations) between observations decrease with their distance in the input space (times, spatial locations, etc.). Is this a reasonable assumption? Why or why not?</li>
<li>Is a sum of two Gaussian variables Gaussian? Is a product of two Gaussian variables Gaussian? If (a,b) have a joint Gaussian distribution, is a|b (a given b) Gaussian? Is a Gaussian?</li>
<li>Repeat the exercise where we observe a data point at <span class="math inline">\(f(x_1) = 1.2\)</span>, but now suppose we additionally observe <span class="math inline">\(f(x_2) = 1.4\)</span>. Let <span class="math inline">\(k(x,x_1) = 0.9\)</span>, and <span class="math inline">\(k(x,x_2) = 0.8\)</span>. Will we be more or less certain about the value of <span class="math inline">\(f(x)\)</span>, than when we had only observed <span class="math inline">\(f(x_1)\)</span>? What is the mean and 95% credible set for our value of <span class="math inline">\(f(x)\)</span> now?</li>
<li>Do you think increasing our estimate of observation noise would increase or decrease our estimate of the length-scale of the ground truth function?</li>
<li>As we move away from the data, suppose the uncertainty in our predictive distribution increases to a point, then stops increasing. Why might that happen?</li>
</ol>
<p>:begin_tab:<code>pytorch</code> <a href="https://discuss.d2l.ai/t/12115">Discussions</a> :end_tab:</p>
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<p><a href="https://github.com/ECLIPSE-Lab/SS23_DataScienceForEM">SS23_DataScienceForEM</a></p>
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button.setAttribute("data-bs-toggle", "tooltip");
button.setAttribute("data-bs-placement", "left");
button.setAttribute("data-bs-title", "Copied!");
tooltip = new bootstrap.Tooltip(button,
{ trigger: "manual",
customClass: "code-copy-button-tooltip",
offset: [0, -8]});
tooltip.show();
}
setTimeout(function() {
if (tooltip) {
tooltip.hide();
button.removeAttribute("data-bs-title");
button.removeAttribute("data-bs-toggle");
button.removeAttribute("data-bs-placement");
}
button.setAttribute("title", currentTitle);
button.classList.remove('code-copy-button-checked');
}, 1000);
// clear code selection
e.clearSelection();
});
function tippyHover(el, contentFn) {
const config = {
allowHTML: true,
content: contentFn,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.closest('section.slide') || el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'light-border',
placement: 'bottom-start'
};
config['offset'] = [0,0];
config['maxWidth'] = 700;
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
if (citeInfo) {
tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {