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<!DOCTYPE html>
<html lang="en"><head>
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<meta name="generator" content="quarto-1.0.36">
<meta name="author" content="Johann de Boer">
<meta name="dcterms.date" content="2022-10-22">
<title>Introduction to Bayesian A/B testing</title>
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<div class="reveal">
<div class="slides">
<section id="title-slide" class="center">
<h1 class="title">Introduction to Bayesian A/B testing</h1>
<p class="author">Johann de Boer</p>
<p class="date">2022-10-22</p>
</section>
<section id="TOC">
<nav role="doc-toc">
<h2 id="toc-title">Agenda</h2>
<ul>
<li><a href="#/setting-the-scene" id="/toc-setting-the-scene">Setting the scene</a></li>
<li><a href="#/priors-and-probability-distributions" id="/toc-priors-and-probability-distributions">Priors and probability distributions</a></li>
<li><a href="#/running-a-bayesian-ab-test" id="/toc-running-a-bayesian-ab-test">Running a Bayesian A/B test</a></li>
<li><a href="#/posterior-analysis" id="/toc-posterior-analysis">Posterior analysis</a></li>
<li><a href="#/summary-and-some-final-remarks" id="/toc-summary-and-some-final-remarks">Summary and some final remarks</a></li>
<li><a href="#/thank-you" id="/toc-thank-you">Thank you!</a></li>
<li><a href="#/when-to-stop-a-bayesian-ab-test" id="/toc-when-to-stop-a-bayesian-ab-test">When to stop a Bayesian A/B test?</a></li>
<li><a href="#/extras" id="/toc-extras">Extras</a></li>
<li><a href="#/game-of-chances" id="/toc-game-of-chances">Game of chances</a></li>
</ul>
</nav>
</section>
<section>
<section id="setting-the-scene" class="title-slide slide level1 center">
<h1>Setting the scene</h1>
</section>
<section id="randomised-control-trials-rcts" class="slide level2">
<h2>Randomised Control Trials (RCTs)</h2>
<p>A simplistic example:</p>
<ul>
<li><p>Users are assigned at <strong>random</strong> to two groups, A and B, with equal probability.</p></li>
<li><p>Let A be our <strong>control</strong> group and B be our <strong>treatment</strong> group.</p></li>
</ul>
<p>We want to know what effect our treatment has.</p>
<aside class="notes">
<p>Early on during an experiment, differences between these groups could simply be due to the random allocation of participants. As the groups get larger, these random differences will diminish, bringing us closer to the difference caused by the treatment.</p>
<p>Applying Bayesian inference effectively gives the experiment a guided head start by including more data (probabilistic data, not real data) in the form of <strong>priors</strong>.</p>
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</section>
<section id="hypothetical-scenario" class="slide level2">
<h2>Hypothetical scenario</h2>
<div class="columns">
<div class="column" style="width:70%;">
<ul>
<li><p>A button on a landing page that takes users to a sign up form.</p></li>
<li><p>At present, the button is labelled “Register your interest”.</p></li>
<li><p>Test whether changing it to “Get started” will result in an increased click-through rate (CTR).</p></li>
<li><p>“Get started” was suggested by an experienced and skilled UX designer.</p></li>
</ul>
</div><div class="column" style="width:30%;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><img data-src="A_B%20Button%20CTA%20test.png" alt="Two button CTA being compared: "Register your interest" vs "Get started""></p>
</figure>
</div>
</div>
</div>
</section></section>
<section>
<section id="priors-and-probability-distributions" class="title-slide slide level1 center">
<h1>Priors and probability distributions</h1>
<p>The key to speeding up your experiment</p>
</section>
<section id="prior-knowledge-and-beliefs" class="slide level2">
<h2>Prior knowledge and beliefs</h2>
<p>Before running an experiment, we form opinions and gather evidence such as:</p>
<ul>
<li><p>The baseline click-through rate of the button (with its current label) and knowledge of any outside variables that affects click-through rate, e.g. seasonality</p></li>
<li><p>Effects we have seen from similar previous experiments</p></li>
<li><p>Qualitative research, such as usability tests, focus groups, and surveys that are related to the test</p></li>
<li><p>Opinions (including critical) from stakeholders and experts</p></li>
</ul>
</section>
<section id="priors-are-probability-distributions" class="slide level2 smaller">
<h2>Priors are probability distributions</h2>
<div class="columns">
<div class="column" style="width:40%;">
<p>Express prior beliefs about the click-through rate of the control group using a <strong>probability distribution</strong>.</p>
<div class="fragment" data-fragment-index="1">
<p>Here’s an example of an extremely uninformative prior – a uniform prior that says any range of click-through rate is as probable as any other equally wide range, i.e. naive.</p>
</div>
</div><div class="column" style="width:60%;">
<div class="fragment" data-fragment-index="1">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-2-1.png" width="960"></p>
</div>
</div>
</div>
</div>
</div>
<div class="callout callout-tip callout-captioned callout-style-default">
<div class="callout-body">
<div class="callout-caption">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<p><strong>Tip</strong></p>
</div>
<div class="callout-content">
<p>The <strong>Beta distribution</strong> is a <strong>probability density function (PDF)</strong> with two <strong>shape parameters</strong>: <span class="math inline">\(B(shape1, shape2)\)</span>. It’s used to describe proportions, such as click-through rate.</p>
</div>
</div>
</div>
<aside class="notes">
<p>The total area under the curve will always add to 100%. That is, the curve represents all possibilities regardless of what shape parameters are used.</p>
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</section>
<section id="something-a-little-more-informative" class="slide level2 smaller">
<h2>Something a little more informative</h2>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-3-1.png" width="960" class="r-stretch"><p>As the curve narrows, notice that the shape parameters of the Beta distribution increase.</p>
</section>
<section id="something-even-more-informative" class="slide level2 smaller">
<h2>Something even more informative</h2>
<div class="cell">
<div class="cell-output-display">
<p><img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-4-1.png" width="1056"></p>
</div>
</div>
<div class="columns">
<div class="column" style="width:75%;">
<p>The shape parameters (<code>shape1</code> and <code>shape2</code>) of the Beta distribution can be considered counts of <strong>successes</strong> and <strong>failures</strong>, respectively. The mean probability of success (i.e. average click-through rate) can be calculated by this formula:</p>
</div><div class="column" style="width:25%;">
<p><span class="math display">\[
\frac{shape1}{shape1 + shape2}
\]</span></p>
</div>
</div>
<aside class="notes">
<p>The shape parameters are actually slightly more than the count of successes and failures, i.e. <span class="math inline">\(successes = \alpha - 1\)</span> and <span class="math inline">\(failures = \beta - 1\)</span>, or <span class="math inline">\(successes = \alpha - 0.5\)</span> and <span class="math inline">\(failures = \beta - 0.5\)</span> if using Jeffreys prior.</p>
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</section>
<section id="lets-say-weve-settled-on-this" class="slide level2 smaller">
<h2>Let’s say we’ve settled on this:</h2>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-5-1.png" width="960" class="r-stretch"><p>The more confident we are about our beliefs, the narrower the curve.</p>
</section>
<section id="what-about-the-treatment-group" class="slide level2">
<h2>What about the treatment group?</h2>
<ul>
<li><p>We expect that the click-through rates of the treatment and control groups will be correlated.</p></li>
<li><p>We’re unsure about how correlated they will be, but we’re not expecting a dramatic difference.</p></li>
<li><p>We’re more confident than not that the treatment will be an improvement, but we’re open to other possibilities.</p></li>
<li><p>We don’t want to bias the experiment results in favour of treatment or control, or towards a conclusion of there being a difference or no difference.</p></li>
</ul>
</section>
<section id="weve-settled-on-these-priors" class="slide level2">
<h2>We’ve settled on these priors:</h2>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-6-1.png" width="960" class="r-stretch"></section></section>
<section>
<section id="running-a-bayesian-ab-test" class="title-slide slide level1 center">
<h1>Running a Bayesian A/B test</h1>
</section>
<section id="prior-agreement" class="slide level2">
<h2>Prior agreement</h2>
<ul>
<li><p>Agreement must be reached on the priors before collecting and analysing data from the experiment.</p></li>
<li><p>Once the priors are agreed to and locked in, we can start the experiment.</p></li>
</ul>
<p>Here’s a summary of the priors we have chosen:</p>
<div class="cell">
<div class="cell-output-display">
<div class="kable-table">
<table>
<thead>
<tr class="header">
<th style="text-align: left;">group</th>
<th style="text-align: right;">shape1</th>
<th style="text-align: right;">shape2</th>
<th style="text-align: left;">mean</th>
<th style="text-align: left;">sd</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">control</td>
<td style="text-align: right;">120</td>
<td style="text-align: right;">288</td>
<td style="text-align: left;">29.4%</td>
<td style="text-align: left;">2.3 p.p.</td>
</tr>
<tr class="even">
<td style="text-align: left;">treatment</td>
<td style="text-align: right;">27</td>
<td style="text-align: right;">60</td>
<td style="text-align: left;">31.0%</td>
<td style="text-align: left;">4.9 p.p.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<aside class="notes">
<p>It’s important to not change your original priors after seeing data collected from the experiment. Doing so is effectively double-dipping, whereby your priors are being influenced by the data you have collected.</p>
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</section>
<section id="lets-run-an-experiment" class="slide level2">
<h2>Let’s run an experiment</h2>
<div class="cell">
</div>
<p>We’ll generate some fake data to mimic a real experiment.</p>
<p>It’ll be rigged though, as we’ll already know the click-through rates for control and treatment, which are:</p>
<ul>
<li><p>Control: 32%</p></li>
<li><p>Treatment: 35%</p></li>
</ul>
<p>That’s a relative uplift of 9.4%.</p>
<p>If we’re successful at applying Bayesian inference then we should hope (but can’t guarantee due to randomness) that the results somewhat match with these expected CTRs.</p>
</section>
<section id="the-next-day" class="slide level2">
<h2>The next day</h2>
<div class="cell">
</div>
<p>Let’s pretend that on average 150 users enter our experiment each day, and we’ve received the following data from day 1:</p>
<div class="cell">
<div class="cell-output-display">
<div class="kable-table">
<table>
<thead>
<tr class="header">
<th style="text-align: left;">group</th>
<th style="text-align: right;">total_users</th>
<th style="text-align: right;">clicked</th>
<th style="text-align: right;">not_clicked</th>
<th style="text-align: left;">CTR</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">control</td>
<td style="text-align: right;">86</td>
<td style="text-align: right;">32</td>
<td style="text-align: right;">54</td>
<td style="text-align: left;">37.2%</td>
</tr>
<tr class="even">
<td style="text-align: left;">treatment</td>
<td style="text-align: right;">81</td>
<td style="text-align: right;">26</td>
<td style="text-align: right;">55</td>
<td style="text-align: left;">32.1%</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<aside class="notes">
<p>Our experiment simulator randomly selects users and assigns them to each group using a Poisson process. It then randomly chooses which users had clicked using Bernoulli trials (e.g. coin flips).</p>
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</section>
<section id="lets-now-incorporate-our-priors" class="slide level2 smaller">
<h2>Let’s now incorporate our priors</h2>
<p><strong>Posteriors</strong> represent your updated beliefs once you’ve incorporated experiment data with your priors. Like priors, posteriors represent your beliefs about the metric of interest, which in our case is click-through rate.</p>
<div class="columns">
<div class="column" style="width:50%;">
<p>For each experiment group, we derive our posterior shape parameters through simple arithmetic addition:</p>
<div class="fragment" data-fragment-index="1">
<ul>
<li>Increment the first shape parameter by the count users who had <strong>clicked</strong></li>
</ul>
</div>
<div class="fragment" data-fragment-index="2">
<ul>
<li>Increment the second shape parameter by the count users who <strong>didn’t click</strong></li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<div class="cell">
</div>
<div class="fragment" data-fragment-index="1">
<div class="cell">
<div class="cell-output-display">
<div class="kable-table">
<table>
<thead>
<tr class="header">
<th style="text-align: left;">count</th>
<th style="text-align: right;">control</th>
<th style="text-align: right;">treatment</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">prior_shape1</td>
<td style="text-align: right;">120</td>
<td style="text-align: right;">27</td>
</tr>
<tr class="even">
<td style="text-align: left;">clicked</td>
<td style="text-align: right;">32</td>
<td style="text-align: right;">26</td>
</tr>
<tr class="odd">
<td style="text-align: left;">posterior_shape1</td>
<td style="text-align: right;">152</td>
<td style="text-align: right;">53</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div class="fragment" data-fragment-index="2">
<div class="cell">
<div class="cell-output-display">
<div class="kable-table">
<table>
<thead>
<tr class="header">
<th style="text-align: left;">count</th>
<th style="text-align: right;">control</th>
<th style="text-align: right;">treatment</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">prior_shape2</td>
<td style="text-align: right;">288</td>
<td style="text-align: right;">60</td>
</tr>
<tr class="even">
<td style="text-align: left;">not_clicked</td>
<td style="text-align: right;">54</td>
<td style="text-align: right;">55</td>
</tr>
<tr class="odd">
<td style="text-align: left;">posterior_shape2</td>
<td style="text-align: right;">342</td>
<td style="text-align: right;">115</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
</div>
<aside class="notes">
<p>The process of incorporating data with priors is called Bayesian updating. The data generated follows a Bernoulli distribution (Binomial with 1 trial). The prior follows a Beta distribution, which is conjugate to the Binomial distribution.</p>
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</section>
<section id="posterior-distribution-of-each-group" class="slide level2">
<h2>Posterior distribution of each group</h2>
<p>We have now updated our beliefs. These posteriors can now be thought of as our new updated priors.</p>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-14-1.png" width="960" class="r-stretch"></section>
<section id="another-six-days-later" class="slide level2">
<h2>Another six days later…</h2>
<p>We’ve collected more data, so let’s again update our priors to form new posteriors for the click-through rates of each group.</p>
<div class="cell">
</div>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-16-1.png" width="960" class="r-stretch"></section>
<section id="another-three-weeks-later" class="slide level2">
<h2>Another three weeks later…</h2>
<div class="cell">
</div>
<p>We’ve now observed a total sample of 4,188 users and a decision is made to end the experiment.</p>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-18-1.png" width="960" class="r-stretch"></section></section>
<section>
<section id="posterior-analysis" class="title-slide slide level1 center">
<h1>Posterior analysis</h1>
<p>Statistical inferences using the posterior distributions</p>
</section>
<section id="monte-carlo-simulation" class="slide level2">
<h2>Monte Carlo simulation</h2>
<div class="columns">
<div class="column" style="width:60%;">
<p>Let’s draw a very large quantity of random samples from our posterior distributions to make inferences about the experiment.</p>
<p>This is called Monte Carlo simulation – named after a casino.</p>
</div><div class="column" style="width:40%;">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="https://creativecommons.org/licenses/by/2.0"><img data-src="Real_Monte_Carlo_Casino.jpg" alt="Monte Carlo Casino, Monaco, France"></a></p>
</figure>
</div>
</div>
</div>
<aside class="notes">
<p>The more samples drawn, the greater the resolution of the inferences you make, but this comes at the cost of computational time and memory. Nowadays, computer processing speed and memory are more than adequate for what we need. Analytical solutions, providing the greatest level of precision, are also sometimes possible.</p>
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<div class="cell">
</div>
</section>
<section id="simulations" class="slide level2 smaller">
<h2>100 simulations</h2>
<p>Let’s start slowly by drawing 100 random samples from our distributions and plot them using histograms…</p>
<div class="columns">
<div class="column" style="width:55%;">
<div class="cell">
</div>
<div class="fragment fade-up">
<p>Here’s some of our Monte Carlo samples:</p>
<div class="cell">
<div class="cell-output-display">
<table>
<thead>
<tr class="header">
<th style="text-align: right;">control</th>
<th style="text-align: right;">treatment</th>
<th style="text-align: right;">uplift</th>
<th style="text-align: left;">beats_control</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.316</td>
<td style="text-align: right;">0.340</td>
<td style="text-align: right;">0.074</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.317</td>
<td style="text-align: right;">0.358</td>
<td style="text-align: right;">0.131</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.310</td>
<td style="text-align: right;">0.347</td>
<td style="text-align: right;">0.119</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.294</td>
<td style="text-align: right;">0.328</td>
<td style="text-align: right;">0.117</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.318</td>
<td style="text-align: right;">0.341</td>
<td style="text-align: right;">0.075</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.301</td>
<td style="text-align: right;">0.347</td>
<td style="text-align: right;">0.154</td>
<td style="text-align: left;">TRUE</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.309</td>
<td style="text-align: right;">0.346</td>
<td style="text-align: right;">0.118</td>
<td style="text-align: left;">TRUE</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div><div class="column" style="width:45%;">
<div class="fragment">
<div class="cell" data-fig.asp="1">
<div class="cell-output-display">
<p><img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-22-1.png" width="480"></p>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="lets-now-beef-it-up-a-bit" class="slide level2">
<h2>Let’s now beef it up a bit…</h2>
<div class="cell">
</div>
<p>We’ll now draw 1,000,000 samples…</p>
<div class="fragment">
<div class="cell">
<div class="cell-output-display">
<p><img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-24-1.png" width="960"></p>
</div>
</div>
</div>
<aside class="notes">
<p>Notice how these histograms follow the same distribution as our posteriors. That is because these samples have been drawn at random according to those posterior distributions.</p>
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</section>
<section id="we-can-now-make-some-inferences" class="slide level2">
<h2>We can now make some inferences</h2>
<p>A summary of our 1,000,000 posterior samples for click-through rate:</p>
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code> control treatment uplift beats_control
Min. :0.2656 Min. :0.2902 Min. :-0.09909 Mode :logical
1st Qu.:0.3014 1st Qu.:0.3309 1st Qu.: 0.06710 FALSE:13815
Median :0.3075 Median :0.3377 Median : 0.09819 TRUE :986185
Mean :0.3076 Mean :0.3378 Mean : 0.09914
3rd Qu.:0.3138 3rd Qu.:0.3446 3rd Qu.: 0.12999
Max. :0.3525 Max. :0.3924 Max. : 0.34028 </code></pre>
</div>
</div>
<ul>
<li>How do these compare to our theoretical CTRs of 32% for control and 35% for treatment, and uplift of 9.4%?</li>
</ul>
<div class="cell">
</div>
<ul>
<li>What is the posterior probability that the CTR of the treatment is greater than that of control? Answer: 98.62%</li>
</ul>
<aside class="notes">
<p>Out of our 1 000 000 simulations, we can see how often the treatment bet control. This tells us the probability that treatment is the winner.</p>
<p>If we filtered our simulations to those where control won and calculated the median CTR uplift, and did the same for cases where treatment won, we can determine the expected losses of choosing either variant as the winner. We should prefer the variant with the lowest expected loss or continue to run the experiment longer to improve our confidence.</p>
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</section>
<section id="posterior-distribution-of-the-ctr-uplift" class="slide level2">
<h2>Posterior distribution of the CTR uplift</h2>
<img data-src="bayes-ab-testing_files/figure-revealjs/unnamed-chunk-27-1.png" width="960" class="r-stretch"></section></section>
<section>
<section id="summary-and-some-final-remarks" class="title-slide slide level1 center">
<h1>Summary and some final remarks</h1>
</section>
<section id="before-starting-an-experiment" class="slide level2">
<h2>Before starting an experiment</h2>
<p>Gather prior knowledge and articulate beliefs:</p>
<ul>
<li><p>Establish a baseline - what do you know about the control group?</p></li>
<li><p>What do you expect the effect of the treatment to be? How sure are you?</p></li>
</ul>
<p>Express those beliefs and knowledge as distributions - these are your priors for your control and treatment groups.</p>
<aside class="notes">
<p>Ensure that the priors encapsulate the collective knowledge and beliefs of all interested parties so that there is agreement. This is to avoid the results from being challenged later.</p>
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</section>
<section id="running-the-experiment" class="slide level2">
<h2>Running the experiment</h2>
<ul>
<li><p>Start the experiment, gather data, and update your priors to form posteriors about the metric of interest</p></li>
<li><p>Draw inferences by running a large number of Monte Carlo simulations using the posterior distributions</p></li>
<li><p>Know when to end the experiment – try to plan for this ahead of running the experiment</p></li>
</ul>
<aside class="notes">
<p>Null-hypothesis significance testing (NHST) is not what Bayesian is for:</p>
<ul>
<li><p>Bayesian tells you the probability of some effect being within some range, given the data. I.e. Given everything we know so far, what are the risks associated with the choices we have?</p></li>
<li><p>NHST tells you the probability of data at least as extreme as what has been observed, given there is no real effect. I.e. How ridiculous would this outcome be if it were due to chance alone?</p></li>
</ul>
<p>NHST is often referred to as the frequentist approach, where decisions are made using p-values and some arbitrary threshold <span class="math inline">\(\alpha\)</span> (i.e. false positive rate).</p>
<p>Unlike NHST, Bayesian A/B testing doesn’t give you a yes/no answer – it instead informs you about the probabilities and risks associated with the choices you have.</p>
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</section></section>
<section id="thank-you" class="title-slide slide level1 center">
<h1>Thank you!</h1>
<p>Further topics that might interest you:</p>
<ul>
<li><p><strong>Bayesian Generalised Linear Models</strong> to better isolate the effect of the treatment from other predictors.</p></li>
<li><p><strong>Survival Analysis</strong>, such as <strong>Kaplan Meier</strong>, to analyse lagged conversion outcomes.</p></li>
</ul>