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<html><head><meta charset="utf-8">
<title>Ebisu</title>
<meta name="description" content="Public-domain library for quiz scheduling with Bayesian statistics.">
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                            }</style></head><body><h1 id="ebisu-intelligent-quiz-scheduling">Ebisu: intelligent quiz scheduling</h1>
<h2 id="important-links">Important links</h2>
<ul>
<li><a href="https://fasiha.github.io/ebisu/">Literate document</a></li>
<li><a href="https://github.com/fasiha/ebisu">GitHub repo</a></li>
<li><a href="https://github.com/fasiha/ebisu/blob/gh-pages/EbisuHowto.ipynb">IPython Notebook crash course</a></li>
<li><a href="https://pypi.python.org/pypi/ebisu/">PyPI package</a></li>
<li><a href="https://fasiha.github.io/#contact">Contact</a></li>
</ul>
<h3 id="table-of-contents">Table of contents</h3>
<!-- TOC depthFrom:2 depthTo:3 withLinks:1 updateOnSave:1 orderedList:0 -->
<ul>
<li><a href="#important-links">Important links</a>
<ul>
<li><a href="#table-of-contents">Table of contents</a></li>
</ul></li>
<li><a href="#introduction">Introduction</a></li>
<li><a href="#quickstart">Quickstart</a></li>
<li><a href="#how-it-works">How it works</a></li>
<li><a href="#the-math">The math</a>
<ul>
<li><a href="#bernoulli-quizzes">Bernoulli quizzes</a></li>
<li><a href="#moving-beta-distributions-through-time">Moving Beta distributions through time</a></li>
<li><a href="#mean-and-variance-of-the-recall-probability-right-now">Mean and variance of the recall probability right now</a></li>
<li><a href="#choice-of-initial-model-parameters">Choice of initial model parameters</a></li>
<li><a href="#updating-the-posterior-with-quiz-results">Updating the posterior with quiz results</a></li>
</ul></li>
<li><a href="#source-code">Source code</a>
<ul>
<li><a href="#core-library">Core library</a></li>
<li><a href="#miscellaneous-functions">Miscellaneous functions</a></li>
<li><a href="#test-code">Test code</a></li>
</ul></li>
<li><a href="#demo-codes">Demo codes</a>
<ul>
<li><a href="#visualizing-half-lives">Visualizing half-lives</a></li>
<li><a href="#why-we-work-with-random-variables">Why we work with random variables</a></li>
</ul></li>
<li><a href="#requirements-for-building-all-aspects-of-this-repo">Requirements for building all aspects of this repo</a></li>
<li><a href="#acknowledgements">Acknowledgements</a></li>
</ul>
<!-- /TOC -->
<h2 id="introduction">Introduction</h2>
<p>Consider a student memorizing a set of facts.</p>
<ul>
<li>Which facts need reviewing?</li>
<li>How does the student’s performance on a review change the fact’s future review schedule?</li>
</ul>
<p>Ebisu is a public-domain library that answers these two questions. It is intended to be used by software developers writing quiz apps, and provides a simple API to deal with these two aspects of scheduling quizzes:</p>
<ul>
<li><code>predictRecall</code> gives the current recall probability for a given fact.</li>
<li><code>updateRecall</code> adjusts the belief about future recall probability given a quiz result.</li>
</ul>
<p>Behind these two simple functions, Ebisu is using a simple yet powerful model of forgetting, a model that is founded on Bayesian statistics.</p>
<p>With this system, quiz applications can hopefully move away from “daily review piles” caused by less flexible scheduling algorithms. For instance, a student might have only five minutes to study today—an app using Ebisu can ensure that only the facts most in danger of being forgotten are reviewed.</p>
<p>Furthermore, Ebisu enables apps to provide an infinite stream of quizzes (nightmare!) for students that are cramming. That is, Ebisu intelligently handles over-reviewing as well as under-reviewing.</p>
<p>This document is a literate source: it contains a detailed mathematical description of the underlying algorithm as well as source code for a Python implementation (requires Scipy and Numpy). Separat implementations in <a href="https://fasiha.github.io/ebisu.js/">JavaScript, Ebisu.js,</a> and <a href="https://github.com/fasiha/ebisu-java">Java, ebisu-java,</a> exist.</p>
<p>The next section is a <a href="#quickstart">Quickstart</a> guide to setup and usage. See this if you know you want to use Ebisu in your app.</p>
<p>Then in the <a href="#how-it-works">How It Works</a> section, I contrast Ebisu to other scheduling algorithms and describe, non-technically, why you should use it.</p>
<p>Then there’s a long <a href="#the-math">Math</a> section that details Ebisu’s algorithm mathematically. If you like Beta-distributed random variables, conjugate priors, and marginalization, this is for you. You’ll also find the key formulas that implement <code>predictRecall</code> and <code>updateRecall</code> here.</p>
<blockquote>
<p>Nerdy details in a nutshell: Ebisu begins by positing a <a href="https://en.wikipedia.org/wiki/Beta_distribution">Beta prior</a> on recall probabilities. As time passes, the recall probability decays exponentially, and Ebisu handles that nonlinearity exactly and analytically—it requires only a few <a href="http://mathworld.wolfram.com/GammaFunction.html">gamma function</a> evaluations to predict the current recall probability. Next, a <em>quiz</em> is modeled as a <a href="%5BBernoulli%20experiment%5D(https://en.wikipedia.org/wiki/Bernoulli_distribution)">Bernoulli trial</a>, whose underlying probability prior is this non-conjugate nonlinearly-transformed Beta. Ebisu approximates the true non-standard posterior with a new Beta distribution by matching its mean and variance. This mean and variance are analytically tractable, and again require a few evaluations of the gamma function.</p>
</blockquote>
<p>Finally, the <a href="#source-code">Source Code</a> section presents the literate source of the library, including several tests to validate the math.</p>
<h2 id="quickstart">Quickstart</h2>
<p><strong>Install</strong> <code>pip3 install ebisu</code> (or <code>pip install ebisu</code> if you only have Python3 🤠—just joking, it works on Python2 also).</p>
<p><strong>Data model</strong> For each fact in your quiz app, you store a model representing a prior distribution. This is a 3-tuple: <code>(alpha, beta, t)</code> and you can create a default model for all newly learned facts with <code>ebisu.defaultModel</code>. (As detailed in the <a href="#choice-of-initial-model-parameters">Choice of initial model parameters</a> section, <code>alpha</code> and <code>beta</code> define a Beta distribution on this fact’s recall probability <code>t</code> time units after it’s most recent review.)</p>
<p><strong>Predict a fact’s current recall probability</strong> <code>ebisu.predictRecall(prior: tuple, tnow: float) -&gt; float</code> where <code>prior</code> is this fact’s model, <code>tnow</code> is the current time elapsed since this fact’s most recent review, and the returned value is a probability between 0 and 1.</p>
<p><strong>Update a fact’s model with quiz results</strong> <code>ebisu.updateRecall(prior: tuple, result: bool, tnow: float) -&gt; tuple</code> where <code>prior</code> and <code>tnow</code> are as above, and where <code>result</code> is true if the student successfully answered the quiz, false otherwise. The returned value is this fact’s new prior model—the old one can be discarded.</p>
<p><strong>IPython Notebook crash course</strong> For a conversational introduction to the API in the context of a mocked quiz app, see this <a href="https://github.com/fasiha/ebisu/blob/gh-pages/EbisuHowto.ipynb">IPython Notebook crash course</a>.</p>
<p><strong>Further information</strong> <a href="https://github.com/fasiha/ebisu/blob/gh-pages/doc/doc.md">Module docstrings</a> in a pinch but full details plus literate source below, under <a href="#source-code">Source code</a>.</p>
<p><strong>Alternative implementations</strong> <a href="https://fasiha.github.io/ebisu.js/">Ebisu.js</a> is a JavaScript port for browser and Node.js. <a href="https://github.com/fasiha/ebisu-java">ebisu-java</a> is for Java and JVM languages.</p>
<h2 id="how-it-works">How it works</h2>
<p>There are many scheduling schemes, e.g.,</p>
<ul>
<li><a href="https://apps.ankiweb.net/">Anki</a>, an open-source Python flashcard app (and a closed-source mobile app),</li>
<li>the <a href="https://www.supermemo.com/help/smalg.htm">SuperMemo</a> family of algorithms (<a href="https://apps.ankiweb.net/docs/manual.html#what-algorithm">Anki’s</a> is a derivative of SM-2),</li>
<li><a href="https://www.memrise.com">Memrise.com</a>, a closed-source webapp,</li>
<li><a href="https://www.duolingo.com/">Duolingo</a> has published a <a href="http://making.duolingo.com/how-we-learn-how-you-learn">blog entry</a> and a <a href="https://github.com/duolingo/halflife-regression">conference paper/code repo</a> on their half-life regression technique,</li>
<li>the Leitner and Pimsleur spacing schemes (also discussed in some length in Duolingo’s paper).</li>
<li>Also worth noting is Michael Mozer’s team’s Bayesian multiscale models, e.g., <a href="http://www.cs.colorado.edu/~mozer/Research/Selected%20Publications/reprints/MozerPashlerCepedaLindseyVul2009.pdf">Mozer, Pashler, Cepeda, Lindsey, and Vul</a>’s 2009 <cite>NIPS</cite> paper and subsequent work.</li>
</ul>
<p>Many of these are inspired by Hermann Ebbinghaus’ discovery of the <a href="https://en.wikipedia.org/w/index.php?title=Forgetting_curve&amp;oldid=766120598#History">exponential forgetting curve</a>, published in 1885, when he was thirty-five. He <a href="https://en.wikipedia.org/w/index.php?title=Hermann_Ebbinghaus&amp;oldid=773908952#Research_on_memory">memorized random</a> consonant–vowel–consonant trigrams (‘PED’, e.g.) and found, among other things, that his recall decayed exponentially with some time-constant.</p>
<p>Anki and SuperMemo use carefully-tuned mechanical rules to schedule a fact’s future review immediately after its current review. The rules can get complicated—I wrote a little <a href="https://gist.github.com/fasiha/31ce46c36371ff57fdbc1254af424174">field guide</a> to Anki’s, with links to the source code—since they are optimized to minimize daily review time while maximizing retention. However, because each fact has simply a date of next review, these algorithms do not gracefully accommodate over- or under-reviewing. Even when used as prescribed, they can schedule many facts for review on one day but few on others. (I must note that all three of these issues—over-reviewing (cramming), under-reviewing, and lumpy reviews—have well-supported solutions in Anki by tweaking the rules and third-party plugins.)</p>
<p>Duolingo’s half-life regression explicitly models the probability of you recalling a fact as <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.811ex" height="2.676ex" style="vertical-align: -0.338ex;" viewBox="0 -1006.6 2501.9 1152.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
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</svg></span>, where Δ is the time since your last review and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 576.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-2-Title">
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</svg></span> is a <em>half-life</em>. In this model, your chances of passing a quiz after <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 576.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-3-Title">
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</svg></span> days is 50%, which drops to 25% after <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.501ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 1077 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-4-Title">
<title id="MathJax-SVG-4-Title">2 h</title>
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</svg></span> days. They estimate this half-life by combining your past performance and fact metadata in a large-scale machine learning technique called half-life regression (a variant of logistic regression or beta regression, more tuned to this forgetting curve). With each fact associated with a half-life, they can predict the likelihood of forgetting a fact if a quiz was given right now. The results of that quiz (for whichever fact was chosen to review) are used to update that fact’s half-life by re-running the machine learning process with the results from the latest quizzes.</p>
<p>The Mozer group’s algorithms also fit a hierarchical Bayesian model that links quiz performance to memory, taking into account inter-fact and inter-student variability, but the training step is again computationally-intensive.</p>
<p>Like Duolingo and Mozer’s approaches, Ebisu explicitly tracks the exponential forgetting curve to provide a list of facts sorted by most to least likely to be forgotten. However, Ebisu formulates the problem very differently—while memory is understood to decay exponentially, Ebisu posits a <em>probability distribution</em> on the half-life and uses quiz results to update its beliefs in a fully Bayesian way. These updates, while a bit more computationally-burdensome than Anki’s scheduler, are much lighter-weight than Duolingo’s industrial-strength approach.</p>
<p>This gives small quiz apps the same intelligent scheduling as Duolingo’s approach—real-time recall probabilities for any fact—but with immediate incorporation of quiz results, even on mobile apps.</p>
<p>To appreciate this further, consider this example. Imagine a fact with half-life of a week: after a week we expect the recall probability to drop to 50%. However, Ebisu can entertain an infinite range of beliefs about this recall probability: it can be very uncertain that it’ll be 50% (the “α=β=3” model below), or it can be very confident in that prediction (“α=β=12” case):</p>
<p><img src="figures/models.png" alt="figures/models.png"></p>
<p>Under either of these models of recall probability, we can ask Ebisu what the expected half-life is after the student is quizzed on this fact a day, a week, or a month after their last review, and whether they passed or failed the quiz:</p>
<p><img src="figures/halflife.png" alt="figures/halflife.png"></p>
<p>If the student correctly answers the quiz, Ebisu expects the new half-life to be greater than a week. If the student answers correctly after just a day, the half-life rises a little bit, since we expected the student to remember this fact that soon after reviewing it. If the student surprises us by <em>failing</em> the quiz just a day after they last reviewed it, the projected half-life drops. The more tentative “α=β=3” model aggressively adjusts the half-life, while the more assured “α=β=12” model is more conservative in its update. (Each fact has an α and β associated with it and I explain what they mean mathematically in the next section. Also, the code for these two charts is <a href="#demo-codes">below</a>.)</p>
<p>Similarly, if the student fails the quiz after a whole month of not reviewing it, this isn’t a surprise—the half-life drops a bit from the initial half-life of a week. If she does surprise us, passing the quiz after a month of not studying it, then Ebisu boosts its expectated half-life—by a lot for the “α=β=3” model, less for the “α=β=12” one.</p>
<blockquote>
<p>Currently, Ebisu treats each fact as independent, very much like Ebbinghaus’ nonsense syllables: it does not understand how facts are related the way Duolingo can with its sentences. However, Ebisu can be used in combination with other techniques to accommodate extra information about relationships between facts.</p>
</blockquote>
<h2 id="the-math">The math</h2>
<h3 id="bernoulli-quizzes">Bernoulli quizzes</h3>
<p>Let’s begin with a quiz. One way or another, we’ve picked a fact to quiz the student on, <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-5-Title">
<title id="MathJax-SVG-5-Title">t</title>
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 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
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</svg></span> days (the units are arbitrary since <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-6-Title">
<title id="MathJax-SVG-6-Title">t</title>
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 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
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</svg></span> can be any positive real number) after her last quiz on it, or since she learned it for the first time.</p>
<p>We’ll model the results of the quiz as a Bernoulli experiment, <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.17ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 7823.2 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-7-Title">
<title id="MathJax-SVG-7-Title">x_t ∼ Bernoulli(p)</title>
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</svg></span>; <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.156ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 928.1 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-8-Title">
<title id="MathJax-SVG-8-Title">x_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-78" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="809" y="-213"></use>
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</svg></span> can be either 1 (success) with probability <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-9-Title">
<title id="MathJax-SVG-9-Title">p_t</title>
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</svg></span>, or 0 (fail) with probability <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.998ex" height="2.509ex" style="vertical-align: -0.671ex;" viewBox="0 -791.3 2582.6 1080.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-10-Title">
<title id="MathJax-SVG-10-Title">1-p_t</title>
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</svg></span>. Let’s think about <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-11-Title">
<title id="MathJax-SVG-11-Title">p_t</title>
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</svg></span> as the recall probability at time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-12-Title">
<title id="MathJax-SVG-12-Title">t</title>
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 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
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</svg></span>—then <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.156ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 928.1 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-13-Title">
<title id="MathJax-SVG-13-Title">x_t</title>
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 <use xlink:href="#MJMATHI-78" x="0" y="0"></use>
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</svg></span> is a coin flip, with a <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-14-Title">
<title id="MathJax-SVG-14-Title">p_t</title>
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</svg></span>-weighted coin.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Beta_distribution">Beta distribution</a> happens to be the <a href="https://en.wikipedia.org/wiki/Conjugate_prior">conjugate prior</a> for the Bernoulli distribution. So if our <em>a priori</em> belief about <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-15-Title">
<title id="MathJax-SVG-15-Title">p_t</title>
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 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
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</svg></span> follow a Beta distribution, that is, if <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.399ex" height="2.843ex" style="vertical-align: -0.838ex; margin-left: -0.089ex;" viewBox="-38.5 -863.1 7491.1 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-16-Title">
<title id="MathJax-SVG-16-Title">p_t ∼ Beta(α_t, β_t)</title>
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</svg></span> for specific <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.314ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 996.1 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-17-Title">
<title id="MathJax-SVG-17-Title">α_t</title>
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</svg></span> and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.142ex" height="2.676ex" style="vertical-align: -0.838ex;" viewBox="0 -791.3 922.1 1152.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-18-Title">
<title id="MathJax-SVG-18-Title">β_t</title>
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 <use xlink:href="#MJMATHI-3B2" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="801" y="-326"></use>
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</svg></span>, then observing the quiz result updates our belief about the recall probability to be: <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="34.843ex" height="2.843ex" style="vertical-align: -0.838ex; margin-left: -0.089ex;" viewBox="-38.5 -863.1 15001.8 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-19-Title">
<title id="MathJax-SVG-19-Title">p_t | x_t ∼ Beta(α_t + x_t, β_t + 1 - x_t).</title>
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<blockquote>
<p><strong>Aside 1</strong> Notice that since <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.156ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 928.1 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-20-Title">
<title id="MathJax-SVG-20-Title">x_t</title>
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</svg></span> is either 1 or 0, the updated parameters <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.642ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 8456.7 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-21-Title">
<title id="MathJax-SVG-21-Title">(α + x_t, β + 1 - x_t)</title>
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</svg></span> are <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.649ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 4154.6 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-22-Title">
<title id="MathJax-SVG-22-Title">(α + 1, β)</title>
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</svg></span> when the student correctly answered the quiz, and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.649ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 4154.6 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-23-Title">
<title id="MathJax-SVG-23-Title">(α, β + 1)</title>
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 <use xlink:href="#MJMAIN-2C" x="1030" y="0"></use>
 <use xlink:href="#MJMATHI-3B2" x="1475" y="0"></use>
 <use xlink:href="#MJMAIN-2B" x="2263" y="0"></use>
 <use xlink:href="#MJMAIN-31" x="3264" y="0"></use>
 <use xlink:href="#MJMAIN-29" x="3765" y="0"></use>
</g>
</svg></span> when she answered incorrectly.</p>
<p><strong>Aside 2</strong> Even if you’re familiar with Bayesian statistics, if you’ve never worked with priors on probabilities, the meta-ness here might confuse you. What the above means is that, before we flipped our <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-24-Title">
<title id="MathJax-SVG-24-Title">p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</svg></span>-weighted coin (before we administered the quiz), we had a specific probability distribution representing the coin’s weighting <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-25-Title">
<title id="MathJax-SVG-25-Title">p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</svg></span>, <em>not</em> just a scalar number. After we observed the result of the coin flip, we updated our belief about the coin’s weighting—it <em>still</em> makes total sense to talk about the probability of something happening after it happens. Said another way, since we’re being Bayesian, something actually happening doesn’t preclude us from maintaining beliefs about what <em>could</em> have happened.</p>
</blockquote>
<p>This is totally ordinary, bread-and-butter Bayesian statistics. However, the major complication arises when the experiment took place not at time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-26-Title">
<title id="MathJax-SVG-26-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span> but <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.894ex" height="2.343ex" style="vertical-align: -0.671ex;" viewBox="0 -719.6 815.4 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-27-Title">
<title id="MathJax-SVG-27-Title">t_2</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
</g>
</svg></span>: we had a Beta prior on <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-28-Title">
<title id="MathJax-SVG-28-Title">p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</svg></span> (probability of recall at time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-29-Title">
<title id="MathJax-SVG-29-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span>) but the test is administered at some other time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.894ex" height="2.343ex" style="vertical-align: -0.671ex;" viewBox="0 -719.6 815.4 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-30-Title">
<title id="MathJax-SVG-30-Title">t_2</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
</g>
</svg></span>.</p>
<p>How can we update our beliefs about the recall probability at time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-31-Title">
<title id="MathJax-SVG-31-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span> to another time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.894ex" height="2.343ex" style="vertical-align: -0.671ex;" viewBox="0 -719.6 815.4 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-32-Title">
<title id="MathJax-SVG-32-Title">t_2</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
</g>
</svg></span>, either earlier or later than <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-33-Title">
<title id="MathJax-SVG-33-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span>?</p>
<h3 id="moving-beta-distributions-through-time">Moving Beta distributions through time</h3>
<p>Our old friend Ebbinghaus comes to our rescue. According to the exponentially-decaying forgetting curve, the probability of recall at time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-34-Title">
<title id="MathJax-SVG-34-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span> is <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.866ex" height="3.176ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -1078.4 4678.3 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-35-Title">
<title id="MathJax-SVG-35-Title">p_t = 2^{-t/h},</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
 <use xlink:href="#MJMAIN-3D" x="1136" y="0"></use>
<g transform="translate(2193,0)">
 <use xlink:href="#MJMAIN-32" x="0" y="0"></use>
<g transform="translate(500,412)">
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="778" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="1140" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-68" x="1640" y="0"></use>
</g>
</g>
 <use xlink:href="#MJMAIN-2C" x="4361" y="0"></use>
</g>
</svg></span> for some notional half-life <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 576.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-36-Title">
<title id="MathJax-SVG-36-Title">h</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-68" x="0" y="0"></use>
</g>
</svg></span>. Let <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.543ex" height="2.676ex" style="vertical-align: -0.671ex;" viewBox="0 -863.1 3678.4 1152.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-37-Title">
<title id="MathJax-SVG-37-Title">t_2 = δ·t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
 <use xlink:href="#MJMAIN-3D" x="1093" y="0"></use>
 <use xlink:href="#MJMATHI-3B4" x="2149" y="0"></use>
 <use xlink:href="#MJMAIN-22C5" x="2816" y="0"></use>
 <use xlink:href="#MJMATHI-74" x="3316" y="0"></use>
</g>
</svg></span>. Then, <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.022ex" height="3.509ex" style="vertical-align: -1.005ex; margin-left: -0.089ex;" viewBox="-38.5 -1078.4 15940.2 1510.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-38-Title">
<title id="MathJax-SVG-38-Title">p_{t_2} = p_{δ t} = 2^{-δt/h} = (2^{-t/h})^δ = (p_t)^δ.</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
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</g>
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</g>
</g>
 <use xlink:href="#MJMAIN-3D" x="4002" y="0"></use>
<g transform="translate(5058,0)">
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<g transform="translate(500,412)">
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 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="778" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="1223" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="1584" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-68" x="2085" y="0"></use>
</g>
</g>
 <use xlink:href="#MJMAIN-3D" x="7818" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="8875" y="0"></use>
<g transform="translate(9264,0)">
 <use xlink:href="#MJMAIN-32" x="0" y="0"></use>
<g transform="translate(500,412)">
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="778" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="1140" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-68" x="1640" y="0"></use>
</g>
</g>
<g transform="translate(11432,0)">
 <use xlink:href="#MJMAIN-29" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="550" y="583"></use>
</g>
 <use xlink:href="#MJMAIN-3D" x="12514" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="13570" y="0"></use>
<g transform="translate(13960,0)">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
<g transform="translate(14819,0)">
 <use xlink:href="#MJMAIN-29" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="550" y="583"></use>
</g>
 <use xlink:href="#MJMAIN-2E" x="15623" y="0"></use>
</g>
</svg></span> That is, to fast-forward or rewind <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-39-Title">
<title id="MathJax-SVG-39-Title">p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</svg></span> to time <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.894ex" height="2.343ex" style="vertical-align: -0.671ex;" viewBox="0 -719.6 815.4 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-40-Title">
<title id="MathJax-SVG-40-Title">t_2</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
</g>
</svg></span>, we raise it to the <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.027ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 3456 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-41-Title">
<title id="MathJax-SVG-41-Title">δ = t_2 / t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="722" y="0"></use>
<g transform="translate(1778,0)">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-32" x="511" y="-213"></use>
</g>
 <use xlink:href="#MJMAIN-2F" x="2593" y="0"></use>
 <use xlink:href="#MJMATHI-74" x="3094" y="0"></use>
</g>
</svg></span> power.</p>
<p>Unfortunately, a Beta-distributed <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-42-Title">
<title id="MathJax-SVG-42-Title">p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</svg></span> becomes not-Beta-distributed when raised to any positive power <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.049ex" height="2.343ex" style="vertical-align: -0.338ex; margin-right: -0.016ex;" viewBox="0 -863.1 451.5 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-43-Title">
<title id="MathJax-SVG-43-Title">δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
</g>
</svg></span>.</p>
<p>In the code snippet below, we start out with <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.593ex" height="2.843ex" style="vertical-align: -0.838ex; margin-left: -0.089ex;" viewBox="-38.5 -863.1 7574.8 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-44-Title">
<title id="MathJax-SVG-44-Title">p_t ∼ Beta(12, 12)</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
 <use xlink:href="#MJMAIN-223C" x="1136" y="0"></use>
 <use xlink:href="#MJMATHI-42" x="2193" y="0"></use>
 <use xlink:href="#MJMATHI-65" x="2952" y="0"></use>
 <use xlink:href="#MJMATHI-74" x="3419" y="0"></use>
 <use xlink:href="#MJMATHI-61" x="3780" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="4310" y="0"></use>
<g transform="translate(4699,0)">
 <use xlink:href="#MJMAIN-31"></use>
 <use xlink:href="#MJMAIN-32" x="500" y="0"></use>
</g>
 <use xlink:href="#MJMAIN-2C" x="5700" y="0"></use>
<g transform="translate(6145,0)">
 <use xlink:href="#MJMAIN-31"></use>
 <use xlink:href="#MJMAIN-32" x="500" y="0"></use>
</g>
 <use xlink:href="#MJMAIN-29" x="7146" y="0"></use>
</g>
</svg></span> and show the distribution of <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.221ex" height="3.176ex" style="vertical-align: -1.005ex; margin-left: -0.089ex;" viewBox="-38.5 -934.9 956.3 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-45-Title">
<title id="MathJax-SVG-45-Title">p_t^δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="712" y="498"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-395"></use>
</g>
</svg></span> for various <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.049ex" height="2.343ex" style="vertical-align: -0.338ex; margin-right: -0.016ex;" viewBox="0 -863.1 451.5 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-46-Title">
<title id="MathJax-SVG-46-Title">δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
</g>
</svg></span>. To make it concrete, imagine <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.84ex" height="2.009ex" style="vertical-align: -0.338ex;" viewBox="0 -719.6 361.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-47-Title">
<title id="MathJax-SVG-47-Title">t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-74" x="0" y="0"></use>
</g>
</svg></span> is seven days. The <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.41ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 5343.2 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-48-Title">
<title id="MathJax-SVG-48-Title">Beta(12, 12)</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-42" x="0" y="0"></use>
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 <use xlink:href="#MJMATHI-74" x="1226" y="0"></use>
 <use xlink:href="#MJMATHI-61" x="1587" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="2117" y="0"></use>
<g transform="translate(2506,0)">
 <use xlink:href="#MJMAIN-31"></use>
 <use xlink:href="#MJMAIN-32" x="500" y="0"></use>
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<g transform="translate(3952,0)">
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</g>
 <use xlink:href="#MJMAIN-29" x="4953" y="0"></use>
</g>
</svg></span> prior on recall probability seven days after the last review is the middle histogram (<span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.293ex" height="2.343ex" style="vertical-align: -0.338ex;" viewBox="0 -863.1 2279.1 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-49-Title">
<title id="MathJax-SVG-49-Title">δ = 1</title>
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 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="722" y="0"></use>
 <use xlink:href="#MJMAIN-31" x="1778" y="0"></use>
</g>
</svg></span>). If the student is quizzed on this fact just two days after last review (<span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.103ex" height="2.343ex" style="vertical-align: -0.338ex;" viewBox="0 -863.1 3058.1 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-50-Title">
<title id="MathJax-SVG-50-Title">δ=0.3</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="722" y="0"></use>
<g transform="translate(1778,0)">
 <use xlink:href="#MJMAIN-30"></use>
 <use xlink:href="#MJMAIN-2E" x="500" y="0"></use>
 <use xlink:href="#MJMAIN-33" x="779" y="0"></use>
</g>
</g>
</svg></span>), that density moves from the middle of the plot to the right, meaning a high probability of recall. However, if the student is quizzed three weeks after review, the original density moves to the left: it’s likely the student will fail the quiz.</p>
<pre class="py"><code class="hljs python"><span class="hljs-keyword">import</span> matplotlib.pyplot <span class="hljs-keyword">as</span> plt
plt.style.use(<span class="hljs-string">'ggplot'</span>)
plt.rcParams[<span class="hljs-string">'svg.fonttype'</span>] = <span class="hljs-string">'none'</span>


<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">generatePis</span><span class="hljs-params">(deltaT, alpha=<span class="hljs-number">12.0</span>, beta=<span class="hljs-number">12.0</span>)</span>:</span>
  <span class="hljs-keyword">import</span> scipy.stats <span class="hljs-keyword">as</span> stats

  piT = stats.beta.rvs(alpha, beta, size=<span class="hljs-number">50</span> * <span class="hljs-number">1000</span>)
  piT2 = piT**deltaT
  plt.hist(piT2, bins=<span class="hljs-number">20</span>, label=<span class="hljs-string">'δ={}'</span>.format(deltaT), alpha=<span class="hljs-number">0.25</span>, normed=<span class="hljs-keyword">True</span>)


[generatePis(p) <span class="hljs-keyword">for</span> p <span class="hljs-keyword">in</span> [<span class="hljs-number">0.3</span>, <span class="hljs-number">1.</span>, <span class="hljs-number">3.</span>]]
plt.xlabel(<span class="hljs-string">'p (recall probability)'</span>)
plt.ylabel(<span class="hljs-string">'Probability(p)'</span>)
plt.title(<span class="hljs-string">'Histograms of p_t^δ for different δ'</span>)
plt.legend(loc=<span class="hljs-number">0</span>)
plt.savefig(<span class="hljs-string">'figures/pidelta.svg'</span>)
plt.savefig(<span class="hljs-string">'figures/pidelta.png'</span>, dpi=<span class="hljs-number">150</span>)
plt.show()</code></pre>
<p><img src="figures/pidelta.png" alt="figures/pidelta.png"></p>
<p>You’ll have to take my word for it that the histograms where <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.293ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 2279.1 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-51-Title">
<title id="MathJax-SVG-51-Title">δ≠1</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
 <use xlink:href="#MJMAIN-2260" x="722" y="0"></use>
 <use xlink:href="#MJMAIN-31" x="1778" y="0"></use>
</g>
</svg></span> are indeed not Beta. For these <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.488ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 640.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-52-Title">
<title id="MathJax-SVG-52-Title">α</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B1" x="0" y="0"></use>
</g>
</svg></span>, <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.332ex" height="2.509ex" style="vertical-align: -0.671ex; margin-right: -0.016ex;" viewBox="0 -791.3 573.5 1080.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-53-Title">
<title id="MathJax-SVG-53-Title">β</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B2" x="0" y="0"></use>
</g>
</svg></span>, and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.049ex" height="2.343ex" style="vertical-align: -0.338ex; margin-right: -0.016ex;" viewBox="0 -863.1 451.5 1008.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-54-Title">
<title id="MathJax-SVG-54-Title">δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-3B4" x="0" y="0"></use>
</g>
</svg></span> used here, they are close to Beta, but especially when over- or under-reviewing, the histograms become skewed and are no longer well-matched by any Beta distribution.</p>
<p>So let’s derive analytically the probability density function (PDF) for <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.221ex" height="3.176ex" style="vertical-align: -1.005ex; margin-left: -0.089ex;" viewBox="-38.5 -934.9 956.3 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-55-Title">
<title id="MathJax-SVG-55-Title">p_t^δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="712" y="498"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-395"></use>
</g>
</svg></span>. Recall the conventional way to obtain the density of a <a href="https://en.wikipedia.org/w/index.php?title=Random_variable&amp;oldid=771423505#Functions_of_random_variables">nonlinearly-transformed random variable</a>: let <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.424ex" height="2.009ex" style="vertical-align: -0.671ex;" viewBox="0 -576.1 2765.7 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-56-Title">
<title id="MathJax-SVG-56-Title">x=p_t</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-78" x="0" y="0"></use>
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<g transform="translate(1906,0)">
 <use xlink:href="#MJMATHI-70" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-74" x="712" y="-213"></use>
</g>
</g>
</svg></span> and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.899ex" height="3.176ex" style="vertical-align: -0.838ex;" viewBox="0 -1006.6 5984.4 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-57-Title">
<title id="MathJax-SVG-57-Title">y = g(x) = x^δ</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-79" x="0" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="775" y="0"></use>
 <use xlink:href="#MJMATHI-67" x="1831" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="2312" y="0"></use>
 <use xlink:href="#MJMATHI-78" x="2701" y="0"></use>
 <use xlink:href="#MJMAIN-29" x="3274" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="3941" y="0"></use>
<g transform="translate(4997,0)">
 <use xlink:href="#MJMATHI-78" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="809" y="513"></use>
</g>
</g>
</svg></span> be the forward transform, so <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.281ex" height="3.176ex" style="vertical-align: -0.838ex;" viewBox="0 -1006.6 5718.2 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-58-Title">
<title id="MathJax-SVG-58-Title">g^{-1}(y) = y^{1/δ}</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
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 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use>
</g>
 <use xlink:href="#MJMAIN-28" x="1485" y="0"></use>
 <use xlink:href="#MJMATHI-79" x="1875" y="0"></use>
 <use xlink:href="#MJMAIN-29" x="2372" y="0"></use>
 <use xlink:href="#MJMAIN-3D" x="3040" y="0"></use>
<g transform="translate(4096,0)">
 <use xlink:href="#MJMATHI-79" x="0" y="0"></use>
<g transform="translate(499,362)">
 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="500" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="1001" y="0"></use>
</g>
</g>
</g>
</svg></span> is its inverse. Then, with <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.289ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 9596.7 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-59-Title">
<title id="MathJax-SVG-59-Title">P_X(x) = Beta(x; α,β)</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-50" x="0" y="0"></use>
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 <use xlink:href="#MJMATHI-65" x="4790" y="0"></use>
 <use xlink:href="#MJMATHI-74" x="5256" y="0"></use>
 <use xlink:href="#MJMATHI-61" x="5618" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="6147" y="0"></use>
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 <use xlink:href="#MJMAIN-29" x="9207" y="0"></use>
</g>
</svg></span>, <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.843ex" height="6.009ex" style="vertical-align: -2.505ex;" viewBox="0 -1508.9 13710.1 2587.3" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-60-Title">
<title id="MathJax-SVG-60-Title">P_{Y}(y) = P_{X}(g^{-1}(y)) · \frac{∂}{∂y} g^{-1}(y),</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-50" x="0" y="0"></use>
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<g transform="translate(3892,0)">
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</g>
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 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use>
</g>
</g>
 <use xlink:href="#MJMAIN-28" x="7113" y="0"></use>
 <use xlink:href="#MJMATHI-79" x="7503" y="0"></use>
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<g transform="translate(9280,0)">
<g transform="translate(120,0)">
<rect stroke="none" width="1149" height="60" x="0" y="220"></rect>
 <use xlink:href="#MJMAIN-2202" x="308" y="676"></use>
<g transform="translate(60,-736)">
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</g>
</g>
</g>
<g transform="translate(10669,0)">
 <use xlink:href="#MJMATHI-67" x="0" y="0"></use>
<g transform="translate(481,412)">
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2212" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="778" y="0"></use>
</g>
</g>
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 <use xlink:href="#MJMAIN-2C" x="13431" y="0"></use>
</g>
</svg></span> and this after some Wolfram Alpha and hand-manipulation becomes <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="44.588ex" height="3.343ex" style="vertical-align: -0.838ex;" viewBox="0 -1078.4 19197.6 1439.2" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-61-Title">
<title id="MathJax-SVG-61-Title">P_{Y}(y) = y^{(α-δ)/δ} · (1-y^{1/δ})^{β-1} / (δ · B(α, β)),</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-50" x="0" y="0"></use>
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<g transform="translate(3892,0)">
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 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="1808" y="0"></use>
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 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="2642" y="0"></use>
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</g>
</g>
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<g transform="translate(9865,0)">
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<g transform="translate(499,412)">
 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="0" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMAIN-2F" x="500" y="0"></use>
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B4" x="1001" y="0"></use>
</g>
</g>
<g transform="translate(11486,0)">
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<g transform="translate(389,412)">
 <use transform="scale(0.707)" xlink:href="#MJMATHI-3B2" x="0" y="0"></use>
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 <use transform="scale(0.707)" xlink:href="#MJMAIN-31" x="1345" y="0"></use>
</g>
</g>
 <use xlink:href="#MJMAIN-2F" x="13281" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="13781" y="0"></use>
 <use xlink:href="#MJMATHI-3B4" x="14171" y="0"></use>
 <use xlink:href="#MJMAIN-22C5" x="14838" y="0"></use>
 <use xlink:href="#MJMATHI-42" x="15338" y="0"></use>
 <use xlink:href="#MJMAIN-28" x="16098" y="0"></use>
 <use xlink:href="#MJMATHI-3B1" x="16487" y="0"></use>
 <use xlink:href="#MJMAIN-2C" x="17128" y="0"></use>
 <use xlink:href="#MJMATHI-3B2" x="17573" y="0"></use>
 <use xlink:href="#MJMAIN-29" x="18140" y="0"></use>
 <use xlink:href="#MJMAIN-29" x="18529" y="0"></use>
 <use xlink:href="#MJMAIN-2C" x="18919" y="0"></use>
</g>
</svg></span> where <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="31.584ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 13598.6 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-62-Title">
<title id="MathJax-SVG-62-Title">B(α, β) = Γ(α) · Γ(β) / Γ(α + β)</title>
<g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true">
 <use xlink:href="#MJMATHI-42" x="0" y="0"></use>
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</svg></span> is <a href="https://en.wikipedia.org/wiki/Beta_function">beta function</a>, also the normalizing denominator in the Beta density (confusing, sorry), and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.909ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 1683 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-63-Title">
<title id="MathJax-SVG-63-Title">Γ(·)</title>
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</svg></span> is the <a href="https://en.wikipedia.org/wiki/Gamma_function">gamma function</a>, a generalization of factorial.</p>
<blockquote>
<p>To check this, type in <code>y^((a-1)/d) * (1 - y^(1/d))^(b-1) / Beta[a,b] * D[y^(1/d), y]</code> at <a href="https://www.wolframalpha.com">Wolfram Alpha</a>.</p>
</blockquote>
<p>Replacing the <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.98ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 852.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-64-Title">
<title id="MathJax-SVG-64-Title">X</title>
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</svg></span>’s and <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.773ex" height="2.009ex" style="vertical-align: -0.171ex;" viewBox="0 -791.3 763.5 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-65-Title">
<title id="MathJax-SVG-65-Title">Y</title>
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</svg></span>’s with our usual variables, we have the probability density for <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.147ex" height="3.176ex" style="vertical-align: -1.005ex; margin-left: -0.089ex;" viewBox="-38.5 -934.9 3507.5 1367.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-66-Title">
<title id="MathJax-SVG-66-Title">p_{t_2} = p_t^δ</title>
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</svg></span> in terms of the original density for <span class="mjpage"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.085ex" height="2.009ex" style="vertical-align: -0.671ex; margin-left: -0.089ex;" viewBox="-38.5 -576.1 897.6 865.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-67-Title">
<title id="MathJax-SVG-67-Title">p_t</title>
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</svg></span>: <span class="mjpage mjpage__block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="32.091ex" height="6.843ex" style="vertical-align: -2.671ex;" viewBox="0 -1796 13817.1 2946.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-68-Title">
<title id="MathJax-SVG-68-Title">P(p_t^δ) = \frac{p^{(α - δ)/δ} · (1-p^{1/δ})^{β-1}}{δ · B(α, β)}.</title>
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