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<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#simple-decision-problems" id="toc-simple-decision-problems" class="nav-link active" data-scroll-target="#simple-decision-problems"><span class="header-section-number">31.1</span> Simple decision problems</a>
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<li><a href="#assessing-the-likelihood-that-a-used-car-will-be-sound" id="toc-assessing-the-likelihood-that-a-used-car-will-be-sound" class="nav-link" data-scroll-target="#assessing-the-likelihood-that-a-used-car-will-be-sound"><span class="header-section-number">31.1.1</span> Assessing the Likelihood That a Used Car Will Be Sound</a></li>
<li><a href="#calculation-without-simulation" id="toc-calculation-without-simulation" class="nav-link" data-scroll-target="#calculation-without-simulation"><span class="header-section-number">31.1.2</span> Calculation without simulation</a></li>
</ul></li>
<li><a href="#probability-interpretation" id="toc-probability-interpretation" class="nav-link" data-scroll-target="#probability-interpretation"><span class="header-section-number">31.2</span> Probability interpretation</a>
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<li><a href="#probability-from-proportion" id="toc-probability-from-proportion" class="nav-link" data-scroll-target="#probability-from-proportion"><span class="header-section-number">31.2.1</span> Probability from proportion</a></li>
<li><a href="#probability-relationships-conditional-probability" id="toc-probability-relationships-conditional-probability" class="nav-link" data-scroll-target="#probability-relationships-conditional-probability"><span class="header-section-number">31.2.2</span> Probability relationships: conditional probability</a></li>
<li><a href="#example-conditional-probability" id="toc-example-conditional-probability" class="nav-link" data-scroll-target="#example-conditional-probability"><span class="header-section-number">31.2.3</span> Example: conditional probability</a></li>
<li><a href="#estimating-driving-risk-for-insurance-purposes" id="toc-estimating-driving-risk-for-insurance-purposes" class="nav-link" data-scroll-target="#estimating-driving-risk-for-insurance-purposes"><span class="header-section-number">31.2.4</span> Estimating Driving Risk for Insurance Purposes</a></li>
<li><a href="#screening-for-disease" id="toc-screening-for-disease" class="nav-link" data-scroll-target="#screening-for-disease"><span class="header-section-number">31.2.5</span> Screening for Disease</a></li>
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<li><a href="#fundamental-problems-in-statistical-practice" id="toc-fundamental-problems-in-statistical-practice" class="nav-link" data-scroll-target="#fundamental-problems-in-statistical-practice"><span class="header-section-number">31.3</span> Fundamental problems in statistical practice</a></li>
<li><a href="#problems-based-on-normal-and-other-distributions" id="toc-problems-based-on-normal-and-other-distributions" class="nav-link" data-scroll-target="#problems-based-on-normal-and-other-distributions"><span class="header-section-number">31.4</span> Problems based on normal and other distributions</a>
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<li><a href="#an-intermediate-problem-in-conditional-probability" id="toc-an-intermediate-problem-in-conditional-probability" class="nav-link" data-scroll-target="#an-intermediate-problem-in-conditional-probability"><span class="header-section-number">31.4.1</span> An Intermediate Problem in Conditional Probability</a></li>
<li><a href="#estimating-the-posterior-distribution" id="toc-estimating-the-posterior-distribution" class="nav-link" data-scroll-target="#estimating-the-posterior-distribution"><span class="header-section-number">31.4.2</span> Estimating the Posterior Distribution</a></li>
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<li><a href="#conclusion" id="toc-conclusion" class="nav-link" data-scroll-target="#conclusion"><span class="header-section-number">31.5</span> Conclusion</a></li>
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<h1 class="title"><span id="sec-bayes-simulation" class="quarto-section-identifier"><span class="chapter-number">31</span> <span class="chapter-title">Bayesian Analysis by Simulation</span></span></h1>
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<blockquote class="blockquote">
<p>This branch of mathematics [probability] is the only one, I believe, in which good writers frequently get results entirely erroneous. <span class="citation" data-cites="peirce1923chance">(<a href="references.html#ref-peirce1923chance" role="doc-biblioref">Peirce 1923</a>, Doctrine of Chances, II)</span></p>
</blockquote>
<p>Bayesian analysis is a way of thinking about problems in probability and statistics that can help one reach otherwise-difficult decisions. It also can sometimes be used in science. The range of its recommended uses is controversial, but this chapter deals only with those uses of Bayesian analysis that are uncontroversial.</p>
<p>Better than defining Bayesian analysis in formal terms is to demonstrate its use. We shall start with the simplest sort of problem, and proceed gradually from there.</p>
<section id="simple-decision-problems" class="level2" data-number="31.1">
<h2 data-number="31.1" class="anchored" data-anchor-id="simple-decision-problems"><span class="header-section-number">31.1</span> Simple decision problems</h2>
<section id="assessing-the-likelihood-that-a-used-car-will-be-sound" class="level3" data-number="31.1.1">
<h3 data-number="31.1.1" class="anchored" data-anchor-id="assessing-the-likelihood-that-a-used-car-will-be-sound"><span class="header-section-number">31.1.1</span> Assessing the Likelihood That a Used Car Will Be Sound</h3>
<p>Consider a problem in estimating the soundness of a used car one considers purchasing (after <span class="citation" data-cites="wonnacott1990introductory">(<a href="references.html#ref-wonnacott1990introductory" role="doc-biblioref">Wonnacott and Wonnacott 1990, 93–94</a>)</span>). Seventy percent of the cars are known to be OK on average, and 30 percent are faulty. Of the cars that <em>are</em> really OK, a mechanic correctly identifies 80 percent as “OK” but says that 20 percent are “faulty”; of those that are faulty, the mechanic correctly identifies 90 percent as faulty and says (incorrectly) that 10 percent are OK.</p>
<p>We wish to know the probability that if the mechanic <em>says</em> a car is “OK,” it <em>really</em> is faulty. Phrased differently, what is the probability of a car being faulty if the mechanic said it was OK?</p>
<p>We can get the desired probabilities directly by simulation without knowing Bayes’ rule, as we shall see. But one must be able to model the physical problem correctly in order to proceed with the simulation; this requirement of a clearly visualized model is a strong point in favor of simulation.</p>
<ol type="1">
<li>Note that we are only interested in outcomes where the mechanic approved a car.</li>
<li>For each car, generate a label of either “faulty” or “working” with probabilities of 0.3 and 0.7, respectively.</li>
<li>For each <em>faulty car</em>, we generate one of two labels, “approved” or “not approved” with probabilities 0.1 and 0.9, respectively.</li>
<li>For each <em>working car</em>, we generate one of two labels, “approved” or “not approved” with probabilities 0.7 and 0.3, respectively.</li>
<li>Out of all cars “approved”, count how many are “faulty”. The ratio between these numbers is our answer.</li>
</ol>
<div class="python">
<div class="callout callout-style-default callout-note callout-titled">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
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<div class="callout-title-container flex-fill">
In-place operators in Python
</div>
</div>
<div class="callout-body-container callout-body">
<p>In the code that follows, we are going to use a new and neat feature of Python, called <em>in-place</em> operations.</p>
<p>We often find ourselves in the situation where we have some variable, say <code>a</code>, and we want to do some operation on the value of <code>a</code> <em>and then store the result in <code>a</code> again</em>.</p>
<p>Of course, we could do something like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The initial value of "a".</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> <span class="dv">10</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set a to get the value of a plus 1.</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>a <span class="op">=</span> a <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result.</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>a</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>11</code></pre>
</div>
</div>
<p>Because this is such a common kind of thing to do, Python has special operators to do this for us, called <em>in-place</em> operators. For the example above, there is an <em>in-place addition</em> operator <code>+=</code> that takes the value of the variable on the left hand side, adds the value on the right hand side, and stores the result in the variable on the left hand side. For example:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The initial value of "b".</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>b <span class="op">=</span> <span class="dv">10</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Set b to get the value of b plus 1.</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Notice the in-place operator "+=".</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>b <span class="op">+=</span> <span class="dv">1</span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result.</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>b</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>11</code></pre>
</div>
</div>
<p>There are also in-place operators for subtraction (e.g. <code>c -= 3</code>), multiplication (e.g. <code>d *= 2</code>), and division (<code>e /= 2</code>).</p>
<p>You will see us using the in-place <code>+=</code> in the code below.</p>
</div>
</div>
</div>
<p>Here is the whole simulation of the car / mechanic problem:</p>
<div class="notebook" name="bayes_cars" title="Bayesian analysis of cars and mechanics">
<p>Start of <code>bayes_cars</code> notebook</p>
<div class="nb-links">
<a class="notebook-link" href="notebooks/bayes_cars.ipynb">Download notebook</a>
<a class="interact-button" href="./interact/lab/index.html?path=bayes_cars.ipynb">Interact</a>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Set up the random number generator.</span></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10000</span> <span class="co"># number of cars</span></span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Counters for number of approved, number of approved and faulty</span></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>approved <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a>approved_and_faulty <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># Decide whether the car is faulty or working, with a probability of</span></span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># 0.3 and 0.7 respectively</span></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a> car <span class="op">=</span> rnd.choice([<span class="st">'faulty'</span>, <span class="st">'working'</span>], p<span class="op">=</span>[<span class="fl">0.3</span>, <span class="fl">0.7</span>])</span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> car <span class="op">==</span> <span class="st">'faulty'</span>:</span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a> <span class="co"># What the mechanic says of a faulty car</span></span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a> mechanic_says <span class="op">=</span> rnd.choice([<span class="st">'approved'</span>, <span class="st">'not approved'</span>], p<span class="op">=</span>[<span class="fl">0.1</span>, <span class="fl">0.9</span>])</span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a> <span class="cf">else</span>:</span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># What the mechanic says of a working car</span></span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a> mechanic_says <span class="op">=</span> rnd.choice([<span class="st">'approved'</span>, <span class="st">'not approved'</span>], p<span class="op">=</span>[<span class="fl">0.7</span>, <span class="fl">0.3</span>])</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> mechanic_says <span class="op">==</span> <span class="st">'approved'</span>:</span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a> approved <span class="op">+=</span> <span class="dv">1</span></span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> car <span class="op">==</span> <span class="st">'faulty'</span>:</span>
<span id="cb7-29"><a href="#cb7-29" aria-hidden="true" tabindex="-1"></a> approved_and_faulty <span class="op">+=</span> <span class="dv">1</span></span>
<span id="cb7-30"><a href="#cb7-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-31"><a href="#cb7-31" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> approved_and_faulty <span class="op">/</span> approved</span>
<span id="cb7-32"><a href="#cb7-32" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-33"><a href="#cb7-33" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion of faulty cars of cars approved:'</span>, np.<span class="bu">round</span>(k, <span class="dv">2</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Proportion of faulty cars of cars approved: 0.06</code></pre>
</div>
</div>
<p>The answer looks to be somewhere between 5 and 6%. The code clearly follows the description step by step, but it is also quite slow. If we can improve the code, we may be able to do our simulation with more cars, and get a more accurate answer.</p>
<p>Let’s use arrays to store the states of all cars in the lot simultaneously:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of cars; we made this number larger by a factor of 100</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">1_000_000</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Generate an array with as many entries as there are cars, each</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="co"># being either 'working' or 'faulty'</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>cars <span class="op">=</span> rnd.choice([<span class="st">'working'</span>, <span class="st">'faulty'</span>], p<span class="op">=</span>[<span class="fl">0.7</span>, <span class="fl">0.3</span>], size<span class="op">=</span>n_trials)</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Count how many cars are working</span></span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a>n_working <span class="op">=</span> np.<span class="bu">sum</span>(cars <span class="op">==</span> <span class="st">'working'</span>)</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a><span class="co"># All the rest are faulty</span></span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a>n_faulty <span class="op">=</span> n_trials <span class="op">-</span> n_working</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a><span class="co"># Create a new array in which to store what a mechanic says</span></span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a><span class="co"># about the car: 'approved' or 'not approved'. Use</span></span>
<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a><span class="co"># dtype=object because we need to store strings.</span></span>
<span id="cb9-17"><a href="#cb9-17" aria-hidden="true" tabindex="-1"></a>mechanic_says <span class="op">=</span> np.zeros(n_trials, dtype<span class="op">=</span><span class="bu">object</span>)</span>
<span id="cb9-18"><a href="#cb9-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-19"><a href="#cb9-19" aria-hidden="true" tabindex="-1"></a><span class="co"># We start with the working cars; what does the mechanic say about them?</span></span>
<span id="cb9-20"><a href="#cb9-20" aria-hidden="true" tabindex="-1"></a><span class="co"># Generate 'approved' or 'not approved' labels with the given probabilities.</span></span>
<span id="cb9-21"><a href="#cb9-21" aria-hidden="true" tabindex="-1"></a>mechanic_says[cars <span class="op">==</span> <span class="st">'working'</span>] <span class="op">=</span> rnd.choice(</span>
<span id="cb9-22"><a href="#cb9-22" aria-hidden="true" tabindex="-1"></a> [<span class="st">'approved'</span>, <span class="st">'not approved'</span>], p<span class="op">=</span>[<span class="fl">0.8</span>, <span class="fl">0.2</span>], size<span class="op">=</span>n_working</span>
<span id="cb9-23"><a href="#cb9-23" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb9-24"><a href="#cb9-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-25"><a href="#cb9-25" aria-hidden="true" tabindex="-1"></a><span class="co"># Similarly, for each faulty car, generate 'approved'/'not approved'</span></span>
<span id="cb9-26"><a href="#cb9-26" aria-hidden="true" tabindex="-1"></a><span class="co"># labels with the given probabilities.</span></span>
<span id="cb9-27"><a href="#cb9-27" aria-hidden="true" tabindex="-1"></a>mechanic_says[cars <span class="op">==</span> <span class="st">'faulty'</span>] <span class="op">=</span> rnd.choice(</span>
<span id="cb9-28"><a href="#cb9-28" aria-hidden="true" tabindex="-1"></a> [<span class="st">'approved'</span>, <span class="st">'not approved'</span>], p<span class="op">=</span>[<span class="fl">0.1</span>, <span class="fl">0.9</span>], size<span class="op">=</span>n_faulty</span>
<span id="cb9-29"><a href="#cb9-29" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb9-30"><a href="#cb9-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-31"><a href="#cb9-31" aria-hidden="true" tabindex="-1"></a><span class="co"># Identify all cars that were approved</span></span>
<span id="cb9-32"><a href="#cb9-32" aria-hidden="true" tabindex="-1"></a><span class="co"># This produces a binary mask, an array that looks like:</span></span>
<span id="cb9-33"><a href="#cb9-33" aria-hidden="true" tabindex="-1"></a><span class="co"># [True, False, False, True, ... ]</span></span>
<span id="cb9-34"><a href="#cb9-34" aria-hidden="true" tabindex="-1"></a>approved <span class="op">=</span> (mechanic_says <span class="op">==</span> <span class="st">'approved'</span>)</span>
<span id="cb9-35"><a href="#cb9-35" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-36"><a href="#cb9-36" aria-hidden="true" tabindex="-1"></a><span class="co"># Identify cars that are faulty AND were approved</span></span>
<span id="cb9-37"><a href="#cb9-37" aria-hidden="true" tabindex="-1"></a>faulty_but_approved <span class="op">=</span> (cars <span class="op">==</span> <span class="st">'faulty'</span>) <span class="op">&</span> approved</span>
<span id="cb9-38"><a href="#cb9-38" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-39"><a href="#cb9-39" aria-hidden="true" tabindex="-1"></a><span class="co"># Count the number of cars that are faulty but approved, as well as</span></span>
<span id="cb9-40"><a href="#cb9-40" aria-hidden="true" tabindex="-1"></a><span class="co"># the total number of cars that were approved</span></span>
<span id="cb9-41"><a href="#cb9-41" aria-hidden="true" tabindex="-1"></a>n_faulty_but_approved <span class="op">=</span> np.<span class="bu">sum</span>(faulty_but_approved)</span>
<span id="cb9-42"><a href="#cb9-42" aria-hidden="true" tabindex="-1"></a>n_approved <span class="op">=</span> np.<span class="bu">sum</span>(approved)</span>
<span id="cb9-43"><a href="#cb9-43" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-44"><a href="#cb9-44" aria-hidden="true" tabindex="-1"></a><span class="co"># Calculate the ratio, which is the answer we seek</span></span>
<span id="cb9-45"><a href="#cb9-45" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> n_faulty_but_approved <span class="op">/</span> n_approved</span>
<span id="cb9-46"><a href="#cb9-46" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-47"><a href="#cb9-47" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion of faulty cars of cars approved:'</span>, np.<span class="bu">round</span>(k, <span class="dv">2</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Proportion of faulty cars of cars approved: 0.05</code></pre>
</div>
</div>
<p>The code now runs much faster, and with a larger number of cars we see that the answer is closer to a 5% chance of a car being broken after it has been approved by a mechanic.</p>
<p>End of <code>bayes_cars</code> notebook</p>
</div>
<!---
End of notebook
-->
</section>
<section id="calculation-without-simulation" class="level3" data-number="31.1.2">
<h3 data-number="31.1.2" class="anchored" data-anchor-id="calculation-without-simulation"><span class="header-section-number">31.1.2</span> Calculation without simulation</h3>
<p>Simulation forces us to model our problem clearly and concretely in code. Such code is most often easier to reason about than opaque statistical methods. Running the simulation gives a good sense of what the correct answer should be. Thereafter, we can still look into different — sometimes more elegant or accurate — ways of modeling and solving the problem.</p>
<p>Let’s examine the following diagram of our car selection:</p>
<p><img src="diagrams/car-tree.png" class="img-fluid"></p>
<p>We see that there are two paths, highlighted, that results in a car being approved by a mechanic. Either a car can be working, and correctly identified as such by a mechanic; or the car can be broken, while the mechanic mistakenly determines it to be working. Our question only pertains to these two paths, so we do not need to study the rest of the tree.</p>
<p>In the long run, in our simulation, about 70% of the cars will end with the label “working”, and about 30% will end up with the label “faulty”. We just took 10000 sample cars above but, in fact, the larger the number of cars we take, the closer we will get to 70% “working” and 30% “faulty”. So, with many samples, we can think of 70% of these samples flowing down the “working” path, and 30% flowing along the “faulty” path.</p>
<p>Now, we want to know, of all the cars approved by a mechanic, how many are faulty:</p>
<p><span class="math display">\[ \frac{\mathrm{cars_{\mathrm{faulty}}}}{\mathrm{cars}_{\mathrm{approved}}} \]</span></p>
<p>We follow the two highlighted paths in the tree:</p>
<ol type="1">
<li>Of a large sample of cars, 30% are faulty. Of these, 10% are approved by a mechanic. That is, 30% * 10% = 3% of all cars.</li>
<li>Of all cars, 70% work. Of these, 80% are approved by a mechanic. That is, 70% * 80% = 56% of all cars.</li>
</ol>
<p>The percentage of faulty cars, out of approved cars, becomes:</p>
<p><span class="math display">\[
3\% / (56\% + 3\%) = 5.08\%
\]</span></p>
<p>Notation-wise, it is a bit easier to calculate these sums using proportions rather than percentages:</p>
<ol type="1">
<li>Faulty cars approved by a mechanic: 0.3 * 0.1 = 0.03</li>
<li>Working cars approved by a mechanic: 0.7 * 0.8 = 0.56</li>
</ol>
<p>Fraction of faulty cars out of approved cars: 0.03 / (0.03 + 0.56) = 0.0508</p>
<p>We see that every time the tree branches, it filters the cars: some go to one branch, the rest to another. In our code, we used the AND (<code>&</code>) operator to find the intersection between faulty AND approved cars, i.e., to filter out from all faulty cars only the cars that were ALSO approved.</p>
</section>
</section>
<section id="probability-interpretation" class="level2" data-number="31.2">
<h2 data-number="31.2" class="anchored" data-anchor-id="probability-interpretation"><span class="header-section-number">31.2</span> Probability interpretation</h2>
<section id="probability-from-proportion" class="level3" data-number="31.2.1">
<h3 data-number="31.2.1" class="anchored" data-anchor-id="probability-from-proportion"><span class="header-section-number">31.2.1</span> Probability from proportion</h3>
<p>In these examples, we often calculate proportions. In the given simulation:</p>
<ul>
<li>How many cars are approved by a mechanic? 59/100.</li>
<li>How many of those 59 were faulty? 3/59.</li>
</ul>
<p>We often also count how commonly events occur: “it rained 4 out of the 10 days”.</p>
<p>An extension of this idea is to <em>predict</em> the probability of an event occurring, based on what we had seen in the past. We can say “out of 100 days, there was some rain on 20 of them; we therefore estimate that the probability of rain occurring is 20/100”. Of course, this is not a complex or very accurate weather model; for that, we’d need to take other factors—such as season—into consideration. Overall, the more observations we have, the better our probability estimates become. We discussed this idea previously in “The Law of Large Numbers”.</p>
<!---
** TODO: REFERENCE SECTION ON LARGE NUMBERS **
-->
<section id="ratios-of-proportions" class="level4" data-number="31.2.1.1">
<h4 data-number="31.2.1.1" class="anchored" data-anchor-id="ratios-of-proportions"><span class="header-section-number">31.2.1.1</span> Ratios of proportions</h4>
<p>At our mechanic’s yard, we can ask “how many red cars here are faulty”? To calculate that, we’d first count the number of red cars, then the number of those red cars that are also broken, then calculate the ratio: <code>red_cars_faulty / red_cars</code>.</p>
<p>We could just as well have worked in percentages: <code>percentage_of_red_cars_broken / percentage_of_cars_that_are_red</code>, since that is <code>(red_cars_broken / 100) / (red_cars / 100)</code>—the same ratio calculated before.</p>
<p>Our point is that the denominator doesn’t matter when calculating ratios, so we could just as well have written:</p>
<p>(red_cars_broken / all_cars) / (red_cars / all_cars)</p>
<p>or</p>
<p><span class="math display">\[
P(\text{cars that are red and that are broken}) / P(\text{red cars})
\]</span></p>
<!---
** TODO: THE ABOVE MAY BE A SUBTLE POINT THAT NEEDS TO EXPANDED, BUT THE TEXT IS GETTING LONG AS-IS **
-->
</section>
</section>
<section id="probability-relationships-conditional-probability" class="level3" data-number="31.2.2">
<h3 data-number="31.2.2" class="anchored" data-anchor-id="probability-relationships-conditional-probability"><span class="header-section-number">31.2.2</span> Probability relationships: conditional probability</h3>
<p>Here’s one way of writing the probability that a car is broken:</p>
<p><span class="math display">\[
P(\text{car is broken})
\]</span></p>
<p>We can shorten “car is broken” to B, and write the same thing as:</p>
<p><span class="math display">\[
P(B)
\]</span></p>
<p>Similarly, we could write the probability that a car is red as:</p>
<p><span class="math display">\[
P(R)
\]</span></p>
<p>We might also want to express the <em>conditional probability</em>, as in the probability that the car is broken, <em>given that</em> we already know that the car is red:</p>
<p><span class="math display">\[
P(\text{car is broken GIVEN THAT car is red})
\]</span></p>
<p>That is getting getting pretty verbose, so we will shorten this as we did above:</p>
<p><span class="math display">\[
P(B \text{ GIVEN THAT } R)
\]</span></p>
<p>To make things even more compact, we write “GIVEN THAT” as a vertical bar <code>|</code> — so the whole thing becomes:</p>
<p><span class="math display">\[
P(B | R)
\]</span></p>
<p>We read this as “the probability that the car is broken given that the car is red”. Such a probability is known as a <em>conditional probability</em>. We discuss these in more detail in <a href="probability_theory_1a.html#sec-cond-uncond" class="quarto-xref"><span>Section 8.13</span></a>.</p>
<p>In our original problem, we ask what the chance is of a car being broken given that a mechanic approved it. As discussed under “Ratios of proportions”, it can be calculated with:</p>
<p><span class="math display">\[\begin{align*}
P(\text{car broken | mechanic approved}) = \\
P(\text{car broken and mechanic approved}) / P(\text{mechanic approved})
\end{align*}\]</span></p>
<p>We have already used <span class="math inline">\(B\)</span> to mean “broken” (above), so let us use <span class="math inline">\(A\)</span> to mean “mechanic approved”. Then we can write the statement above in a more compact way:</p>
<p><span class="math display">\[
P(B | A) = P(B \text{ and } A) / P(A)
\]</span></p>
<p>To put this generally, conditional probabilities for two events <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span> can be written as:</p>
<p><span class="math inline">\(P(X | Y) = P(X \text{ and } Y) / P(Y)\)</span></p>
<p>Where (again) <span class="math inline">\(\text{ and }\)</span> means that <em>both</em> events occur.</p>
</section>
<section id="example-conditional-probability" class="level3" data-number="31.2.3">
<h3 data-number="31.2.3" class="anchored" data-anchor-id="example-conditional-probability"><span class="header-section-number">31.2.3</span> Example: conditional probability</h3>
<p>Let’s discuss a very relevant example. You get a Covid test, and the test is negative. Now, you would like to know what the chance is of you having Covid.</p>
<p>We have the following information:</p>
<ul>
<li>1.5% of people in your area have Covid</li>
<li>The false positive rate of the tests (i.e., that they detect Covid when it is absent) is very low at 0.5%</li>
<li>The false negative rate (i.e., that they fail to detect Covid when it is present) is quite high at 40%</li>
</ul>
<p><img src="diagrams/covid-tree.png" class="img-fluid"></p>
<p>Again, we start with our simulation.</p>
<div class="notebook" name="bayes_covid" title="Bayesian analysis of Covid test result">
<p>Start of <code>bayes_covid</code> notebook</p>
<div class="nb-links">
<a class="notebook-link" href="notebooks/bayes_covid.ipynb">Download notebook</a>
<a class="interact-button" href="./interact/lab/index.html?path=bayes_covid.ipynb">Interact</a>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="co"># The number of people.</span></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">1_000_000</span></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a><span class="co"># For each person, generate a True or False label,</span></span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a><span class="co"># indicating that they have / don't have Covid.</span></span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a>person_has_covid <span class="op">=</span> rnd.choice(</span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a> [<span class="va">True</span>, <span class="va">False</span>], p<span class="op">=</span>[<span class="fl">0.015</span>, <span class="fl">0.985</span>],</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>n_trials</span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-15"><a href="#cb11-15" aria-hidden="true" tabindex="-1"></a><span class="co"># Calculate the numbers of people with and without Covid.</span></span>
<span id="cb11-16"><a href="#cb11-16" aria-hidden="true" tabindex="-1"></a>n_with_covid <span class="op">=</span> np.<span class="bu">sum</span>(person_has_covid)</span>
<span id="cb11-17"><a href="#cb11-17" aria-hidden="true" tabindex="-1"></a>n_without_covid <span class="op">=</span> n_trials <span class="op">-</span> n_with_covid</span>
<span id="cb11-18"><a href="#cb11-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-19"><a href="#cb11-19" aria-hidden="true" tabindex="-1"></a><span class="co"># In this array, we will store, for each person, whether they</span></span>
<span id="cb11-20"><a href="#cb11-20" aria-hidden="true" tabindex="-1"></a><span class="co"># had a positive or a negative test.</span></span>
<span id="cb11-21"><a href="#cb11-21" aria-hidden="true" tabindex="-1"></a>test_result <span class="op">=</span> np.zeros(n_trials, dtype<span class="op">=</span><span class="bu">bool</span>)</span>
<span id="cb11-22"><a href="#cb11-22" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-23"><a href="#cb11-23" aria-hidden="true" tabindex="-1"></a><span class="co"># Draw test results for people with Covid.</span></span>
<span id="cb11-24"><a href="#cb11-24" aria-hidden="true" tabindex="-1"></a>test_result[person_has_covid] <span class="op">=</span> rnd.choice(</span>
<span id="cb11-25"><a href="#cb11-25" aria-hidden="true" tabindex="-1"></a> [<span class="va">True</span>, <span class="va">False</span>], p<span class="op">=</span>[<span class="fl">0.6</span>, <span class="fl">0.4</span>],</span>
<span id="cb11-26"><a href="#cb11-26" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>n_with_covid</span>
<span id="cb11-27"><a href="#cb11-27" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb11-28"><a href="#cb11-28" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-29"><a href="#cb11-29" aria-hidden="true" tabindex="-1"></a><span class="co"># Draw test results for people without Covid.</span></span>
<span id="cb11-30"><a href="#cb11-30" aria-hidden="true" tabindex="-1"></a><span class="co"># ~person_has_covid` flips all Boolean values from FALSE to TRUE</span></span>
<span id="cb11-31"><a href="#cb11-31" aria-hidden="true" tabindex="-1"></a><span class="co"># and from TRUE to FALSE.</span></span>
<span id="cb11-32"><a href="#cb11-32" aria-hidden="true" tabindex="-1"></a>test_result[<span class="op">~</span>person_has_covid] <span class="op">=</span> rnd.choice(</span>
<span id="cb11-33"><a href="#cb11-33" aria-hidden="true" tabindex="-1"></a> [<span class="va">True</span>, <span class="va">False</span>], p<span class="op">=</span>[<span class="fl">0.005</span>, <span class="fl">0.995</span>],</span>
<span id="cb11-34"><a href="#cb11-34" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>n_without_covid</span>
<span id="cb11-35"><a href="#cb11-35" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb11-36"><a href="#cb11-36" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-37"><a href="#cb11-37" aria-hidden="true" tabindex="-1"></a><span class="co"># Get the Covid statuses of all those with negative tests</span></span>
<span id="cb11-38"><a href="#cb11-38" aria-hidden="true" tabindex="-1"></a><span class="co"># (`test_result` is a Boolean mask, like `[True, False, False, True, ...]`,</span></span>
<span id="cb11-39"><a href="#cb11-39" aria-hidden="true" tabindex="-1"></a><span class="co"># and `~test_result` flips all Boolean values to `[False, True, True, False, ...]`.</span></span>
<span id="cb11-40"><a href="#cb11-40" aria-hidden="true" tabindex="-1"></a>covid_status_negative_test <span class="op">=</span> person_has_covid[<span class="op">~</span>test_result]</span>
<span id="cb11-41"><a href="#cb11-41" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-42"><a href="#cb11-42" aria-hidden="true" tabindex="-1"></a><span class="co"># Now, count how many with Covid had a negative test results.</span></span>
<span id="cb11-43"><a href="#cb11-43" aria-hidden="true" tabindex="-1"></a>n_with_covid_and_negative_test <span class="op">=</span> np.<span class="bu">sum</span>(covid_status_negative_test)</span>
<span id="cb11-44"><a href="#cb11-44" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-45"><a href="#cb11-45" aria-hidden="true" tabindex="-1"></a><span class="co"># And how many people, overall, had negative test results.</span></span>
<span id="cb11-46"><a href="#cb11-46" aria-hidden="true" tabindex="-1"></a>n_with_negative_test <span class="op">=</span> <span class="bu">len</span>(covid_status_negative_test)</span>
<span id="cb11-47"><a href="#cb11-47" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-48"><a href="#cb11-48" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> n_with_covid_and_negative_test <span class="op">/</span> n_with_negative_test</span>
<span id="cb11-49"><a href="#cb11-49" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-50"><a href="#cb11-50" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion with Covid of those with negative test:'</span>, np.<span class="bu">round</span>(k, <span class="dv">4</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Proportion with Covid of those with negative test: 0.0061</code></pre>
</div>
</div>
<p>This gives around 0.006 or 0.6%.</p>
<p>End of <code>bayes_covid</code> notebook</p>
</div>
<!---
End of notebook.
-->
<p>Now that we have a rough indication of what the answer should be, let’s try and calculate it directly, based on the tree of information shown earlier.</p>
<p>We will use these abbreviations:</p>
<ul>
<li><span class="math inline">\(C^+\)</span> means Covid positive (you do actually have Covid).</li>
<li><span class="math inline">\(C^-\)</span> means Covid negative (you do <em>not</em> actually have Covid).</li>
<li><span class="math inline">\(T^+\)</span> means the Covid <em>test</em> was positive.</li>
<li><span class="math inline">\(T^-\)</span> means the Covid <em>test</em> was negative.</li>
</ul>
<p>For example <span class="math inline">\(P(C^+ | T^-)\)</span> is the probability (<span class="math inline">\(P\)</span>) that you do actually have Covid (<span class="math inline">\(C^+\)</span>) <em>given that</em> (<span class="math inline">\(|\)</span>) the test was negative (<span class="math inline">\(T^-\)</span>).</p>
<p>We would like to know the probability of having Covid <em>given that</em> your test was negative (<span class="math inline">\(P(C^+ | T^-)\)</span>). Using the conditional probability relationship from above, we can write:</p>
<p><span class="math display">\[
P(C^+ | T^-) = P(C^+ \text{ and } T^-) / P(T^-)
\]</span></p>
<p>We see from the tree diagram that <span class="math inline">\(P(C^+ \text{ and } T^-) = P(T^- | C^+) * P(C^+) = .4 * .015 = 0.006\)</span>.</p>
<!---
**TODO: ADD REFERENCE TO SUMMATION OF MUTUALLY EXCLUSIVE PROBABILITIES**
-->
<p>We observe that <span class="math inline">\(P(T^-) = P(T^- \text{ and } C^-) + P(T^- \text{ and } C^+)\)</span>, i.e. that we can obtain a negative test result through two paths, having Covid or not having Covid. We expand these further as conditional probabilities:</p>
<p><span class="math inline">\(P(T^- \text{ and } C^-) = P(T^- | C^-) * P(C^-)\)</span></p>
<p>and</p>
<p><span class="math inline">\(P(T^- \text{ and } C^+) = P(T^- | C^+) * P(C^+)\)</span>.</p>
<p>We can now calculate</p>
<p><span class="math display">\[
P(T^-) = P(T^- | C^-) * P(C^-) + P(T^- | C^+) * P(C^+)
\]</span></p>
<p><span class="math display">\[
= .995 * .985 + .4 * .015 = 0.986
\]</span></p>
<p>The answer, then, is:</p>
<p><span class="math inline">\(P(C^+ | T^-) = 0.006 / 0.986 = 0.0061\)</span> or 0.61%.</p>
<p>This matches very closely our simulation result, so we have some confidence that we have done the calculation correctly.</p>
</section>
<section id="estimating-driving-risk-for-insurance-purposes" class="level3" data-number="31.2.4">
<h3 data-number="31.2.4" class="anchored" data-anchor-id="estimating-driving-risk-for-insurance-purposes"><span class="header-section-number">31.2.4</span> Estimating Driving Risk for Insurance Purposes</h3>
<p>Another sort of introductory problem, following after <span class="citation" data-cites="feller1968introduction">(<a href="references.html#ref-feller1968introduction" role="doc-biblioref">Feller 1968</a>, p 122)</span>:</p>
<div class="notebook" name="bayes_accidents" title="Bayesian analysis for insurance premium">
<p>Start of <code>bayes_accidents</code> notebook</p>
<div class="nb-links">
<a class="notebook-link" href="notebooks/bayes_accidents.ipynb">Download notebook</a>
<a class="interact-button" href="./interact/lab/index.html?path=bayes_accidents.ipynb">Interact</a>
</div>
<p>A mutual insurance company charges its members according to the risk of having an car accident. It is known that there are two classes of people — 80 percent of the population with good driving judgment and with a probability of .06 of having an accident each year, and 20 percent with poor judgment and a probability of .6 of having an accident each year. The company’s policy is to charge $100 for each percent of risk, i. e., a driver with a probability of .6 should pay 60*$100 = $6000.</p>
<p>If nothing is known of a driver except that they had an accident last year, what fee should they pay?</p>
<p>Another way to phrase this question is: given that a driver had an accident last year, what is the probability of them having an accident overall?</p>
<p>We will proceed as follows:</p>
<ol type="1">
<li>Generate a population of N people. Label each as <code>good driver</code> or <code>poor driver</code>.</li>
<li>Simulate the last year for each person: did they have an accident or not?</li>
<li>Select only the ones that had an accident last year.</li>
<li>Among those, calculate what their average risk is of making an accident. This will indicate the appropriate insurance premium.</li>
</ol>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">100_000</span></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a>cost_per_percent <span class="op">=</span> <span class="dv">100</span></span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a>people <span class="op">=</span> rnd.choice(</span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a> [<span class="st">'good driver'</span>, <span class="st">'poor driver'</span>], p<span class="op">=</span>[<span class="fl">0.8</span>, <span class="fl">0.2</span>],</span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>n_trials</span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a>good_driver <span class="op">=</span> (people <span class="op">==</span> <span class="st">'good driver'</span>)</span>
<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a>poor_driver <span class="op">=</span> <span class="op">~</span>good_driver</span>
<span id="cb13-15"><a href="#cb13-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-16"><a href="#cb13-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Did they have an accident last year?</span></span>
<span id="cb13-17"><a href="#cb13-17" aria-hidden="true" tabindex="-1"></a>had_accident <span class="op">=</span> np.zeros(n_trials, dtype<span class="op">=</span><span class="bu">bool</span>)</span>
<span id="cb13-18"><a href="#cb13-18" aria-hidden="true" tabindex="-1"></a>had_accident[good_driver] <span class="op">=</span> rnd.choice(</span>
<span id="cb13-19"><a href="#cb13-19" aria-hidden="true" tabindex="-1"></a> [<span class="va">True</span>, <span class="va">False</span>], p<span class="op">=</span>[<span class="fl">0.06</span>, <span class="fl">0.94</span>],</span>
<span id="cb13-20"><a href="#cb13-20" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>np.<span class="bu">sum</span>(good_driver)</span>
<span id="cb13-21"><a href="#cb13-21" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb13-22"><a href="#cb13-22" aria-hidden="true" tabindex="-1"></a>had_accident[poor_driver] <span class="op">=</span> rnd.choice(</span>
<span id="cb13-23"><a href="#cb13-23" aria-hidden="true" tabindex="-1"></a> [<span class="va">True</span>, <span class="va">False</span>], p<span class="op">=</span>[<span class="fl">0.6</span>, <span class="fl">0.4</span>],</span>
<span id="cb13-24"><a href="#cb13-24" aria-hidden="true" tabindex="-1"></a> size<span class="op">=</span>np.<span class="bu">sum</span>(poor_driver)</span>
<span id="cb13-25"><a href="#cb13-25" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb13-26"><a href="#cb13-26" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-27"><a href="#cb13-27" aria-hidden="true" tabindex="-1"></a>ppl_with_accidents <span class="op">=</span> people[had_accident]</span>
<span id="cb13-28"><a href="#cb13-28" aria-hidden="true" tabindex="-1"></a>n_good_driver_accidents <span class="op">=</span> np.<span class="bu">sum</span>(ppl_with_accidents <span class="op">==</span> <span class="st">'good driver'</span>)</span>
<span id="cb13-29"><a href="#cb13-29" aria-hidden="true" tabindex="-1"></a>n_poor_driver_accidents <span class="op">=</span> np.<span class="bu">sum</span>(ppl_with_accidents <span class="op">==</span> <span class="st">'poor driver'</span>)</span>
<span id="cb13-30"><a href="#cb13-30" aria-hidden="true" tabindex="-1"></a>n_all_with_accidents <span class="op">=</span> n_good_driver_accidents <span class="op">+</span> n_poor_driver_accidents</span>
<span id="cb13-31"><a href="#cb13-31" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-32"><a href="#cb13-32" aria-hidden="true" tabindex="-1"></a>avg_risk_percent <span class="op">=</span> ((n_good_driver_accidents <span class="op">*</span> <span class="fl">0.06</span> <span class="op">+</span></span>
<span id="cb13-33"><a href="#cb13-33" aria-hidden="true" tabindex="-1"></a> n_poor_driver_accidents <span class="op">*</span> <span class="fl">0.6</span>) <span class="op">/</span></span>
<span id="cb13-34"><a href="#cb13-34" aria-hidden="true" tabindex="-1"></a> n_all_with_accidents <span class="op">*</span> <span class="dv">100</span>)</span>
<span id="cb13-35"><a href="#cb13-35" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-36"><a href="#cb13-36" aria-hidden="true" tabindex="-1"></a>premium <span class="op">=</span> avg_risk_percent <span class="op">*</span> cost_per_percent</span>
<span id="cb13-37"><a href="#cb13-37" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-38"><a href="#cb13-38" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Premium is:'</span>, np.<span class="bu">round</span>(premium))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Premium is: 4462.0</code></pre>
</div>
</div>
<p>The answer should be around 4450 USD.</p>
<p>End of <code>bayes_accidents</code> notebook</p>
</div>
<!---
End of notebook.
-->
</section>
<section id="screening-for-disease" class="level3" data-number="31.2.5">
<h3 data-number="31.2.5" class="anchored" data-anchor-id="screening-for-disease"><span class="header-section-number">31.2.5</span> Screening for Disease</h3>
<!---
**TODO: SHALL WE REMOVE THIS PROBLEM, OR INTEGRATE PARTS OF ITS DESCRIPTION WITH THE COVID EXAMPLE?**
-->
<p>This is a classic Bayesian problem (quoted by Tversky and Kahneman <span class="citation" data-cites="tversky1982evidential">(<a href="references.html#ref-tversky1982evidential" role="doc-biblioref">1982, 154</a>)</span>, from Cascells <em>et al.</em> <span class="citation" data-cites="cascells1978interpretation">(<a href="references.html#ref-cascells1978interpretation" role="doc-biblioref">1978, 999</a>)</span>):</p>
<blockquote class="blockquote">
<p>If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?</p>
</blockquote>
<p>Tversky and Kahneman note that among the respondents — students and staff at Harvard Medical School — “the most common response, given by almost half of the participants, was 95%” — very much the wrong answer.</p>
<p>To obtain an answer by simulation, we may rephrase the question above with (hypothetical) absolute numbers as follows:</p>
<p>If a test to detect a disease whose prevalence has been estimated to be about 100,000 in the population of 100 million persons over age 40 (that is, about 1 in a thousand) has been observed to have a false positive rate of 60 in 1200 observations, and never gives a negative result if a person really has the disease, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?</p>
<p>If the raw numbers are not available, the problem can be phrased in such terms as “about 1 case in 1000” and “about 5 false positives in 100 cases.”</p>
<p>One may obtain an answer as follows:</p>
<ol type="1">
<li><p>Construct bucket A with 999 white beads and 1 black bead, and bucket B with 95 green beads and 5 red beads. A more complete problem that also discusses false negatives would need a third bucket.</p></li>
<li><p>Pick a bead from bucket A. If black, record “T,” replace the bead, and end the trial. If white, continue to step 3.</p></li>
<li><p>If a white bead is drawn from bucket A, select a bead from bucket B. If red, record “F” and replace the bead, and if green record “N” and replace the bead.</p></li>
<li><p>Repeat steps 2-4 perhaps 10,000 times, and in the results count the proportion of “T”s to (“T”s plus “F”s) ignoring the “N”s).</p>
<p>Of course 10,000 draws would be tedious, but even after a few hundred draws a person would be likely to draw the correct conclusion that the proportion of “T”s to (“T”s plus “F”s) would be small. And it is easy with a computer to do 10,000 trials very quickly.</p>
<p>Note that the respondents in the Cascells <em>et al.</em> study were not naive; the medical staff members were supposed to understand statistics. Yet most doctors and other personnel offered wrong answers. If simulation can do better than the standard deductive method, then simulation would seem to be the method of choice. And only one piece of training for simulation is required: Teach the habit of saying “I’ll simulate it” and then actually doing so.</p></li>
</ol>
</section>
</section>
<section id="fundamental-problems-in-statistical-practice" class="level2" data-number="31.3">
<h2 data-number="31.3" class="anchored" data-anchor-id="fundamental-problems-in-statistical-practice"><span class="header-section-number">31.3</span> Fundamental problems in statistical practice</h2>
<p>Box and Tiao <span class="citation" data-cites="box1992bayesian">(<a href="references.html#ref-box1992bayesian" role="doc-biblioref">1992</a>)</span> begin their classic exposition of Bayesian statistics with the analysis of a famous problem first published by Fisher <span class="citation" data-cites="fisher1959statistical">(<a href="references.html#ref-fisher1959statistical" role="doc-biblioref">1959, 18</a>)</span>.</p>
<blockquote class="blockquote">
<p>…there are mice of two colors, black and brown. The black mice are of two genetic kinds, homozygotes (<em>BB</em>) and heterozygotes (<em>Bb</em>), and the brown mice are of one kind (<em>bb</em>). It is known from established genetic theory that the probabilities associated with offspring from various matings are as listed in <a href="#tbl-mice-genetics" class="quarto-xref">Table <span>31.1</span></a>.</p>
</blockquote>
<p>(See <span class="citation" data-cites="box1992bayesian">(<a href="references.html#ref-box1992bayesian" role="doc-biblioref">Box and Tiao 1992, 12–14</a>)</span>).</p>
<div id="tbl-mice-genetics" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-mice-genetics-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 31.1: Probabilities for Genetic Character of Mice Offspring
</figcaption>
<div aria-describedby="tbl-mice-genetics-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<thead>
<tr class="header">
<th></th>
<th style="text-align: left;">BB (black)</th>
<th style="text-align: left;">Bb (black)</th>
<th>bb (brown)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>BB mated with bb</td>
<td style="text-align: left;">0</td>
<td style="text-align: left;">1</td>
<td>0</td>