Rearranging notes
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<chapter xml:id="ch-notes">
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<chapter xml:id="ch-notes">
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<title>Class Notes</title>
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<title>Class Notes</title>
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<xi:include href="./notes/week01.ptx" />
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<xi:include href="./notes/1-13.ptx" />
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<xi:include href="./notes/1-15.ptx" />
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</chapter>
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</chapter>
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<chapter xml:id="quizzes">
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<chapter xml:id="quizzes">
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@@ -1,11 +1,11 @@
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<?xml version="1.0" encoding="UTF-8"?>
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<?xml version="1.0" encoding="UTF-8"?>
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<section xml:id="notes-week-01">
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<section xml:id="notes-01-13">
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<title>Week 1</title>
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<title>Tuesday, Jan 13</title>
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<introduction>
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<introduction>
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<p>
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<p>
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This is an outline of the topics we covered in the first week of class.
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This is an outline of the topics we covered in class.
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These notes are <em>not</em> a substitute for your own note-taking.
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These notes are <em>not</em> a substitute for your own note-taking.
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I highly recommend that you take your own notes during class.
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I highly recommend that you take your own notes during class.
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If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
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If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
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@@ -13,10 +13,7 @@
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</introduction>
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</introduction>
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<subsection>
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<subsection xml:id="subsec-Sets">
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<title>Tuesday 1/13</title>
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<subsubsection xml:id="subsubsec-Sets">
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<title>Sec 1.1: Sets</title>
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<title>Sec 1.1: Sets</title>
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<p>
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<p>
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@@ -223,9 +220,9 @@
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</figure>
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</figure>
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</sidebyside>
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</sidebyside>
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</sbsgroup>
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</sbsgroup>
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</subsubsection>
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</subsection>
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<subsubsection xml:id="subsubsec-Probability">
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<subsection xml:id="subsec-Probability">
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<title>Sec 1.2: Probability</title>
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<title>Sec 1.2: Probability</title>
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<p>
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<p>
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@@ -516,280 +513,5 @@
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</p>
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</p>
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</statement>
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</statement>
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</example>
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</example>
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</subsubsection>
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</subsection>
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</subsection>
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</section>
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<subsection>
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<title>Thursday 1/15</title>
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<subsubsection xml:id="subsubsec-Conditional-Probability">
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<title>Conditional Probability</title>
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<p>
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Question: How does evidence (e.g., knowledge of one event occurring) change our knowledge of probabilities for other events?
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</p>
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<example>
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<statement>
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<p>
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Roll a fair D6 two times.
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Let <m>A = \{\text{sum } \geq 10\}</m> and <m>B = \{\text{first roll is } 6\}</m>.
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<m>A</m> feels more likely if we already know <m>B</m> has occurred.
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</p>
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</statement>
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</example>
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<definition xml:id="def-conditional-probability">
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<statement>
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<p>
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The <term>conditional probability</term> of <m>A</m> given <m>B</m> is: ,
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<md>
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<mrow> \Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} </mrow>
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</md>
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</p>
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<figure xml:id="fig-conditional-probability">
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<caption><m>\Pr(A \mid B)</m> tells the proportion of <m>B</m> which is overlapped by <m>A</m>.</caption>
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<image width="50%">
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<description>
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<p>
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Two overlapping circles representing events <m>A</m> and <m>B</m> sit inside a rectangle representing the sample space <m>\Omega</m>.
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The circle labeled <m>B</m> is shaded.
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The portion of that circle which is overlapped by the <m>A</m> circle is also filled in with slanted lines.
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(180:1.75cm) circle (2.5cm)}
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\def\secondcircle{(0:1.75cm) circle (2.5cm)}
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\fill [gray!30] \secondcircle;
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\begin{scope}
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\clip \firstcircle;
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\clip \secondcircle;
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\fill [pattern=north east lines] \firstcircle;
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\end{scope}
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\draw \firstcircle node[text=black] {$A$};
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\draw \secondcircle node[text=black] {$B$};
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\draw (-5, -3) rectangle (5, 3) node [text=black,right] {$\Omega$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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</statement>
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</definition>
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<example xml:id="example-rolls-conditional">
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<statement>
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<p>
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Continuing from the previous example, <m>|\Omega| = 36</m>.
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<md>
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<mrow> A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} </mrow>
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<mrow> B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
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<mrow> A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\} </mrow>
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</md>
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So <m>\Pr(A) = \frac{6}{36}, \Pr(B) = \frac{6}{36}, \text{and } \Pr(A\cap B) = \frac{3}{36}</m>.
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Then:
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<md>
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<mrow> \Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}. </mrow>
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</md>
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Notice that <m>\Pr(A\mid B)</m> is significantly larger than <m>\Pr(A)</m>.
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</p>
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</statement>
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</example>
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</subsubsection>
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<subsubsection xml:id="subsubsec-Diagnostic-Testing">
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<title>Diagnostic Testing</title>
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<p>
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Setup: A patient takes a diagnostic test.
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Let <m>P</m> be the event that they test positive.
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Let <m>D</m> be the event that they have the disease.
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</p>
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<definition xml:id="def-sensitivity-specificity">
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<statement>
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<p>
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The <term>sensitivity</term> of a diagnostic test is <m>\Pr(P \mid D)</m>.
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The <term>specificity</term> of a diagnostic test is <m>\Pr(P^c \mid D^c)</m>.
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</p>
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</statement>
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</definition>
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<p>
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But, what the patient really wants to know is <m>\Pr(D \mid P)</m>.
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</p>
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<example>
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<statement>
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<p>
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A disease has a prevalence of 1%.
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A test has sensitivity of 90% and specificity of 91%.
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For a patient who gets a positive test result, what is the probability that they have the disease?
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</p>
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<p>
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<md>
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<mrow> \text{A) } 9/10 \amp \amp \text{B) } 8/10 \amp \amp \text{C) } 1/10 \amp \amp \text{D) } 1/100 </mrow>
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</md>
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</p>
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</statement>
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<answer>
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<p>
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C!
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</p>
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</answer>
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</example>
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<theorem xml:id="thm-Bayes-v1">
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<title>Bayes' Theorem (v1)</title>
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<statement>
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<p>
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For events <m>A, B</m> with nonzero probability:
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<md>
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<mrow> \Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)} </mrow>
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</md>
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</p>
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</statement>
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</theorem>
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<p>
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For example, if a patient sees a positive diagnostic test result, they might try to calculate:
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<md>
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<mrow> \Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)} </mrow>
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</md>
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<m>\Pr(P \mid D)</m> is the sensitivity. <m>\Pr(D)</m> could be the prevalence. We don't have direct access to <m>\Pr(P)</m>.
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</p>
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<p>
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Observation: <m>\Omega = D \cup D^c</m>, so <m>P = (P\cap D) \cup (P\cap D^c)</m>.
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</p>
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<figure xml:id="fig-P-breakdown">
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<caption></caption>
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<image width="50%">
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<description>
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<p>
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</p>
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</description>
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<latex-image>
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\begin{tikzpicture}
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\def\firstcircle{(0, 0) circle (2)}
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\def\leftside{(-3, -3) rectangle (-0.5, 3)}
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\def\rightside{(-0.5, -3) rectangle (4, 3)}
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\begin{scope}
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\clip\leftside;
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\fill [gray!50] \firstcircle;
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\end{scope}
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\begin{scope}
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\clip\rightside;
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\fill [pattern=north east lines] \firstcircle;
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\end{scope}
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\draw (0, 0) circle (2);
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\node at (2.5, 0) {$P$};
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\draw (-0.5, 3) to (-0.5, -3);
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\node at (-1.5, -3.5) {$D$};
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\node at (1.5, -3.5) {$D^c$};
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\draw (-3, -3) rectangle (4, 3) node [text=black,right] {$\Omega$};
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\end{tikzpicture}
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</latex-image>
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</image>
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</figure>
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<p>
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Observation 2:
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<md>
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<mrow> \Pr(P \mid D) \amp \frac{\Pr(P\cap D)}{\Pr(D)} \amp \amp \Rightarrow \amp \Pr(P \cap D) \amp = \Pr(P\mid D)\Pr(D) </mrow>
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<mrow> \Pr(P \mid D^c) \amp \frac{\Pr(P\cap D^c)}{\Pr(D^c)} \amp \amp \Rightarrow \amp \Pr(P \cap D^c) \amp = \Pr(P\mid D^c)\Pr(D^c) </mrow>
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</md>
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So:
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<md>
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<mrow> \Pr(P) = \Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c) </mrow>
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</md>
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</p>
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<theorem xml:id="thm-Bayes-v2">
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<title>Bayes' Theorem (v2)</title>
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<statement>
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<p>
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<md>
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<mrow> \Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A \mid B)\Pr(B) + \Pr(A \mid B^c)\Pr(B^c)} </mrow>
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</md>
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</p>
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</statement>
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</theorem>
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<example>
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<statement>
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<p>
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Continuing from the previous example:
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<md>
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<mrow> \Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)} </mrow>
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<mrow> \amp = \frac{(0.9)(0.01)}{(0.9)(0.01) + (1 - 0.91)(1 - 0.01)} </mrow>
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<mrow> \amp \approx 0.092 </mrow>
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</md>
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What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease?
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<md>
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<mrow> \Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + \Pr(P^c\mid D)\Pr(D)} </mrow>
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<mrow> \amp = \frac{(0.91)(1 - 0.01)}{(0.91)(1 - 0.01) + (1 - 0.9)(0.01)} </mrow>
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<mrow> \amp \approx 0.999 </mrow>
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</md>
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</p>
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</statement>
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</example>
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</subsubsection>
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<subsubsection xml:id="subsubsec-Independent-Events">
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<title>Independent Events</title>
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<p>
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Question: <m>\Pr(A \mid B)</m> is supposed to capture how information about <m>B</m> affects the probability of <m>A</m>.
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What if it doesn't?
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</p>
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<definition xml:id="def-independent-events">
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<statement>
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<p>
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Events <m>A, B</m> are <term>independent</term> if <m>\Pr(A \mid B) = \Pr(A)</m>.
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</p>
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</statement>
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</definition>
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<p>
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Observation: If <m>A, B</m> have nonzero probability and are independent, then:
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<md>
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<mrow> \Pr(A\mid B) \amp = \Pr(A) </mrow>
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<mrow> \frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) </mrow>
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<mrow> \Pr(A\cap B) \amp = \Pr(A)\Pr(B) </mrow>
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</md>
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We can take this last equation as a definition of independence.
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</p>
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<example xml:id="example-rolls-independent">
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<statement>
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<p>
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Continuing <xref ref="example-rolls-conditional"/>, recall <m>A = \{\text{sum} \geq 10\}</m> and <m>B = \{\text{1st roll is } 6\}</m>.
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We found that <m>\Pr(A \mid B) \neq \Pr(A)</m>, so <m>A</m> and <m>B</m> are not independent.
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</p>
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<p>
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Now consider the event <m>C = \{\text{sum} = 7\}</m>.
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We have:
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<md>
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<mrow> C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} </mrow>
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<mrow> \Pr(C) \amp = \frac{6}{36} = \frac{1}{6} </mrow>
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<mrow> B\cap C \amp = \{(6, 1)\} </mrow>
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<mrow> \Pr(B\cap C) \amp = \frac{1}{36} </mrow>
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<mrow> \text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C) </mrow>
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</md>
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Therefore events <m>B, C</m> are independent.
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</p>
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</statement>
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</example>
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</subsubsection>
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</subsection>
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</section>
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@@ -0,0 +1,286 @@
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<?xml version="1.0" encoding="UTF-8"?>
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||||||
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||||||
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<section xml:id="notes-01-15">
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||||||
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<title>Thursday 1/15</title>
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||||||
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||||||
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<introduction>
|
||||||
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<p>
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||||||
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This is an outline of the topics we covered in class.
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||||||
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These notes are <em>not</em> a substitute for your own note-taking.
|
||||||
|
I highly recommend that you take your own notes during class.
|
||||||
|
If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
|
||||||
|
</p>
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||||||
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</introduction>
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||||||
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||||||
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||||||
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<subsection xml:id="subsec-Conditional-Probability">
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||||||
|
<title>Sec 1.3: Conditional Probability</title>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Question: How does evidence (e.g., knowledge of one event occurring) change our knowledge of probabilities for other events?
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<example>
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
Roll a fair D6 two times.
|
||||||
|
Let <m>A = \{\text{sum } \geq 10\}</m> and <m>B = \{\text{first roll is } 6\}</m>.
|
||||||
|
<m>A</m> feels more likely if we already know <m>B</m> has occurred.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</example>
|
||||||
|
|
||||||
|
<definition xml:id="def-conditional-probability">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
The <term>conditional probability</term> of <m>A</m> given <m>B</m> is: ,
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<figure xml:id="fig-conditional-probability">
|
||||||
|
<caption><m>\Pr(A \mid B)</m> tells the proportion of <m>B</m> which is overlapped by <m>A</m>.</caption>
|
||||||
|
<image width="50%">
|
||||||
|
<description>
|
||||||
|
<p>
|
||||||
|
Two overlapping circles representing events <m>A</m> and <m>B</m> sit inside a rectangle representing the sample space <m>\Omega</m>.
|
||||||
|
The circle labeled <m>B</m> is shaded.
|
||||||
|
The portion of that circle which is overlapped by the <m>A</m> circle is also filled in with slanted lines.
|
||||||
|
</p>
|
||||||
|
</description>
|
||||||
|
<latex-image>
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\def\firstcircle{(180:1.75cm) circle (2.5cm)}
|
||||||
|
\def\secondcircle{(0:1.75cm) circle (2.5cm)}
|
||||||
|
\fill [gray!30] \secondcircle;
|
||||||
|
\begin{scope}
|
||||||
|
\clip \firstcircle;
|
||||||
|
\clip \secondcircle;
|
||||||
|
\fill [pattern=north east lines] \firstcircle;
|
||||||
|
\end{scope}
|
||||||
|
\draw \firstcircle node[text=black] {$A$};
|
||||||
|
\draw \secondcircle node[text=black] {$B$};
|
||||||
|
\draw (-5, -3) rectangle (5, 3) node [text=black,right] {$\Omega$};
|
||||||
|
\end{tikzpicture}
|
||||||
|
</latex-image>
|
||||||
|
</image>
|
||||||
|
</figure>
|
||||||
|
</statement>
|
||||||
|
</definition>
|
||||||
|
|
||||||
|
<example xml:id="example-rolls-conditional">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
Continuing from the previous example, <m>|\Omega| = 36</m>.
|
||||||
|
<md>
|
||||||
|
<mrow> A \amp = \{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} </mrow>
|
||||||
|
<mrow> B \amp = \{(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} </mrow>
|
||||||
|
<mrow> A \cap B \amp = \{(6, 4), (6, 5), (6, 6)\} </mrow>
|
||||||
|
</md>
|
||||||
|
So <m>\Pr(A) = \frac{6}{36}, \Pr(B) = \frac{6}{36}, \text{and } \Pr(A\cap B) = \frac{3}{36}</m>.
|
||||||
|
Then:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(A \mid B) = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{3/36}{6/36} = \frac{3}{6} = \frac{1}{2}. </mrow>
|
||||||
|
</md>
|
||||||
|
Notice that <m>\Pr(A\mid B)</m> is significantly larger than <m>\Pr(A)</m>.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</example>
|
||||||
|
</subsection>
|
||||||
|
|
||||||
|
|
||||||
|
<subsection xml:id="subsec-Diagnostic-Testing">
|
||||||
|
<title>Diagnostic Testing</title>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Setup: A patient takes a diagnostic test.
|
||||||
|
Let <m>P</m> be the event that they test positive.
|
||||||
|
Let <m>D</m> be the event that they have the disease.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<definition xml:id="def-sensitivity-specificity">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
The <term>sensitivity</term> of a diagnostic test is <m>\Pr(P \mid D)</m>.
|
||||||
|
The <term>specificity</term> of a diagnostic test is <m>\Pr(P^c \mid D^c)</m>.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</definition>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
But, what the patient really wants to know is <m>\Pr(D \mid P)</m>.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<example>
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
A disease has a prevalence of 1%.
|
||||||
|
A test has sensitivity of 90% and specificity of 91%.
|
||||||
|
For a patient who gets a positive test result, what is the probability that they have the disease?
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
<md>
|
||||||
|
<mrow> \text{A) } 9/10 \amp \amp \text{B) } 8/10 \amp \amp \text{C) } 1/10 \amp \amp \text{D) } 1/100 </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
|
||||||
|
<answer>
|
||||||
|
<p>
|
||||||
|
C!
|
||||||
|
</p>
|
||||||
|
</answer>
|
||||||
|
</example>
|
||||||
|
|
||||||
|
<theorem xml:id="thm-Bayes-v1">
|
||||||
|
<title>Bayes' Theorem (v1)</title>
|
||||||
|
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
For events <m>A, B</m> with nonzero probability:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A)} </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</theorem>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
For example, if a patient sees a positive diagnostic test result, they might try to calculate:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(D \mid P) = \frac{\Pr(P \mid D)\Pr(D)}{\Pr(P)} </mrow>
|
||||||
|
</md>
|
||||||
|
<m>\Pr(P \mid D)</m> is the sensitivity. <m>\Pr(D)</m> could be the prevalence. We don't have direct access to <m>\Pr(P)</m>.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Observation: <m>\Omega = D \cup D^c</m>, so <m>P = (P\cap D) \cup (P\cap D^c)</m>.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<figure xml:id="fig-P-breakdown">
|
||||||
|
<caption></caption>
|
||||||
|
<image width="50%">
|
||||||
|
<description>
|
||||||
|
<p>
|
||||||
|
</p>
|
||||||
|
</description>
|
||||||
|
<latex-image>
|
||||||
|
\begin{tikzpicture}
|
||||||
|
\def\firstcircle{(0, 0) circle (2)}
|
||||||
|
\def\leftside{(-3, -3) rectangle (-0.5, 3)}
|
||||||
|
\def\rightside{(-0.5, -3) rectangle (4, 3)}
|
||||||
|
\begin{scope}
|
||||||
|
\clip\leftside;
|
||||||
|
\fill [gray!50] \firstcircle;
|
||||||
|
\end{scope}
|
||||||
|
\begin{scope}
|
||||||
|
\clip\rightside;
|
||||||
|
\fill [pattern=north east lines] \firstcircle;
|
||||||
|
\end{scope}
|
||||||
|
\draw (0, 0) circle (2);
|
||||||
|
\node at (2.5, 0) {$P$};
|
||||||
|
\draw (-0.5, 3) to (-0.5, -3);
|
||||||
|
\node at (-1.5, -3.5) {$D$};
|
||||||
|
\node at (1.5, -3.5) {$D^c$};
|
||||||
|
\draw (-3, -3) rectangle (4, 3) node [text=black,right] {$\Omega$};
|
||||||
|
\end{tikzpicture}
|
||||||
|
</latex-image>
|
||||||
|
</image>
|
||||||
|
</figure>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Observation 2:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(P \mid D) \amp \frac{\Pr(P\cap D)}{\Pr(D)} \amp \amp \Rightarrow \amp \Pr(P \cap D) \amp = \Pr(P\mid D)\Pr(D) </mrow>
|
||||||
|
<mrow> \Pr(P \mid D^c) \amp \frac{\Pr(P\cap D^c)}{\Pr(D^c)} \amp \amp \Rightarrow \amp \Pr(P \cap D^c) \amp = \Pr(P\mid D^c)\Pr(D^c) </mrow>
|
||||||
|
</md>
|
||||||
|
So:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(P) = \Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c) </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<theorem xml:id="thm-Bayes-v2">
|
||||||
|
<title>Bayes' Theorem (v2)</title>
|
||||||
|
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(B \mid A) = \frac{\Pr(A \mid B)\Pr(B)}{\Pr(A \mid B)\Pr(B) + \Pr(A \mid B^c)\Pr(B^c)} </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</theorem>
|
||||||
|
|
||||||
|
<example>
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
Continuing from the previous example:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + \Pr(P\mid D^c)\Pr(D^c)} </mrow>
|
||||||
|
<mrow> \amp = \frac{(0.9)(0.01)}{(0.9)(0.01) + (1 - 0.91)(1 - 0.01)} </mrow>
|
||||||
|
<mrow> \amp \approx 0.092 </mrow>
|
||||||
|
</md>
|
||||||
|
What if the patient got a negative test result instead? In that case, what is the probaiblity they do not have the disease?
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + \Pr(P^c\mid D)\Pr(D)} </mrow>
|
||||||
|
<mrow> \amp = \frac{(0.91)(1 - 0.01)}{(0.91)(1 - 0.01) + (1 - 0.9)(0.01)} </mrow>
|
||||||
|
<mrow> \amp \approx 0.999 </mrow>
|
||||||
|
</md>
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</example>
|
||||||
|
</subsection>
|
||||||
|
|
||||||
|
|
||||||
|
<subsection xml:id="subsec-Independent-Events">
|
||||||
|
<title>Sec 1.4: Independent Events</title>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Question: <m>\Pr(A \mid B)</m> is supposed to capture how information about <m>B</m> affects the probability of <m>A</m>.
|
||||||
|
What if it doesn't?
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<definition xml:id="def-independent-events">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
Events <m>A, B</m> are <term>independent</term> if <m>\Pr(A \mid B) = \Pr(A)</m>.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</definition>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Observation: If <m>A, B</m> have nonzero probability and are independent, then:
|
||||||
|
<md>
|
||||||
|
<mrow> \Pr(A\mid B) \amp = \Pr(A) </mrow>
|
||||||
|
<mrow> \frac{\Pr(A\cap B)}{\Pr(B)} \amp = \Pr(A) </mrow>
|
||||||
|
<mrow> \Pr(A\cap B) \amp = \Pr(A)\Pr(B) </mrow>
|
||||||
|
</md>
|
||||||
|
We can take this last equation as a definition of independence.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<example xml:id="example-rolls-independent">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
Continuing <xref ref="example-rolls-conditional"/>, recall <m>A = \{\text{sum} \geq 10\}</m> and <m>B = \{\text{1st roll is } 6\}</m>.
|
||||||
|
We found that <m>\Pr(A \mid B) \neq \Pr(A)</m>, so <m>A</m> and <m>B</m> are not independent.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
Now consider the event <m>C = \{\text{sum} = 7\}</m>.
|
||||||
|
We have:
|
||||||
|
<md>
|
||||||
|
<mrow> C \amp = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} </mrow>
|
||||||
|
<mrow> \Pr(C) \amp = \frac{6}{36} = \frac{1}{6} </mrow>
|
||||||
|
<mrow> B\cap C \amp = \{(6, 1)\} </mrow>
|
||||||
|
<mrow> \Pr(B\cap C) \amp = \frac{1}{36} </mrow>
|
||||||
|
<mrow> \text{therefore: } \Pr(C\mid B) \amp = \frac{\Pr(C\cap B)}{\Pr(B)} = \frac{1/36}{1/6} = \frac{1}{6} = \Pr(C) </mrow>
|
||||||
|
</md>
|
||||||
|
Therefore events <m>B, C</m> are independent.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</example>
|
||||||
|
</subsection>
|
||||||
|
</section>
|
||||||
@@ -0,0 +1,41 @@
|
|||||||
|
<?xml version="1.0" encoding="UTF-8"?>
|
||||||
|
|
||||||
|
<section xml:id="notes-01-20">
|
||||||
|
<title>Tuesday 1/20</title>
|
||||||
|
|
||||||
|
<introduction>
|
||||||
|
<p>
|
||||||
|
This is an outline of the topics we covered in class.
|
||||||
|
These notes are <em>not</em> a substitute for your own note-taking.
|
||||||
|
I highly recommend that you take your own notes during class.
|
||||||
|
If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.
|
||||||
|
</p>
|
||||||
|
</introduction>
|
||||||
|
|
||||||
|
|
||||||
|
<subsection xml:id="subsec-Discrete-Random-Variables">
|
||||||
|
<title>Sec 2.1: Random Variables</title>
|
||||||
|
|
||||||
|
<definition xml:id="def-RV">
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
A <term>random variable</term> is a function <m>X \colon \Omega \to \R</m>.
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</definition>
|
||||||
|
|
||||||
|
<p>
|
||||||
|
The idea is that <m>X</m> is a variable representing a real number value which depends on the outcome of an experiment.
|
||||||
|
</p>
|
||||||
|
|
||||||
|
<example>
|
||||||
|
<statement>
|
||||||
|
<p>
|
||||||
|
An experiment consists of planting 50 seeds in a garden, then growing them for 3 months.
|
||||||
|
Let <m>H_i</m> be the height of plant <m>i</m>.
|
||||||
|
|
||||||
|
</p>
|
||||||
|
</statement>
|
||||||
|
</example>
|
||||||
|
</subsection>
|
||||||
|
</section>
|
||||||
@@ -1,28 +0,0 @@
|
|||||||
<?xml version="1.0" encoding="UTF-8"?>
|
|
||||||
|
|
||||||
<section xml:id="notes-week-02">
|
|
||||||
<title>Week 2</title>
|
|
||||||
|
|
||||||
<subsection>
|
|
||||||
<title>Monday</title>
|
|
||||||
|
|
||||||
<p>
|
|
||||||
</p>
|
|
||||||
</subsection>
|
|
||||||
|
|
||||||
|
|
||||||
<subsection>
|
|
||||||
<title>Wednesday</title>
|
|
||||||
|
|
||||||
<p>
|
|
||||||
</p>
|
|
||||||
</subsection>
|
|
||||||
|
|
||||||
|
|
||||||
<subsection>
|
|
||||||
<title>Friday</title>
|
|
||||||
|
|
||||||
<p>
|
|
||||||
</p>
|
|
||||||
</subsection>
|
|
||||||
</section>
|
|
||||||
Reference in New Issue
Block a user