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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<div class="toc-title-box"><a href="sec-Likelihood.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Likelihood</span></a></div>
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<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Central Limit Theorem</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
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|
||
</ul>
|
||
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<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
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<ul class="structural toc-item-list">
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<li class="toc-item toc-appendix">
|
||
<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="title">Colophon</span></a></div></li>
|
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|
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Likelihood"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Likelihood</span>
|
||
</h2>
|
||
<div class="para" id="sec-Likelihood-2">So far, we’ve been concerned with probability theory. Starting with a probability distribution and some parameter values, we’ve tried to answer questions like: What’s the probability of seeing certain experimental results? Statistics is concerned with going in the other direction: Upon seeing the experimental results, can we determine the type of underlying probability distribution? Can we determine its parameters?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-estimator"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="def-estimator-1-1">An <dfn class="terminology">estimator</dfn> is a value of a parameter computed from a sample of data.<div class="autopermalink" data-description="Paragraph"><a href="#def-estimator-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 4.1.1"><a href="#def-estimator" title="Copy heading and permalink for Definition 4.1.1" aria-label="Copy heading and permalink for Definition 4.1.1">🔗</a></div></article><article class="example example-like" id="sec-Likelihood-4"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="sec-Likelihood-4-1">Suppose we find a coin on the street and don’t know whether or not it’s fair. We want to know the probability <span class="process-math">\(p\)</span> of the coin coming up heads. We might, for example, flip the coin <span class="process-math">\(n\)</span> times and count the number <span class="process-math">\(k\)</span> of heads. Then, we’ll estimate <span class="process-math">\(p = \frac{k}{n}\text{.}\)</span> We’ll refer to this as a <dfn class="terminology">common sense</dfn> estimator. (Other distributions and parameter types will have different notions of "common sense".)<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-4-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.1.2"><a href="#sec-Likelihood-4" title="Copy heading and permalink for Example 4.1.2" aria-label="Copy heading and permalink for Example 4.1.2">🔗</a></div></article><div class="para" id="sec-Likelihood-5">An estimator is, itself, a random variable: it produces a numerical value based on the results of an experiment. We’ll use notation like <span class="process-math">\(\est{p}\)</span> for a random variable which is an estimator for a parameter <span class="process-math">\(p\text{.}\)</span> (Similarly, <span class="process-math">\(\est{\lambda}\)</span> would denote an estimator for a parameter called <span class="process-math">\(\lambda\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-unbiased"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="def-unbiased-1-1">An estimator <span class="process-math">\(\est{p}\)</span> is called <dfn class="terminology">unbiased</dfn> if <span class="process-math">\(\E(\est{p}) = p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-unbiased-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 4.1.3"><a href="#def-unbiased" title="Copy heading and permalink for Definition 4.1.3" aria-label="Copy heading and permalink for Definition 4.1.3">🔗</a></div></article><article class="example example-like" id="sec-Likelihood-7"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="sec-Likelihood-7-1-1">
|
||
<div class="para">Suppose we have a coin with parameter <span class="process-math">\(p\text{,}\)</span> which we’ll flip <span class="process-math">\(n\)</span> times and count the number <span class="process-math">\(k\)</span> of heads. We use the unbiased estimator <span class="process-math">\(\est{p} = \frac{k}{n}\text{.}\)</span> In this case, notice that <span class="process-math">\(k \sim \Bin(n, p)\text{,}\)</span> so we know <span class="process-math">\(\E(k) = np\text{,}\)</span> although we don’t know the value of <span class="process-math">\(p\text{.}\)</span> (We probably do know the value of <span class="process-math">\(n\text{;}\)</span> after all, we’re flipping the coin!) Now:</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-7-1-1-9">
|
||
\begin{gather*}
|
||
\E(\est{p}) = \E\left(\frac{k}{n}\right) = \frac{1}{n} \cdot \E(k) = \frac{1}{n} \cdot np = p.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">It’s worth pausing for a moment to be appropriately impressed with ourselves. We still don’t know the true value of <span class="process-math">\(p\text{.}\)</span> But we managed to show that our common sense method of estimating <span class="process-math">\(p\)</span> gives, on average, the correct value.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.1.4"><a href="#sec-Likelihood-7" title="Copy heading and permalink for Example 4.1.4" aria-label="Copy heading and permalink for Example 4.1.4">🔗</a></div></article><div class="para" id="sec-Likelihood-8">We’d like to be able to collect some data and use that data to estimate the values of whatever parameters our distribution has. Perhaps as a starting point, it would be good to identify the single most likely value of a parameter:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-likelihood"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="def-likelihood-1-1">
|
||
<div class="para">Suppose we collect some data to estimate a parameter <span class="process-math">\(p\text{.}\)</span> The <dfn class="terminology">likelihood function</dfn> is:</div>
|
||
<div class="displaymath process-math" id="def-likelihood-1-1-3">
|
||
\begin{gather*}
|
||
\L(p) = \Pr(\text{data} \mid \text{parameter value is } p).
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">The value of <span class="process-math">\(p\)</span> which maximizes the function <span class="process-math">\(\L(p)\)</span> is called the <dfn class="terminology">maximum likelihood estimation</dfn>, or <dfn class="terminology">MLE</dfn>, of <span class="process-math">\(p\text{.}\)</span>
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#def-likelihood-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 4.1.5"><a href="#def-likelihood" title="Copy heading and permalink for Definition 4.1.5" aria-label="Copy heading and permalink for Definition 4.1.5">🔗</a></div></article><article class="example example-like" id="sec-Likelihood-10"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.6</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="sec-Likelihood-10-1-1">
|
||
<div class="para">Suppose we flip a coin 100 times. Let <span class="process-math">\(N\)</span> be the number of heads. Then <span class="process-math">\(N\sim \Bin(100, p)\)</span> for some unknown value of <span class="process-math">\(p\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-10-1-1-4">
|
||
\begin{align*}
|
||
\Pr(N = k) \amp = b(k; 100, p) = {100 \choose k} p^k (1 - p)^{100 - k}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">In this case, <span class="process-math">\(k\)</span> is the data that we collect, and <span class="process-math">\(p\)</span> is the value of the parameter. Suppose we see <span class="process-math">\(52\)</span> heads. Then:</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-10-1-1-8">
|
||
\begin{gather*}
|
||
\L(p) = {100 \choose 52} p^{52} (1 - p)^{48}.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">If we want to know the most likely value of the parameter <span class="process-math">\(p\text{,}\)</span> then we should maximize <span class="process-math">\(\L(p)\)</span> over the interval <span class="process-math">\(0 \leq p \leq 1\text{.}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-10-1-1-12">
|
||
\begin{align*}
|
||
\L'(p) \amp = {100 \choose 52} \left[ 52 p^{51}(1 - p)^{48} + p^{52} 48 (1 - p)^{47}(-1)\right] \\
|
||
\amp = {100 \choose 52} p^{51} (1 - p)^{47} \left[ 52 (1 - p) - 48p \right] \\
|
||
\amp = {100 \choose 52} p^{51} (1 - p)^{47} \left[ 52 - 100 p \right].
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">We can see that <span class="process-math">\(\L'(p) = 0\)</span> when <span class="process-math">\(p = 0, 1, \frac{52}{100}\text{,}\)</span> and the endpoints of the interval we’re maximizing over are <span class="process-math">\(p = 0, 1\text{,}\)</span> so we can build a table:</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="sec-Likelihood-10-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Looking for the Maximum<div class="autopermalink" data-description="Table 4.1.7: Looking for the Maximum"><a href="#sec-Likelihood-10-1-2" title="Copy heading and permalink for Table 4.1.7: Looking for the Maximum" aria-label="Copy heading and permalink for Table 4.1.7: Looking for the Maximum">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(p\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\L(p)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(52/100\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(\gt 0\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0\)</span></td>
|
||
</tr>
|
||
</table></div></figure> <div class="para" id="sec-Likelihood-10-1-3">Note that we don’t need to know the exact value of <span class="process-math">\(\L(52/100)\)</span> in order to see that it’s strictly positive, and therefore the maximum value of <span class="process-math">\(\L(p)\text{.}\)</span> So, the MLE of <span class="process-math">\(p\)</span> is <span class="process-math">\(52/100\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-10-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.1.6"><a href="#sec-Likelihood-10" title="Copy heading and permalink for Example 4.1.6" aria-label="Copy heading and permalink for Example 4.1.6">🔗</a></div></article><div class="para" id="sec-Likelihood-11">We can see in the previous example that the MLE of <span class="process-math">\(p\)</span> is also the common sense estimation of <span class="process-math">\(p\text{.}\)</span> This will be the case in general for the binomial distribution, so we won’t need to redo this work over and over:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-11" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="fact theorem-like" id="fact-MLE-binomial"><h3 class="heading">
|
||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.8</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Binomial MLE.</span>
|
||
</h3>
|
||
<div class="para" id="fact-MLE-binomial-2-1">If we see <span class="process-math">\(k\)</span> heads in <span class="process-math">\(n\)</span> coin flips, then the MLE of the bias <span class="process-math">\(p\)</span> is <span class="process-math">\(\frac{k}{n}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#fact-MLE-binomial-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Fact 4.1.8: Binomial MLE"><a href="#fact-MLE-binomial" title="Copy heading and permalink for Fact 4.1.8: Binomial MLE" aria-label="Copy heading and permalink for Fact 4.1.8: Binomial MLE">🔗</a></div></article><div class="para" id="sec-Likelihood-13">As we remember from Calculus 1, maximizing a continuous function works slightly differently over a closed interval (like the previous example) or an open interval (like the next example).<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-13" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="sec-Likelihood-14"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.9</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="sec-Likelihood-14-1-1">
|
||
<div class="para">Suppose a radioactive material emits particles as it decays. Let <span class="process-math">\(N\)</span> count the particles emitted. Then <span class="process-math">\(N \sim \Poiss(\lambda)\)</span> for some unknown rate parameter <span class="process-math">\(\lambda\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-14-1-1-4">
|
||
\begin{gather*}
|
||
\Pr(N = k) = p(k; \lambda) = \frac{\lambda^k}{k!} e^{-\lambda}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Suppose we observe a sample of material for 1 hour and count 8 particles emitted. Then:</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-14-1-1-5">
|
||
\begin{gather*}
|
||
\L(\lambda) = \frac{\lambda^8}{8!} e^{-\lambda}
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">To maximize <span class="process-math">\(\L(\lambda)\)</span> over the interval <span class="process-math">\(0 \lt \lambda \lt \infty\text{,}\)</span> we start by finding critical points.</div>
|
||
<div class="displaymath process-math" id="sec-Likelihood-14-1-1-8">
|
||
\begin{align*}
|
||
\L'(\lambda) \amp = \frac{1}{8!} \left[ 8 \lambda^7 e^{-\lambda} + \lambda^8 e^{-\lambda} (-1)\right] \\
|
||
\amp = \frac{1}{8!} \lambda^7 e^{-\lambda} \left[ 8 - e^{-\lambda} \right]
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">The only critical point is <span class="process-math">\(\lambda = 8\text{,}\)</span> but we have not justified that this is the location of a global maximum. A critical point is only a potential location of a local minimum or maximum. But with a bit more justification: observe that <span class="process-math">\(\L'\)</span> is positive on the interval <span class="process-math">\((0, 8)\)</span> and negative on the interval <span class="process-math">\((8, \infty)\text{.}\)</span> Therefore the function <span class="process-math">\(\L\)</span> increases on <span class="process-math">\((0, 8)\)</span> and decreases on <span class="process-math">\((8, \infty)\text{.}\)</span> So it must reach its maximum at <span class="process-math">\(\lambda = 8\text{,}\)</span> which is therefore the MLE.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-14-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.1.9"><a href="#sec-Likelihood-14" title="Copy heading and permalink for Example 4.1.9" aria-label="Copy heading and permalink for Example 4.1.9">🔗</a></div></article><div class="para" id="sec-Likelihood-15">As with the binomial distribution, the calculation will be essentially the same regardless of the specific number of particles observed, so:<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-15" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="fact theorem-like" id="fact-MLE-Poisson"><h3 class="heading">
|
||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.10</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Poisson MLE.</span>
|
||
</h3>
|
||
<div class="para" id="fact-MLE-Poisson-2-1">If we observe a Poisson process and see <span class="process-math">\(k\)</span> events occur, then the MLE of the rate parameter <span class="process-math">\(\lambda\)</span> is <span class="process-math">\(k\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#fact-MLE-Poisson-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Fact 4.1.10: Poisson MLE"><a href="#fact-MLE-Poisson" title="Copy heading and permalink for Fact 4.1.10: Poisson MLE" aria-label="Copy heading and permalink for Fact 4.1.10: Poisson MLE">🔗</a></div></article><div class="para" id="sec-Likelihood-17">The situation for continuous random variables is similar, but slightly different. To build the likelihood function, we should use the pdf of the continuous random variable. So the likelihood function will give the probabiliy density given the data collected, rather than the probability itself.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-17" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="example-exponential-MLE"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.11</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-exponential-MLE-1-1">
|
||
<div class="para">Suppose we observe a cell and measure the time <span class="process-math">\(T\)</span> until a toxin molecule leaves the cell. Then <span class="process-math">\(T \sim \Exp(\lambda)\)</span> for some unknown rate parameter <span class="process-math">\(\lambda\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="example-exponential-MLE-1-1-4">
|
||
\begin{gather*}
|
||
f(t) = \lambda e^{-\lambda t}.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Suppose we observe a molecule leave the cell at <span class="process-math">\(t = 0.3\)</span> minutes. Then:</div>
|
||
<div class="displaymath process-math" id="example-exponential-MLE-1-1-6">
|
||
\begin{align*}
|
||
L(\lambda) \amp = \lambda e^{-0.3\lambda} \\
|
||
L'(\lambda) \amp = e^{-0.3\lambda} + \lambda e^{-0.3t} (-0.3) \\
|
||
\amp = e^{-0.3\lambda}\left[ 1 - 0.3\lambda\right]
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">The critical point is <span class="process-math">\(\lambda = \frac{1}{0.3} \approx 3.33\text{.}\)</span> Since <span class="process-math">\(\L' \gt 0\)</span> on <span class="process-math">\((0, 3.33)\)</span> and <span class="process-math">\(\L' \lt 0\)</span> on <span class="process-math">\((3.33, \infty)\text{,}\)</span> there is a global maximum at <span class="process-math">\(\lambda \approx 3.33\text{,}\)</span> which is therefore the MLE.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="example-exponential-MLE-1-2">The specific time <span class="process-math">\(0.3\)</span> minutes doesn’t particularly matter in this calculation. Whatever the time <span class="process-math">\(t\text{,}\)</span> essentially the same calculation will result in a MLE of <span class="process-math">\(\lambda = 1/t\text{.}\)</span> But what if we collect multiple pieces of data?<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="example-exponential-MLE-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Time Data<div class="autopermalink" data-description="Table 4.1.12: Time Data"><a href="#example-exponential-MLE-1-3" title="Copy heading and permalink for Table 4.1.12: Time Data" aria-label="Copy heading and permalink for Table 4.1.12: Time Data">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r0 l0 t0 lines">Molecule</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Waiting Time</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Rate Estimation</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3.33\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.8\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1.25\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.5\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2.00\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.6\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1.67\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.9\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1.11\)</span></td>
|
||
</tr>
|
||
</table></div></figure> <div class="para" id="example-exponential-MLE-1-4">How should take all of this data into account in our maximum likelihood estimation? We might consider taking the average of all of the separate rate estimations, which would give <span class="process-math">\(1.87\text{.}\)</span> Is this the most likely? We need some mathematical justification.<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para logical" id="example-exponential-MLE-1-5">
|
||
<div class="para">To account for multiple, independent data points, we should multiply the probability densities for each in the creation of our likelihood function:</div>
|
||
<div class="displaymath process-math" id="example-exponential-MLE-1-5-1">
|
||
\begin{align*}
|
||
\L(\lambda) \amp = \left(\lambda e^{-0.3\lambda}\right)\left(\lambda e^{-0.8\lambda}\right)\left(\lambda e^{-0.5\lambda}\right)\left(\lambda e^{-0.6\lambda}\right)\left(\lambda e^{-0.9\lambda}\right) \\
|
||
\amp = \lambda^5 e^{-0.3\lambda - 0.8\lambda - 0.5\lambda - 0.6\lambda - 0.9\lambda } \\
|
||
\amp = \lambda^5 e^{-3.1\lambda}
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Now we can find the maximum:</div>
|
||
<div class="displaymath process-math" id="example-exponential-MLE-1-5-2">
|
||
\begin{align*}
|
||
\L'(\lambda) \amp = 5 \lambda^4 e^{-3.1\lambda} + \lambda^5 e^{-3.1\lambda} (-3.1) \\
|
||
\amp = 5 \lambda^4 e^{-3.1\lambda} + \lambda^5 e^{-3.1\lambda} (-3.1) \\
|
||
\amp = \lambda^4 e^{-3.1\lambda} \left[5 - 3.1\lambda \right]
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">The only critical point is <span class="process-math">\(5/3.1 \approx 1.61\text{.}\)</span> Since <span class="process-math">\(\L' \gt 0\)</span> on <span class="process-math">\((0, 1.61)\)</span> and <span class="process-math">\(\L' \lt 0\)</span> on <span class="process-math">\((1.61, \infty)\text{,}\)</span> there is a global maximum at <span class="process-math">\(\lambda = 1.61\text{,}\)</span> which is therefore the MLE.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="example-exponential-MLE-1-6">It may seem less clear how to generalize this calculation for other tables of data. Observe that the value <span class="process-math">\(3.1\)</span> is the sum of the five times in the table, so <span class="process-math">\(3.1/5\)</span> is the average time. The MLE turned out to be the reciprocal of the average time (just as the MLE with only one data point was the reciprocal of that one time). Notice that this does <em class="emphasis">not</em> match the guess we made previously of averaging the individual rate estimations for each data point.<div class="autopermalink" data-description="Paragraph"><a href="#example-exponential-MLE-1-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 4.1.11"><a href="#example-exponential-MLE" title="Copy heading and permalink for Example 4.1.11" aria-label="Copy heading and permalink for Example 4.1.11">🔗</a></div></article><article class="fact theorem-like" id="fact-MLE-exponential"><h3 class="heading">
|
||
<span class="type">Fact</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1.13</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exponential MLE.</span>
|
||
</h3>
|
||
<div class="para" id="fact-MLE-exponential-2-1">If we observe a Poisson process and see events occur after waiting times <span class="process-math">\(t_1, t_2, \dotsc, t_n\text{,}\)</span> then the MLE of the rate parameter <span class="process-math">\(\lambda\)</span> is <span class="process-math">\(\frac{n}{t_1 + t_2 + \dotsb + t_n}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#fact-MLE-exponential-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Fact 4.1.13: Exponential MLE"><a href="#fact-MLE-exponential" title="Copy heading and permalink for Fact 4.1.13: Exponential MLE" aria-label="Copy heading and permalink for Fact 4.1.13: Exponential MLE">🔗</a></div></article><section class="exercises" id="sec-Likelihood-20"><h3 class="heading hide-type">
|
||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||
</h3>
|
||
<article class="exercise exercise-like" id="sec-Likelihood-20-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Likelihood-20-1-1-1">Suppose a coin has an unknown probability of coming up heads. We perform the experiment in five independent trials, during which it takes 4, 5, 4, 3, and 6 flips to see our first heads in each trial. What is the maximum likelihood estimation for the probability of the coin coming up heads on a flip?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 4.1.1"><a href="#sec-Likelihood-20-1" title="Copy heading and permalink for Exercise 4.1.1" aria-label="Copy heading and permalink for Exercise 4.1.1">🔗</a></div></article><article class="exercise exercise-like" id="sec-Likelihood-20-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Likelihood-20-2-1-1">Suppose a coin has an unknown probability of coming up heads. We perform the experiment in <span class="process-math">\(n\)</span> independent trials, during which it takes <span class="process-math">\(k_1, k_2, \dotsc, k_n\)</span> flips to see our first heads in each trial. Find a "common sense" MLE formula for the geometric distribution.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="solutions"><details id="sec-Likelihood-20-2-2" class="hint solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Hint</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="hint solution-like knowl__content">
|
||
<div class="para" id="sec-Likelihood-20-2-2-1">You <em class="emphasis">could</em> set up a calculation analogous to <a href="sec-Likelihood.html#example-exponential-MLE" class="xref" data-knowl="./knowl/xref/example-exponential-MLE.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 4.1.11">Example 4.1.11</a>. Or, you could consider <a href="sec-Likelihood.html#fact-MLE-binomial" class="xref" data-knowl="./knowl/xref/fact-MLE-binomial.html" data-reveal-label="Reveal" data-close-label="Close" title="Fact 4.1.8: Binomial MLE">Fact 4.1.8</a>.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Hint 4.1.2.1"><a href="#sec-Likelihood-20-2-2" title="Copy heading and permalink for Hint 4.1.2.1" aria-label="Copy heading and permalink for Hint 4.1.2.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 4.1.2"><a href="#sec-Likelihood-20-2" title="Copy heading and permalink for Exercise 4.1.2" aria-label="Copy heading and permalink for Exercise 4.1.2">🔗</a></div></article><article class="exercise exercise-like" id="sec-Likelihood-20-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Likelihood-20-3-1-1">A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 4.1.3"><a href="#sec-Likelihood-20-3" title="Copy heading and permalink for Exercise 4.1.3" aria-label="Copy heading and permalink for Exercise 4.1.3">🔗</a></div></article><article class="exercise exercise-like" id="sec-Likelihood-20-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Likelihood-20-4-1-1">A radioactive material emits particles at an unknown probabilistic rate <span class="process-math">\(\lambda\)</span> particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function <span class="process-math">\(\mathcal{L}(\lambda)\)</span> based on this data. What is the maximum likelihood estimation for <span class="process-math">\(\lambda\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 4.1.4"><a href="#sec-Likelihood-20-4" title="Copy heading and permalink for Exercise 4.1.4" aria-label="Copy heading and permalink for Exercise 4.1.4">🔗</a></div></article><article class="exercise exercise-like" id="sec-Likelihood-20-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="sec-Likelihood-20-5-1-1">Suppose a parameter <span class="process-math">\(\theta\)</span> takes values in <span class="process-math">\([0, 1]\)</span> with likelihood function <span class="process-math">\(\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-Likelihood-20-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Exercise 4.1.5"><a href="#sec-Likelihood-20-5" title="Copy heading and permalink for Exercise 4.1.5" aria-label="Copy heading and permalink for Exercise 4.1.5">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.1"><a href="#sec-Likelihood-20" title="Copy heading and permalink for Exercises 4.1" aria-label="Copy heading and permalink for Exercises 4.1">🔗</a></div></section><div class="autopermalink" data-description="Section 4.1: Likelihood"><a href="#sec-Likelihood" title="Copy heading and permalink for Section 4.1: Likelihood" aria-label="Copy heading and permalink for Section 4.1: Likelihood">🔗</a></div></section></div>
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