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27 changes: 14 additions & 13 deletions schedule/slides/15-LDA-and-QDA.html
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<h2>15 LDA and QDA</h2>
<p><span class="secondary">Stat 406</span></p>
<p><span class="secondary">Geoff Pleiss, Trevor Campbell</span></p>
<p>Last modified – 09 October 2023</p>
<p>Last modified – 14 October 2024</p>
<p><span class="math display">\[
\DeclareMathOperator*{\argmin}{argmin}
\DeclareMathOperator*{\argmax}{argmax}
Expand All @@ -424,33 +424,34 @@ <h2>15 LDA and QDA</h2>
\newcommand{\bls}{\widehat{\beta}_{ols}}
\newcommand{\blt}{\widehat{\beta}^L_{s}}
\newcommand{\bll}{\widehat{\beta}^L_{\lambda}}
\newcommand{\U}{\mathbf{U}}
\newcommand{\D}{\mathbf{D}}
\newcommand{\V}{\mathbf{V}}
\]</span></p>
</section>
<section id="last-time" class="slide level2">
<h2>Last time</h2>
<p>We showed that with two classes, the <span class="secondary">Bayes’ classifier</span> is</p>
<p><span class="math display">\[g_*(X) = \begin{cases}
1 &amp; \textrm{ if } \frac{p_1(X)}{p_0(X)} &gt; \frac{1-\pi}{\pi} \\
<p><span class="math display">\[g_*(x) = \begin{cases}
1 &amp; \textrm{ if } \frac{p_1(x)}{p_0(x)} &gt; \frac{1-\pi}{\pi} \\
0 &amp; \textrm{ otherwise}
\end{cases}\]</span></p>
<p>where <span class="math inline">\(p_1(X) = Pr(X \given Y=1)\)</span>, <span class="math inline">\(p_0(X) = Pr(X \given Y=0)\)</span> and <span class="math inline">\(\pi = Pr(Y=1)\)</span></p>
<p>where <span class="math inline">\(p_1(x) = \Pr(X=x \given Y=1)\)</span>, <span class="math inline">\(p_0(x) = \Pr(X=x \given Y=0)\)</span> and <span class="math inline">\(\pi = \Pr(Y=1)\)</span></p>
<div class="fragment">
<p>For more than two classes.</p>
<p><span class="math display">\[g_*(X) =
\argmax_k \frac{\pi_k p_k(X)}{\sum_k \pi_k p_k(X)}\]</span></p>
<p>where <span class="math inline">\(p_k(X) = Pr(X \given Y=k)\)</span> and <span class="math inline">\(\pi_k = P(Y=k)\)</span></p>
<p>For more than two classes:</p>
<p><span class="math display">\[g_*(x) =
\argmax_k \frac{\pi_k p_k(x)}{\sum_k \pi_k p_k(x)}\]</span></p>
<p>where <span class="math inline">\(p_k(x) = \Pr(X=x \given Y=k)\)</span> and <span class="math inline">\(\pi_k = P(Y=k)\)</span></p>
</div>
</section>
<section id="estimating-these" class="slide level2">
<h2>Estimating these</h2>
<p>Let’s make some assumptions:</p>
<ol type="1">
<li><span class="math inline">\(Pr(X\given Y=k) = \mbox{N}(\mu_k,\Sigma_k)\)</span></li>
<li><span class="math inline">\(\Pr(X=x\given Y=k) = \mbox{N}(x; \mu_k,\Sigma_k)\)</span></li>
<li><span class="math inline">\(\Sigma_k = \Sigma_{k'} = \Sigma\)</span></li>
</ol>
<div class="fragment">
<p>This leads to <span class="secondary">Linear Discriminant Analysis</span> (LDA), one of the oldest classifiers</p>
</div>
</section>
<section id="lda" class="slide level2">
<h2>LDA</h2>
Expand Down Expand Up @@ -606,10 +607,10 @@ <h2></h2>
</section>
<section id="naïve-bayes" class="slide level2">
<h2>Naïve Bayes</h2>
<p>Assume that <span class="math inline">\(Pr(X | Y = k) = Pr(X_1 | Y = k)\cdots Pr(X_p | Y = k)\)</span>.</p>
<p>Assume that <span class="math inline">\(\Pr(X=x | Y = k) = \Pr(X_1=x_1 | Y = k)\cdots \Pr(X_p=x_p | Y = k)\)</span>.</p>
<p>That is, conditional on the class, the feature distribution is independent.</p>
<div class="fragment">
<p>If we further assume that <span class="math inline">\(Pr(X_j | Y = k)\)</span> is Gaussian,</p>
<p>If we further assume that <span class="math inline">\(\Pr(X_j=x_j | Y = k)\)</span> is Gaussian,</p>
<p>This is the same as QDA but with <span class="math inline">\(\Sigma_k\)</span> Diagonal.</p>
</div>
<div class="fragment">
Expand Down
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