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<title>Quiz-1 | Revision | Week-4</title>
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</head>
<body>
<div class="slide titlepage">
<h1 class="title">Quiz-1 | Revision | Week-4</h1>
<p class="subtitle">Machine Learning Techniques</p>
</div>
<div id="unsupervised-learning" class="slide section level1">
<h1>Unsupervised Learning</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<div>
<ul class="incremental">
<li>Representation Learning
<ul class="incremental">
<li>PCA</li>
<li>Kernel-PCA</li>
</ul></li>
<li>Clustering
<ul class="incremental">
<li>K-means or Lloyd’s algorithm</li>
</ul></li>
<li>Estimation
<ul class="incremental">
<li>MLE</li>
<li>Bayesian estimation</li>
<li>GMMs</li>
<li>EM algorithm</li>
</ul></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
</div>
</div>
</div>
<div id="theme" class="slide section level1">
<h1>Theme</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
\Large \text{Compression} \rightarrow \text{Comprehension}
\]</span></p>
</div><div class="column" style="width:0%;">
</div>
</div>
</div>
<div id="estimation" class="slide section level1">
<h1>Estimation</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<div>
<ul class="incremental">
<li><p>Model the data-generation process</p>
<ul class="incremental">
<li>Where does the data come from?</li>
<li>What is the process that generates the data?</li>
<li>What are the parameters that govern this process?</li>
</ul>
<p><br></p></li>
<li><p>How is it different from clustering and representation learning?</p>
<ul class="incremental">
<li>Present <span class="math inline">\(\rightarrow\)</span> Future: Start with what is given and try to best explain some patterns in it.</li>
<li>Present <span class="math inline">\(\rightarrow\)</span> Past: Explore the process that could have given rise to the data.</li>
</ul></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="generative-story" class="slide section level1">
<h1>Generative story</h1>
<div class="columns" align="center">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
1, \quad 0, \quad 1, \quad 1, \quad 0, \quad 1
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="generative-story-1" class="slide section level1">
<h1>Generative story</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br> <span class="math display">\[
1, \quad 0, \quad 1, \quad 1, \quad 0, \quad 1
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>Coin flip</li>
<li>Bernoulli distribution</li>
<li><span class="math inline">\(x \sim Br(p)\)</span></li>
<li><span class="math inline">\(x\)</span> is sampled from a Bernoulli distribution with parameter <span class="math inline">\(p\)</span></li>
</ul>
</div>
</div>
</div>
</div>
<div id="generative-story-2" class="slide section level1">
<h1>Generative story</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
165, \quad 173, \quad 150, \quad 185, \quad 178, \quad 171
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="generative-story-3" class="slide section level1">
<h1>Generative story</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br> <span class="math display">\[
165, \quad 173, \quad 150, \quad 185, \quad 178, \quad 171
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>Gaussian distribution</li>
<li><span class="math inline">\(x \sim \mathcal{N}(\mu, \sigma^2)\)</span></li>
<li><span class="math inline">\(x\)</span> is sampled from a Gaussian distribution with parameters <span class="math inline">\(\mu, \sigma^2\)</span></li>
</ul>
</div>
</div>
</div>
</div>
<div id="parameter-estimation" class="slide section level1">
<h1>Parameter estimation</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>Data</li>
<li>Model</li>
<li>Estimate the parameters</li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="parameter-estimation-1" class="slide section level1">
<h1>Parameter estimation</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<ul>
<li>Data</li>
<li>Model</li>
<li>Estimate the parameters</li>
</ul>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p>Methods</p>
<div>
<ul class="incremental">
<li>MLE</li>
<li>Bayesian estimation</li>
</ul>
</div>
</div>
</div>
</div>
<div id="mle" class="slide section level1">
<h1>MLE</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p>Depends on two things</p>
<div>
<ul class="incremental">
<li>Data</li>
<li>Model</li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-1" class="slide section level1">
<h1>MLE</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p>Depends on two things</p>
<ul>
<li>Data</li>
<li>Model</li>
</ul>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
\large L(D; \theta)
\]</span></p>
</div>
</div>
</div>
<div id="assumptions-in-mle" class="slide section level1">
<h1>Assumptions in MLE</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p>Two assumptions</p>
<ul>
<li>Independence</li>
<li>Identical distribution</li>
</ul>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br></p>
<p><u>Independence</u> <span class="math display">\[
P(x_i\ |\ x_j) = P(x_i), \quad i \neq j
\]</span> <u>Identically distributed</u> <span class="math display">\[
P(x_i) = P(x_j), \quad \forall i, j \in [1, n]
\]</span></p>
</div>
</div>
</div>
<div id="mle-for-bernoulli" class="slide section level1">
<h1>MLE for Bernoulli</h1>
<div class="columns" align="center">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
1,\quad 0,\quad 1,\quad 1,\quad 0
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-bernoulli-1" class="slide section level1">
<h1>MLE for Bernoulli</h1>
<div class="columns" align="center">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
1,\quad 0,\quad 1,\quad 1,\quad 0
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><u>Likelihood</u> <span class="math display">\[
L(D; p) = p^3(1 - p)^2
\]</span></p>
</div>
</div>
</div>
<div id="mle-for-bernoulli-2" class="slide section level1">
<h1>MLE for Bernoulli</h1>
<div class="columns" align="center">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
1,\quad 0,\quad 1,\quad 1,\quad 0
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><u>Likelihood</u> <span class="math display">\[
L(D; p) = p^3(1 - p)^2
\]</span> <u>Log-likelihood</u> <span class="math display">\[
\begin{aligned}
l(D; p) &= \log L(D; p)\\\\
&= 3 \log p + 2 \log(1 - p)
\end{aligned}
\]</span></p>
</div>
</div>
</div>
<div id="mle-for-bernoulli-3" class="slide section level1">
<h1>MLE for Bernoulli</h1>
<div class="columns" align="center">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\begin{aligned}
l(D; p) &= 3 \log p + 2 \log(1 - p)\\\\
l^{\prime}(D; p) &= 0\\\\
\cfrac{3}{p} - \cfrac{2}{1 - p} &= 0\\\\
\hat{p}_{\tiny ML} &= \cfrac{3}{5}
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-bernoulli-4" class="slide section level1">
<h1>MLE for Bernoulli</h1>
<div class="columns" align="center">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\Large \hat{p}_{\tiny ML} = \cfrac{1}{n} \cdot \sum \limits_{i = 1}^{n} x_i
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-gaussian" class="slide section level1">
<h1>MLE for Gaussian</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\large L(D; \mu, \sigma^2) = \prod \limits_{i = 1}^{n} f(x_i; \mu, \sigma^2)
\]</span> where <span class="math display">\[
\small f(x; \mu, \sigma^2) = \cfrac{1}{\sqrt{2 \pi} \sigma} \exp \left[ \cfrac{-(x - \mu)^2}{2 \sigma^2} \right]
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-gaussian-1" class="slide section level1">
<h1>MLE for Gaussian</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
l(D; \mu, \sigma^2) = - n \log (\sqrt{2 \pi} \sigma) + \cfrac{1}{2 \sigma^2} \sum \limits_{i = 1}^{n} -(x_i - \mu)^2
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-gaussian-2" class="slide section level1">
<h1>MLE for Gaussian</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\Large \hat{\mu}_{\tiny ML} = \cfrac{1}{n} \cdot \sum \limits_{i = 1}^{n} x_i
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="bayesian-modeling" class="slide section level1">
<h1>Bayesian modeling</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>Prior</li>
<li>Data</li>
<li>Likelihood</li>
<li>Evidence</li>
<li>Posterior</li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="bayes-rule" class="slide section level1">
<h1>Bayes’ rule</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
\text{Posterior} = \cfrac{\text{Likelihood}}{\text{Evidence}} \cdot \text{Prior}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="bayes-rule-1" class="slide section level1">
<h1>Bayes’ rule</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
\text{Posterior} = \cfrac{\text{Likelihood}}{\text{Evidence}} \cdot \text{Prior}
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
P(\theta\ |\ D) = \cfrac{P(D\ |\ \theta)}{P(D)} \cdot P(\theta)
\]</span></p>
</div>
</div>
</div>
<div id="beta" class="slide section level1">
<h1>Beta</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p><br> <span class="math display">\[
\text{Beta}(\alpha, \beta) = \cfrac{1}{Z} \cdot p^{\alpha - 1} \cdot (1 - p)^{\beta - 1} \quad\quad \alpha, \beta > 0
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="conjugate-prior" class="slide section level1">
<h1>Conjugate prior</h1>
<p><br></p>
<div>
<ul class="incremental">
<li>Beta distribution is a conjugate prior for the Bernoulli distribution</li>
<li><span class="math inline">\(n_1\)</span> ones and <span class="math inline">\(n_0\)</span> zeros</li>
<li>Prior is <span class="math inline">\(\text{Beta}(\alpha, \beta)\)</span></li>
<li>Posterior is <span class="math inline">\(\text{Beta}(\alpha + n_1, \beta + n_0)\)</span></li>
</ul>
</div>
</div>
<div id="point-estimates" class="slide section level1">
<h1>Point estimates</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>MAP
<ul class="incremental">
<li>mode of the posterior</li>
<li>differentiate posterior w.r.t <span class="math inline">\(p\)</span> and set it to zero</li>
</ul></li>
<li>Expected value
<ul class="incremental">
<li><span class="math inline">\(\cfrac{\alpha}{\alpha + \beta}\)</span></li>
</ul></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="why-go-for-gmms" class="slide section level1">
<h1>Why go for GMMs</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br></p>
<p><br></p>
<div>
<ul class="incremental">
<li>Model more complex situations</li>
<li>Multi-modal datasets</li>
<li>Latent variable models</li>
</ul>
</div>
</div><div class="column" style="width:50%;">
<p><img src="images/img_008.png" width="500" /></p>
</div>
</div>
</div>
<div id="gmm-generative-story" class="slide section level1">
<h1>GMM: generative story</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p>Two steps</p>
<div>
<ul class="incremental">
<li>Pick a mixture</li>
<li>Sample a point from the mixture</li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="gmm-setup" class="slide section level1">
<h1>GMM: setup</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p>Two steps</p>
<div>
<ul class="incremental">
<li><span class="math inline">\(K\)</span> mixutres: <span class="math inline">\([\pi_1, \cdots, \pi_k, \cdots, \pi_K]\)</span></li>
<li>Mixture means: <span class="math inline">\([\mu_1, \cdots, \mu_k, \cdots, \mu_K]\)</span></li>
<li>Mixture variances: <span class="math inline">\([\sigma_1^2, \cdots, \sigma_k^2, \cdots, \sigma_K^2]\)</span></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="question" class="slide section level1">
<h1>Question</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br></p>
<p>What is the value of the following expression? <span class="math display">\[
\sum \limits_{i = 1}^{K} \pi_k
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="gmm-generative-process" class="slide section level1">
<h1>GMM: generative process</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p>For generating a data-point <span class="math inline">\(x_i\)</span>:</p>
<div>
<ul class="incremental">
<li>Sample a mixture index based on the probabilities <span class="math inline">\([\pi_1, \cdots, \pi_K]\)</span></li>
<li>Call this index <span class="math inline">\(k\)</span></li>
<li><span class="math inline">\(z_i = k\)</span></li>
<li>Sample a point from the <span class="math inline">\(k^{th}\)</span> Gaussian</li>
<li><span class="math inline">\(x_i \sim \mathcal{N}(\mu_k, \sigma_k^2)\)</span></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="gmm-density" class="slide section level1">
<h1>GMM: density</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\Large f_{\tiny GMM}(x) = \sum \limits_{k = 1}^{K} P(z = k) \cdot f(x\ |\ z = k)
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="gmm-density-1" class="slide section level1">
<h1>GMM: density</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\Large f_{\tiny GMM}(x) = \sum \limits_{k = 1}^{K} \pi_k \cdot f(x\ |\ z = k)
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-a-gmm" class="slide section level1">
<h1>MLE for a GMM</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\begin{aligned}
L(D; \theta) &= \prod \limits_{i = 1}^{n} f_{\tiny GMM} (x_i)\\\\
l(D; \theta) &= \sum \limits_{i = 1}^{n} \log f_{\tiny GMM}(x_i)
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-a-gmm-1" class="slide section level1">
<h1>MLE for a GMM</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><span class="math display">\[
\begin{aligned}
l(D; \theta) &= \sum \limits_{i = 1}^{n} \log f_{\tiny GMM}(x_i)\\\\
&= \sum \limits_{i = 1}^{n} \log \left[ \sum \limits_{k = 1}^{K} \pi_k f(x_i\ |\ z_i = k) \right]
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-a-gmm-2" class="slide section level1">
<h1>MLE for a GMM</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><span class="math display">\[
\begin{aligned}
l(D; \theta) &= \sum \limits_{i = 1}^{n} \log f_{\tiny GMM}(x_i)\\\\
&= \sum \limits_{i = 1}^{n} \log \left[ \sum \limits_{k = 1}^{K} \pi_k f(x_i\ |\ z_i = k) \right]\\\\
&= \sum \limits_{i = 1}^{n} \log \left[ \sum \limits_{k = 1}^{K} \lambda_k^{i} \cfrac{\pi_k}{\lambda_k^{i}} f(x_i\ |\ z_i = k) \right]
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="mle-for-a-gmm-3" class="slide section level1">
<h1>MLE for a GMM</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><span class="math display">\[
\begin{aligned}
l(D; \theta) &= \sum \limits_{i = 1}^{n} \log f_{\tiny GMM}(x_i)\\\\
&= \sum \limits_{i = 1}^{n} \log \left[ \sum \limits_{k = 1}^{K} \pi_k f(x_i\ |\ z_i = k) \right]\\\\
&= \sum \limits_{i = 1}^{n} \log \left[ \sum \limits_{k = 1}^{K} \lambda_k^{i} \cfrac{\pi_k}{\lambda_k^{i}} f(x_i\ |\ z_i = k) \right]\\\\\
&\geq \sum \limits_{i = 1}^{n} \sum \limits_{k = 1}^{K} \lambda_k^{i} \log \left[ \cfrac{\pi_k}{\lambda_k^{i}} f(x_i\ |\ z_i = k) \right]\quad \quad \text{Jensen's inequality}
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="modified-log-likelihood" class="slide section level1">
<h1>Modified log-likelihood</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\Large ml(D; \theta, \lambda) = \sum \limits_{i = 1}^{n} \sum \limits_{k = 1}^{K} \lambda_k^{i} \log \left[ \cfrac{\pi_k}{\lambda_k^{i}} f(x_i\ |\ z_i = k) \right]\\\
\]</span> where</p>
<p><br> <span class="math display">\[
0 \leq \lambda_k^{i} \leq 1 \quad \text{and} \quad \sum \limits_{k = 1}^{K} \lambda_k^{i} = 1
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="em" class="slide section level1">
<h1>EM</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p>Parameter estimation for GMMs</p>
<div>
<ul class="incremental">
<li><p>Intialize <span class="math inline">\(\theta^{0}\)</span> using Lloyd’s with <span class="math inline">\(k = K\)</span></p></li>
<li><p>Until convergence</p>
<ul class="incremental">
<li>E-step
<ul class="incremental">
<li>use <span class="math inline">\(\theta^{(t)}\)</span> to estimate <span class="math inline">\(\lambda^{(t)}\)</span></li>
</ul></li>
<li>M-step
<ul class="incremental">
<li>use <span class="math inline">\(\lambda^{(t)}\)</span> to estimate <span class="math inline">\(\theta^{(t+1)}\)</span></li>
</ul></li>
</ul></li>
</ul>
</div>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="e-step" class="slide section level1">
<h1>E-step</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
\begin{aligned}
\lambda_k^{i} &= P(z_i = k\ |\ x_i)\\\\
&= \cfrac{P(z_i = k) \cdot f(x_i\ |\ z_i = k)}{\sum \limits_{l = 1}^{K} P(z_i = l) \cdot f(x_i\ |\ z_i = l)}
\end{aligned}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
<div id="m-step" class="slide section level1">
<h1>M-step</h1>
<div class="columns" align="left">
<div class="column" style="width:50%;">
<p><br> <span class="math display">\[
\Large \begin{aligned}
\mu_k = \cfrac{\sum \limits_{i = 1}^{n} \lambda_k^{i} x_i}{\sum \limits_{i = 1}^{n} \lambda_k^{i}}
\end{aligned}
\]</span></p>
</div><div class="column" style="width:50%;">
<p><br> <span class="math display">\[
\Large \begin{aligned}
\sigma_k^{2} = \cfrac{\sum \limits_{i = 1}^{n} \lambda_k^{i} (x_i - \mu_k)^2}{\sum \limits_{i = 1}^{n} \lambda_k^{i}}
\end{aligned}
\]</span></p>
</div>
</div>
<p><span class="math display">\[
\Large \pi_k = \cfrac{1}{n} \cdot \sum \limits_{i = 1}^{n} \lambda_k^{i}
\]</span></p>
</div>
<div id="hard-clustering" class="slide section level1">
<h1>Hard clustering</h1>
<div class="columns" align="left">
<div class="column" style="width:100%;">
<p><br></p>
<p><br> <span class="math display">\[
z_i = \operatorname*{argmax} \limits_{k} \quad \lambda_k^{i}
\]</span></p>
</div><div class="column" style="width:0%;">
<p><br></p>
<p><br></p>
</div>
</div>
</div>
</body>
</html>