Update documentation

Author: arturtoshev

_images/1f50f557496d0fb9c5b38a4bed0b874e67abff5fed7ae8b399d996d8f6c22494.png

@@ -643,13 +643,13 @@ $$
 Without discussing all the details, we will now briefly sketch how Gaussian Processes can be adapted to classification. Consider for simplicity the 2-class problem:
 
 $$
-0 < y < 1, \quad h(x) = \text{sigmoid}(\varphi(x))
+0 < y < 1, \quad h(x) = \text{sigmoid}(\varphi(x)).
 $$
 
-the PDF for $y$ conditioned on the feature $\varphi(x)$ then follows a Bernoulli-distribution:
+The PDF for $y$ conditioned on the feature $\varphi(x)$ then follows a Bernoulli-distribution:
 
 $$
-p(y | \varphi) = \text{sigmoid}(\varphi)^{y} \left( 1 - \text{sigmoid}(\varphi) \right)^{1-y}, \quad i=1, \ldots, n
+p(y | \varphi) = \text{sigmoid}(\varphi)^{y} \left( 1 - \text{sigmoid}(\varphi) \right)^{1-y}, \quad i=1, \ldots, n.
 $$
 
 Finding the predictive PDF for unseen data $p(y^{(n+1)}|y)$, given the training data
@@ -679,13 +679,13 @@ $$
 hence giving us
 
 $$
-p(\tilde{\varphi}) = \mathcal{N}( \tilde{\varphi}; 0, K + \nu I)
+p(\tilde{\varphi}) = \mathcal{N}( \tilde{\varphi}; 0, K + \nu I),
 $$
 
 where $K_{ij} = k(x^{(i)}, x^{(j)})$, i.e., a Grammian matrix generated by the kernel functions from the feature map $\varphi(x)$.
 
 > - Note that we do **NOT** include an explicit noise term in the data covariance as we assume that all sample data have been correctly classified.
-> - For numerical reasons, we introduce a noise-like form that improves the conditioning of $K + \mu I$
+> - For numerical reasons, we introduce a noise-like form $\nu I$ that improves the conditioning of $K$
 > - For two-class classification it is sufficient to predict $p(y^{(n+1)} = 1 | y)$ as $p(y^{(n+1)} = 0 | y) = 1 - p(y^{(n+1)} = 1 | y)$
 
 Using the PDF $p(y=1|\varphi) = \text{sigmoid}(\varphi(x))$ we obtain the predictive PDF:

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@@ -664,8 +664,8 @@ <h3><span class="section-number">2.1.4. </span>Bayesian Linear Regression Model<
 <p>The model below essentially makes the following prior assumptions:</p>
 <div class="math notranslate nohighlight">
 \[y \approx h(x) = wx + b + \epsilon, \quad \text{with:}\]</div>
-<div class="amsmath math notranslate nohighlight" id="equation-0db22e97-6b0a-46e0-a476-53d8ecd3575d">
-<span class="eqno">(2.30)<a class="headerlink" href="#equation-0db22e97-6b0a-46e0-a476-53d8ecd3575d" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-b60eee56-96e4-4e84-8e5b-6e3421e3a6c3">
+<span class="eqno">(2.30)<a class="headerlink" href="#equation-b60eee56-96e4-4e84-8e5b-6e3421e3a6c3" title="Permalink to this equation">#</a></span>\[\begin{align}
 y_i &amp;\sim \mathcal{N}(\mu, \sigma^2)\\
 \mu &amp;= w \cdot x_i + b\\
 w &amp;\sim \mathcal{N}(0,1^2)\\

_images/3fe9e36024586af07da99c14267a9e9f94e619515aad4ffbc454b01b9206df14.png

@@ -999,12 +999,12 @@ <h2><span class="section-number">7.4. </span>GP for Classification<a class="head
 <p>Without discussing all the details, we will now briefly sketch how Gaussian Processes can be adapted to classification. Consider for simplicity the 2-class problem:</p>
 <div class="math notranslate nohighlight">
 \[
-0 &lt; y &lt; 1, \quad h(x) = \text{sigmoid}(\varphi(x))
+0 &lt; y &lt; 1, \quad h(x) = \text{sigmoid}(\varphi(x)).
 \]</div>
-<p>the PDF for <span class="math notranslate nohighlight">\(y\)</span> conditioned on the feature <span class="math notranslate nohighlight">\(\varphi(x)\)</span> then follows a Bernoulli-distribution:</p>
+<p>The PDF for <span class="math notranslate nohighlight">\(y\)</span> conditioned on the feature <span class="math notranslate nohighlight">\(\varphi(x)\)</span> then follows a Bernoulli-distribution:</p>
 <div class="math notranslate nohighlight">
 \[
-p(y | \varphi) = \text{sigmoid}(\varphi)^{y} \left( 1 - \text{sigmoid}(\varphi) \right)^{1-y}, \quad i=1, \ldots, n
+p(y | \varphi) = \text{sigmoid}(\varphi)^{y} \left( 1 - \text{sigmoid}(\varphi) \right)^{1-y}, \quad i=1, \ldots, n.
 \]</div>
 <p>Finding the predictive PDF for unseen data <span class="math notranslate nohighlight">\(p(y^{(n+1)}|y)\)</span>, given the training data</p>
 <div class="math notranslate nohighlight">
@@ -1031,13 +1031,13 @@ <h2><span class="section-number">7.4. </span>GP for Classification<a class="head
 <p>hence giving us</p>
 <div class="math notranslate nohighlight">
 \[
-p(\tilde{\varphi}) = \mathcal{N}( \tilde{\varphi}; 0, K + \nu I)
+p(\tilde{\varphi}) = \mathcal{N}( \tilde{\varphi}; 0, K + \nu I),
 \]</div>
 <p>where <span class="math notranslate nohighlight">\(K_{ij} = k(x^{(i)}, x^{(j)})\)</span>, i.e., a Grammian matrix generated by the kernel functions from the feature map <span class="math notranslate nohighlight">\(\varphi(x)\)</span>.</p>
 <blockquote>
 <div><ul class="simple">
 <li><p>Note that we do <strong>NOT</strong> include an explicit noise term in the data covariance as we assume that all sample data have been correctly classified.</p></li>
-<li><p>For numerical reasons, we introduce a noise-like form that improves the conditioning of <span class="math notranslate nohighlight">\(K + \mu I\)</span></p></li>
+<li><p>For numerical reasons, we introduce a noise-like form <span class="math notranslate nohighlight">\(\nu I\)</span> that improves the conditioning of <span class="math notranslate nohighlight">\(K\)</span></p></li>
 <li><p>For two-class classification it is sufficient to predict <span class="math notranslate nohighlight">\(p(y^{(n+1)} = 1 | y)\)</span> as <span class="math notranslate nohighlight">\(p(y^{(n+1)} = 0 | y) = 1 - p(y^{(n+1)} = 1 | y)\)</span></p></li>
 </ul>
 </div></blockquote>