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rezaarezvan · Sep 16, 2024 · 7e21329 · 7e21329
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diff --git a/cityu/cs4487/cs4487_2/index.html b/cityu/cs4487/cs4487_2/index.html
@@ -252,6 +252,9 @@ <h4 id="geometric-interpretation">Geometric Interpretation</h4>
 \begin{align*}
 f_{NB}(\mathbf{x}) &#x26;= \underset{c \in \{1, 2\}}{\arg\max} p(y = c) \prod_{j = 1}^{2} p(x_j | y = c) \newline
 &#x26;= \underset{c \in \{1, 2\}}{\arg\max} \log p(y = c) + \sum_{j = 1}^{2} \log p(x_j | y = c) \newline
+&#x26;= \underset{c \in \{1, 2\}}{\arg\max} \log p(y = c) + \sum_{j = 1}^{2} \log \mathcal{N}(x_j; \mu_{j | c}, \sigma_{j | c}^2) \newline
+&#x26;= \underset{c \in \{1, 2\}}{\arg\max} \log p(y = c) + \sum_{j = 1}^{2} \log \left[ \frac{1}{\sqrt{2\pi\sigma_{j | c}^2}} \exp\left(-\frac{(x_j - \mu_{j | c})^2}{2\sigma_{j | c}^2}\right) \right] \newline
+&#x26;= \underset{c \in \{1, 2\}}{\arg\max} \log p(y = c) + \sum_{j = 1}^{2} \log \left[\left(2\pi\sigma_{j | c}^2\right)^{-\frac{1}{2}} \exp\left(-\frac{(x_j - \mu_{j | c})^2}{2\sigma_{j | c}^2}\right) \right] \newline
 &#x26;= \underset{c \in \{1, 2\}}{\arg\max} \log p(y = c) + \sum_{j = 1}^{2} \left( -\frac{1}{2} \log(2\pi\sigma_{j | c}^2) - \frac{(x_j - \mu_{j | c})^2}{2\sigma_{j | c}^2} \right)
 \end{align*}
 $$</p>
@@ -278,13 +281,13 @@ <h3 id="naive-bayes-for-boolean-vectors">Naive Bayes for Boolean Vectors</h3>
 $$
 p(x_j = 0 | y = c) = 1 - \varphi_{j | c}
 $$</p>
-<p>Therefore, the MLE parameters are $\varphi{j | c} = M_{j | c} / M_c$, where,
+<p>Therefore, the MLE parameters are $\varphi_{j | c} = M_{j | c} / M_c$, where,
 $M_{j | c}$ is the number of training examples in class $c$ that contain feature $x_j$.
 $M_c$ is the number of training examples in class $c$.</p>
 <p>We therefore get class-conditional distributions,
 $$
-p(x_1, \ldots, x_N | y = c) = \prod_{j = 1}^{N} p(x_j | y = c) = \prod_{j = 1}^{N} p(x_j | y = c) \newline
-\log p(x_1, \ldots, x_N | y = c) = \sum_{j = 1}^{N} \log p(x_j | y = c) = \sum_{j = 1}^{N} \log p(x_j | y = c)
+p(x_1, \ldots, x_N | y = c) = \prod_{j = 1}^{N} p(x_j | y = c) \newline
+\log p(x_1, \ldots, x_N | y = c) = \sum_{j = 1}^{N} \log p(x_j | y = c)
 $$</p>
 <p>So, for an input example $\mathbf{x}$, the log-probabilities of features present given $y = c$ adds.</p>
 <h4 id="smoothing">Smoothing</h4>
@@ -305,7 +308,7 @@ <h3 id="multinomial-naive-bayes">Multinomial Naive Bayes</h3>
 <p>Then we need to use a multinomial (or categorical) distribution as class-conditional,
 $$
 p(x_j = k | y = c) = \varphi_{x, j | c} \newline
-\sum_{x in \chi_j} \varphi_{x, j | c} = 1
+\sum_{x \in \chi_j} \varphi_{x, j | c} = 1
 $$</p>
 <p>Our MLE parameter, $\varphi_{x, j | c} = M_{x, j | c} / M_{j | c}$, where,
 $M_{x, j | c}$ is the number of training examples in class $c$ that have feature $x_j = x$.
@@ -346,7 +349,7 @@ <h4 id="geometric-interpretation-1">Geometric Interpretation</h4>
 $$</p>
 <p>By cancelling out the common terms, we get,
 $$
-\log \frac{p(y = 1)}{p(y = 2)} - \frac{1}{2} \mathbf{mu}_1^T \mathbf{\Sigma}^{-1} \mathbf{\mu}_1 + \frac{1}{2} \mathbf{mu}_2^T \mathbf{\Sigma}^{-1} \mathbf{\mu}_2 + (\mathbf{\mu}_1 - \mathbf{\mu}_2)^T \mathbf{\Sigma}^{-1} \mathbf{x} = 0
+\log \frac{p(y = 1)}{p(y = 2)} - \frac{1}{2} \mathbf{\mu}_1^T \mathbf{\Sigma}^{-1} \mathbf{\mu}_1 + \frac{1}{2} \mathbf{\mu}_2^T \mathbf{\Sigma}^{-1} \mathbf{\mu}_2 + (\mathbf{\mu}_1 - \mathbf{\mu}_2)^T \mathbf{\Sigma}^{-1} \mathbf{x} = 0
 $$</p>
 <p>This shows that the decision boundary is linear in $\mathbf{x}$.</p>
 <h3 id="naive-bayes-vs-lda">Naive Bayes vs LDA</h3>

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diff --git a/pagefind/pagefind-entry.json b/pagefind/pagefind-entry.json
@@ -1 +1 @@
-{"version":"1.1.1","languages":{"en":{"hash":"en_70812422c9","wasm":"en","page_count":147}}}
+{"version":"1.1.1","languages":{"en":{"hash":"en_a057c762d2","wasm":"en","page_count":147}}}
diff --git a/pagefind/pagefind.en_70812422c9.pf_meta b/pagefind/pagefind.en_70812422c9.pf_meta
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diff --git a/posts/bucketlist/index.html b/posts/bucketlist/index.html
@@ -221,7 +221,7 @@
 </button> </nav> </div> </div> </header> <main>  <div class="mx-auto max-w-screen-sm px-3"> <div class="animate"> <a href="/posts" class="not-prose group relative flex w-fit flex-nowrap rounded border border-black/15 py-1.5 pl-7 pr-3 transition-colors duration-300 ease-in-out hover:bg-black/5 hover:text-black focus-visible:bg-black/5 focus-visible:text-black dark:border-white/20 dark:hover:bg-white/5 dark:hover:text-white dark:focus-visible:bg-white/5 dark:focus-visible:text-white"> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24" class="absolute left-2 top-1/2 size-4 -translate-y-1/2 fill-none stroke-current stroke-2"> <line x1="5" y1="12" x2="19" y2="12" class="translate-x-2 scale-x-0 transition-transform duration-300 ease-in-out group-hover:translate-x-0 group-hover:scale-x-100 group-focus-visible:translate-x-0 group-focus-visible:scale-x-100"></line> <polyline points="12 5 5 12 12 19" class="translate-x-1 transition-transform duration-300 ease-in-out group-hover:translate-x-0 group-focus-visible:translate-x-0"></polyline> </svg> <div class="text-sm"> Back to posts </div> </a> </div> <div class="my-10 space-y-1"> <div class="animate flex items-center gap-1.5"> <div class="font-base text-sm"> <time datetime="2022-07-31T15:21:34.000Z"> July 31, 2022 </time> </div> 
 &bull;
 <div class="font-base text-sm">
-Last modified:  <time datetime="2024-07-13T21:29:20.000Z"> July 14, 2024 </time> </div> 
+Last modified:  <time datetime="2024-09-14T11:18:10.000Z"> September 14, 2024 </time> </div> 
 &bull;
 <div class="font-base text-sm"> 2 min read </div> </div> <h1 class="animate text-3xl font-semibold text-black dark:text-white"> My personal bucket list </h1> </div> <article class="animate"> <p>One day each and one of us will die - this is a universal truth. I am writing this to keep my self accountable and remind my future self to recognize where I was and where I’ve gotten to.</p>
 <ul>