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2 changes: 1 addition & 1 deletion posts/kbhangelman_syndrome/index.html
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cause of Angelman Syndrome Angelman Syndrome is primarily caused by the UBE3A and the ubiquitin proteasome system. Poly-ubiquitin chain asks to discard cells."><meta name=author content="Houjun Liu"><link rel=stylesheet href=/css/global.css><link rel=stylesheet href=/css/syntax.css></head><body><div class=center-clearfix><header><span id=header-name onclick='window.location.href="/"' style=cursor:pointer>Houjun Liu</span><div id=socialpanel><a href=https://www.jemoka.com/search/ class=header-social id=header-search><i class="ic fa-solid fa-magnifying-glass"></i></i></a>
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<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Angelman Syndrome</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#cause-of-angelman-syndrome--kbhangelman-syndrome-dot-md>cause of <a href=HAHAHUGOSHORTCODE76s1HBHB>Angelman Syndrome</a></a></li></ul></nav></aside><main><article><div><p><a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a> is a syndrome is ~1 in 15000, clinically recognizable, developmental delay syndrome.</p><h2 id=cause-of-angelman-syndrome--kbhangelman-syndrome-dot-md>cause of <a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a></h2><p><a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a> is primarily caused by the <a href>UBE3A</a> and the <a href>ubiquitin proteasome system.</a> Poly-<a href>ubiquitin</a> chain asks to discard cells.</p></div></article></main><footer><p id=footer>&copy; 2019-2024 Houjun Liu. Licensed CC BY-NC-SA 4.0.</p></footer></div></body></html>
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Angelman Syndrome</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#cause-of-angelman-syndrome--kbhangelman-syndrome-dot-md>cause of <a href=HAHAHUGOSHORTCODE83s1HBHB>Angelman Syndrome</a></a></li></ul></nav></aside><main><article><div><p><a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a> is a syndrome is ~1 in 15000, clinically recognizable, developmental delay syndrome.</p><h2 id=cause-of-angelman-syndrome--kbhangelman-syndrome-dot-md>cause of <a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a></h2><p><a href=/posts/kbhangelman_syndrome/>Angelman Syndrome</a> is primarily caused by the <a href>UBE3A</a> and the <a href>ubiquitin proteasome system.</a> Poly-<a href>ubiquitin</a> chain asks to discard cells.</p></div></article></main><footer><p id=footer>&copy; 2019-2024 Houjun Liu. Licensed CC BY-NC-SA 4.0.</p></footer></div></body></html>
2 changes: 1 addition & 1 deletion posts/kbhbasis/index.html
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\begin{equation} v = a_1v_1+ \dots + a_{n}v_{n} \end{equation}"><meta name=author content="Houjun Liu"><link rel=stylesheet href=/css/global.css><link rel=stylesheet href=/css/syntax.css></head><body><div class=center-clearfix><header><span id=header-name onclick='window.location.href="/"' style=cursor:pointer>Houjun Liu</span><div id=socialpanel><a href=https://www.jemoka.com/search/ class=header-social id=header-search><i class="ic fa-solid fa-magnifying-glass"></i></i></a>
<a href=https://github.com/Jemoka/ class=header-social id=header-github><i class="ic fa-brands fa-github"></i></a>
<a href=https://maly.io/@jemoka class=header-social id=header-twitter><i class="ic fa-brands fa-mastodon"></i></a>
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>basis</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#constituents>constituents</a></li><li><a href=#requirements>requirements</a></li><li><a href=#additional-information>additional information</a><ul><li><a href=#criteria-for-basis--kbhbasis-dot-md>criteria for <a href=HAHAHUGOSHORTCODE138s7HBHB>basis</a></a></li><li><a href=#dualing-basis-construction>Dualing Basis Construction</a></li></ul></li></ul></nav></aside><main><article><div><p>A basis is a list of <a href=/posts/kbhvector/>vector</a>s in \(V\) that <a href=/posts/kbhspan/#spans>spans</a> \(V\) and is <a href=/posts/kbhlinear_independence/>linearly independent</a></p><h2 id=constituents>constituents</h2><ul><li>a LIST! of <a href=/posts/kbhvector/>vector</a>s in <a href=/posts/kbhvector_space/>vector space</a> \(V\)</li></ul><h2 id=requirements>requirements</h2><ul><li>the list is&mldr;<ul><li><a href=/posts/kbhlinear_independence/>linear independent</a></li><li><a href=/posts/kbhspan/#spans>spans</a> \(V\)</li></ul></li></ul><h2 id=additional-information>additional information</h2><h3 id=criteria-for-basis--kbhbasis-dot-md>criteria for <a href=/posts/kbhbasis/>basis</a></h3><p>A list \(v_1, \dots v_{n}\) of vectors in \(V\) is a <a href=/posts/kbhbasis/>basis</a> of \(V\) <a href=/posts/kbhequivalence/>IFF</a> every \(v \in V\) can be written uniquely as:</p><p>\begin{equation}
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>basis</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#constituents>constituents</a></li><li><a href=#requirements>requirements</a></li><li><a href=#additional-information>additional information</a><ul><li><a href=#criteria-for-basis--kbhbasis-dot-md>criteria for <a href=HAHAHUGOSHORTCODE135s7HBHB>basis</a></a></li><li><a href=#dualing-basis-construction>Dualing Basis Construction</a></li></ul></li></ul></nav></aside><main><article><div><p>A basis is a list of <a href=/posts/kbhvector/>vector</a>s in \(V\) that <a href=/posts/kbhspan/#spans>spans</a> \(V\) and is <a href=/posts/kbhlinear_independence/>linearly independent</a></p><h2 id=constituents>constituents</h2><ul><li>a LIST! of <a href=/posts/kbhvector/>vector</a>s in <a href=/posts/kbhvector_space/>vector space</a> \(V\)</li></ul><h2 id=requirements>requirements</h2><ul><li>the list is&mldr;<ul><li><a href=/posts/kbhlinear_independence/>linear independent</a></li><li><a href=/posts/kbhspan/#spans>spans</a> \(V\)</li></ul></li></ul><h2 id=additional-information>additional information</h2><h3 id=criteria-for-basis--kbhbasis-dot-md>criteria for <a href=/posts/kbhbasis/>basis</a></h3><p>A list \(v_1, \dots v_{n}\) of vectors in \(V\) is a <a href=/posts/kbhbasis/>basis</a> of \(V\) <a href=/posts/kbhequivalence/>IFF</a> every \(v \in V\) can be written uniquely as:</p><p>\begin{equation}
v = a_1v_1+ \dots + a_{n}v_{n}
\end{equation}</p><p>where \(a_1, \dots, a_{n} \in \mathbb{F}\).</p><h4 id=forward-direction>forward direction</h4><p>Suppose we have \(v_1, \dots, v_{n}\) as the <a href=/posts/kbhbasis/>basis</a> in \(V\). We desire that \(v_1, \dots v_{n}\) uniquely constructs each \(v \in V\).</p><p>By definition, they <a href=/posts/kbhspan/>span</a> \(V\) and are <a href=/posts/kbhlinear_independence/>linear independent</a> in \(V\).</p><p>Because of the <a href=/posts/kbhspan/#spans>spanning</a> quality, there exists <em>at least</em> one set of \(a_1, \dots, a_{n} \in \mathbb{F}\) such that we can write:</p><p>\begin{equation}
v \in V = a_1v_1+ \dots + a_{n}v_{n}
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2 changes: 1 addition & 1 deletion posts/kbhbaysian_network/index.html
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\begin{equation} \prod_{i=1}^{n} p(X_{i} \mid pa(x_{i})) \end{equation}"><meta name=author content="Houjun Liu"><link rel=stylesheet href=/css/global.css><link rel=stylesheet href=/css/syntax.css></head><body><div class=center-clearfix><header><span id=header-name onclick='window.location.href="/"' style=cursor:pointer>Houjun Liu</span><div id=socialpanel><a href=https://www.jemoka.com/search/ class=header-social id=header-search><i class="ic fa-solid fa-magnifying-glass"></i></i></a>
<a href=https://github.com/Jemoka/ class=header-social id=header-github><i class="ic fa-brands fa-github"></i></a>
<a href=https://maly.io/@jemoka class=header-social id=header-twitter><i class="ic fa-brands fa-mastodon"></i></a>
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Baysian Network</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#conditional-independence>conditional independence</a><ul><li><a href=#checking-for-conditional-independence>checking for conditional independence</a></li></ul></li><li><a href=#parameter-learning--kbhparameter-learning-dot-md--in-baysian-network--kbhbaysian-network-dot-md><a href=HAHAHUGOSHORTCODE146s7HBHB>parameter learning</a> in <a href=HAHAHUGOSHORTCODE146s8HBHB>Baysian Network</a></a></li></ul></nav></aside><main><article><div><p>A <a href=/posts/kbhbaysian_network/>Baysian Network</a> is composed of:</p><ol><li>a directed, acyclic graph</li><li>a set of <a href=/posts/kbhprobability/#conditional-probability>conditional probabilities</a> acting as <a href=/posts/kbhfactor/>factor</a>s.</li></ol><p>You generally want arrows to go in the direction of causality.</p><figure><img src=/ox-hugo/2023-09-28_10-20-23_screenshot.png></figure><p>Via the chain rule of Bayes nets, we can write this equivalently as:</p><p>\begin{equation}
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Baysian Network</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#conditional-independence>conditional independence</a><ul><li><a href=#checking-for-conditional-independence>checking for conditional independence</a></li></ul></li><li><a href=#parameter-learning--kbhparameter-learning-dot-md--in-baysian-network--kbhbaysian-network-dot-md><a href=HAHAHUGOSHORTCODE144s7HBHB>parameter learning</a> in <a href=HAHAHUGOSHORTCODE144s8HBHB>Baysian Network</a></a></li></ul></nav></aside><main><article><div><p>A <a href=/posts/kbhbaysian_network/>Baysian Network</a> is composed of:</p><ol><li>a directed, acyclic graph</li><li>a set of <a href=/posts/kbhprobability/#conditional-probability>conditional probabilities</a> acting as <a href=/posts/kbhfactor/>factor</a>s.</li></ol><p>You generally want arrows to go in the direction of causality.</p><figure><img src=/ox-hugo/2023-09-28_10-20-23_screenshot.png></figure><p>Via the chain rule of Bayes nets, we can write this equivalently as:</p><p>\begin{equation}
(P(B) \cdot P(S)) \cdot P(E \mid B,S) \cdot P(D \mid E) \cdot P(C \mid E)
\end{equation}</p><p>generally, for \(n\) different variables,</p><p>\begin{equation}
\prod_{i=1}^{n} p(X_{i} \mid pa(x_{i}))
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2 changes: 1 addition & 1 deletion posts/kbhbaysian_parameter_learning/index.html
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To do this, we desire:"><meta name=author content="Houjun Liu"><link rel=stylesheet href=/css/global.css><link rel=stylesheet href=/css/syntax.css></head><body><div class=center-clearfix><header><span id=header-name onclick='window.location.href="/"' style=cursor:pointer>Houjun Liu</span><div id=socialpanel><a href=https://www.jemoka.com/search/ class=header-social id=header-search><i class="ic fa-solid fa-magnifying-glass"></i></i></a>
<a href=https://github.com/Jemoka/ class=header-social id=header-github><i class="ic fa-brands fa-github"></i></a>
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<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Baysian Parameter Learning</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#bayesian-parameter-learning-on-binary-distributions>Bayesian Parameter Learning on Binary Distributions</a><ul><li><a href=#beta-distribution>Beta Distribution</a></li><li><a href=#total-probability-in-beta-distributions>Total Probability in beta distributions</a></li><li><a href=#choosing-a-prior>Choosing a prior</a></li></ul></li><li><a href=#dirichlet-distribution>Dirichlet Distribution</a></li><li><a href=#expectation--kbhexpectation-dot-md--of-a-distribution><a href=HAHAHUGOSHORTCODE148s14HBHB>expectation</a> of a distribution</a></li></ul></nav></aside><main><article><div><p>We treat this as an inference problem in <a href=/posts/kbhnaive_bayes/>Naive Bayes</a>: <strong>observations are independent from each other</strong>.</p><p>Instead of trying to compute a \(\theta\) that works for <a href=/posts/kbhmaximum_likelihood_parameter_learning/>Maximum Likelihood Parameter Learning</a>, what we instead do is try to understand what \(\theta\) can be in terms of a distribution.</p><p>That is, we want to get some:</p><figure><img src=/ox-hugo/2023-10-05_10-22-12_screenshot.png></figure><p>&ldquo;for each value of \(\theta\), what&rsquo;s the chance that that is the actual value&rdquo;</p><p>To do this, we desire:</p><p>\begin{equation}
<a href=https://www.reddit.com/user/Jemoka/ class=header-social id=header-reddit><i class="ic fa-brands fa-reddit"></i></a></div></header><div id=title><h1>Baysian Parameter Learning</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#bayesian-parameter-learning-on-binary-distributions>Bayesian Parameter Learning on Binary Distributions</a><ul><li><a href=#beta-distribution>Beta Distribution</a></li><li><a href=#total-probability-in-beta-distributions>Total Probability in beta distributions</a></li><li><a href=#choosing-a-prior>Choosing a prior</a></li></ul></li><li><a href=#dirichlet-distribution>Dirichlet Distribution</a></li><li><a href=#expectation--kbhexpectation-dot-md--of-a-distribution><a href=HAHAHUGOSHORTCODE146s14HBHB>expectation</a> of a distribution</a></li></ul></nav></aside><main><article><div><p>We treat this as an inference problem in <a href=/posts/kbhnaive_bayes/>Naive Bayes</a>: <strong>observations are independent from each other</strong>.</p><p>Instead of trying to compute a \(\theta\) that works for <a href=/posts/kbhmaximum_likelihood_parameter_learning/>Maximum Likelihood Parameter Learning</a>, what we instead do is try to understand what \(\theta\) can be in terms of a distribution.</p><p>That is, we want to get some:</p><figure><img src=/ox-hugo/2023-10-05_10-22-12_screenshot.png></figure><p>&ldquo;for each value of \(\theta\), what&rsquo;s the chance that that is the actual value&rdquo;</p><p>To do this, we desire:</p><p>\begin{equation}
p(\theta | D)
\end{equation}</p><p>&ldquo;what&rsquo;s the probability of theta being at a certain value given the observations we had.&rdquo;</p><p>And to obtain the actual the actual value, we calculate the <a href=/posts/kbhexpectation/>expectation</a> of this distribution:</p><p>\begin{equation}
\hat{\theta} = \mathbb{E}[\theta] = \int \theta p(\theta | D) \dd{\theta}
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