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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>
<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>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=HAHAHUGOSHORTCODE107s1HBHB>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/kbhargmax/index.html
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additional information argmax of log see argmax of log"><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>argmax</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#finding-argmax>finding argmax</a><ul><li><a href=#direct-optimization--kbhoptimization-dot-md>direct <a href=HAHAHUGOSHORTCODE93s0HBHB>optimization</a></a></li><li><a href=#gradient-ascent>gradient ascent</a></li></ul></li><li><a href=#additional-information>additional information</a><ul><li><a href=#argmax-of-log--kbhmaximum-likelihood-parameter-learning-dot-md><a href=HAHAHUGOSHORTCODE93s1HBHB>argmax of log</a></a></li></ul></li></ul></nav></aside><main><article><div><p>function that returns the input that maximizes the expression.</p><h2 id=finding-argmax>finding argmax</h2><h3 id=direct-optimization--kbhoptimization-dot-md>direct <a href=/posts/kbhoptimization/>optimization</a></h3><p>Typical maximization system. Take derivative, set it to 0, solve, plug in, solve. THis is pretty bad during times are not differentiable.</p><h3 id=gradient-ascent>gradient ascent</h3><p>We take steps following the direction</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>argmax</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#finding-argmax>finding argmax</a><ul><li><a href=#direct-optimization--kbhoptimization-dot-md>direct <a href=HAHAHUGOSHORTCODE95s0HBHB>optimization</a></a></li><li><a href=#gradient-ascent>gradient ascent</a></li></ul></li><li><a href=#additional-information>additional information</a><ul><li><a href=#argmax-of-log--kbhmaximum-likelihood-parameter-learning-dot-md><a href=HAHAHUGOSHORTCODE95s1HBHB>argmax of log</a></a></li></ul></li></ul></nav></aside><main><article><div><p>function that returns the input that maximizes the expression.</p><h2 id=finding-argmax>finding argmax</h2><h3 id=direct-optimization--kbhoptimization-dot-md>direct <a href=/posts/kbhoptimization/>optimization</a></h3><p>Typical maximization system. Take derivative, set it to 0, solve, plug in, solve. THis is pretty bad during times are not differentiable.</p><h3 id=gradient-ascent>gradient ascent</h3><p>We take steps following the direction</p><p>\begin{equation}
\theta_{1j} = \theta_{0j} + \eta \pdv{LL(\theta_{0})}{\theta_{0j}}
\end{equation}</p><h2 id=additional-information>additional information</h2><h3 id=argmax-of-log--kbhmaximum-likelihood-parameter-learning-dot-md><a href=/posts/kbhmaximum_likelihood_parameter_learning/#argmax-of-log>argmax of log</a></h3><p>see <a href=/posts/kbhmaximum_likelihood_parameter_learning/#argmax-of-log>argmax of log</a></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/kbhautism/index.html
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autism gene signature The gene signature of autism can be measured in clean and quantitative assays."><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>autism</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#autism--kbhautism-dot-md--gene-signature><a href=HAHAHUGOSHORTCODE110s2HBHB>autism</a> gene signature</a></li></ul></nav></aside><main><article><div><p>autism is a spectrum disorder that are caused by both environmental and genetic factors.</p><p>Key Question: how can different <a href=/posts/kbhchromatin/>chromatin</a> regulators lead to the same set of symptoms named &ldquo;<a href=/posts/kbhautism/>autism</a>&rdquo;.</p><h2 id=autism--kbhautism-dot-md--gene-signature><a href=/posts/kbhautism/>autism</a> gene signature</h2><p>The gene signature of autism can be measured in clean and quantitative assays.</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>autism</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#autism--kbhautism-dot-md--gene-signature><a href=HAHAHUGOSHORTCODE111s2HBHB>autism</a> gene signature</a></li></ul></nav></aside><main><article><div><p>autism is a spectrum disorder that are caused by both environmental and genetic factors.</p><p>Key Question: how can different <a href=/posts/kbhchromatin/>chromatin</a> regulators lead to the same set of symptoms named &ldquo;<a href=/posts/kbhautism/>autism</a>&rdquo;.</p><h2 id=autism--kbhautism-dot-md--gene-signature><a href=/posts/kbhautism/>autism</a> gene signature</h2><p>The gene signature of autism can be measured in clean and quantitative assays.</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=HAHAHUGOSHORTCODE136s7HBHB>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=HAHAHUGOSHORTCODE134s7HBHB>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/kbhbernoulli_random_variable/index.html
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Which emulates the behavior of your function at \(0\) and \(1\) and we kinda don&rsquo;t care any other place."><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>Bernoulli distribution</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=#properties-of-bernoulli-distribution--kbhbernoulli-random-variable-dot-md>properties of <a href=HAHAHUGOSHORTCODE156s4HBHB>Bernoulli distribution</a></a></li><li><a href=#bernoulli--kbhbernoulli-random-variable-dot-md--as-indicator><a href=HAHAHUGOSHORTCODE156s7HBHB>Bernoulli</a> as indicator</a></li><li><a href=#mle-for-bernouli>MLE for Bernouli</a></li></ul></li></ul></nav></aside><main><article><div><p>Consider a case where there&rsquo;s only a single <a href=/posts/kbhbinary_number_system/#base-2>binary</a> outcome:</p><ul><li>&ldquo;success&rdquo;, with probability \(p\)</li><li>&ldquo;failure&rdquo;, with probability \(1-p\)</li></ul><h2 id=constituents>constituents</h2><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>Bernoulli distribution</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=#properties-of-bernoulli-distribution--kbhbernoulli-random-variable-dot-md>properties of <a href=HAHAHUGOSHORTCODE152s4HBHB>Bernoulli distribution</a></a></li><li><a href=#bernoulli--kbhbernoulli-random-variable-dot-md--as-indicator><a href=HAHAHUGOSHORTCODE152s7HBHB>Bernoulli</a> as indicator</a></li><li><a href=#mle-for-bernouli>MLE for Bernouli</a></li></ul></li></ul></nav></aside><main><article><div><p>Consider a case where there&rsquo;s only a single <a href=/posts/kbhbinary_number_system/#base-2>binary</a> outcome:</p><ul><li>&ldquo;success&rdquo;, with probability \(p\)</li><li>&ldquo;failure&rdquo;, with probability \(1-p\)</li></ul><h2 id=constituents>constituents</h2><p>\begin{equation}
X \sim Bern(p)
\end{equation}</p><h2 id=requirements>requirements</h2><p>the <a href=/posts/kbhprobability_mass_function/>probability mass function</a>:</p><p>\begin{equation}
P(X=k) =
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