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2 changes: 1 addition & 1 deletion posts/kbhalpha_vector/index.html
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At every belief \(b\), there is a policy which has the highest \(U(b)\) at that \(b\) given be the alpha vector formulation."><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>alpha vector</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#additional-information>Additional Information</a><ul><li><a href=#top-action>top action</a></li><li><a href=#optimal-value-function-for-pomdp--kbhconditional-plan-dot-md--with-alpha-vector--kbhalpha-vector-dot-md><a href=HAHAHUGOSHORTCODE64s9HBHB>optimal value function for POMDP</a> with <a href=HAHAHUGOSHORTCODE64s10HBHB>alpha vector</a></a></li><li><a href=#one-step-lookahead-in-pomdp>one-step lookahead in POMDP</a></li><li><a href=#alpha-vector--kbhalpha-vector-dot-md--pruning><a href=HAHAHUGOSHORTCODE64s16HBHB>alpha vector</a> pruning</a></li></ul></li></ul></nav></aside><main><article><div><p>Recall, from <a href=/posts/kbhconditional_plan/#id-6f19368f-74b5-4606-a882-ec9bc5619873-conditional-plan-evaluation>conditional plan evaluation</a>, we had that:</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>alpha vector</h1><span class=tagbox></span></div><aside id=toc><h1 id=toc-title>table of contents</h1><nav id=TableOfContents><ul><li><a href=#additional-information>Additional Information</a><ul><li><a href=#top-action>top action</a></li><li><a href=#optimal-value-function-for-pomdp--kbhconditional-plan-dot-md--with-alpha-vector--kbhalpha-vector-dot-md><a href=HAHAHUGOSHORTCODE63s9HBHB>optimal value function for POMDP</a> with <a href=HAHAHUGOSHORTCODE63s10HBHB>alpha vector</a></a></li><li><a href=#one-step-lookahead-in-pomdp>one-step lookahead in POMDP</a></li><li><a href=#alpha-vector--kbhalpha-vector-dot-md--pruning><a href=HAHAHUGOSHORTCODE63s16HBHB>alpha vector</a> pruning</a></li></ul></li></ul></nav></aside><main><article><div><p>Recall, from <a href=/posts/kbhconditional_plan/#id-6f19368f-74b5-4606-a882-ec9bc5619873-conditional-plan-evaluation>conditional plan evaluation</a>, we had that:</p><p>\begin{equation}
U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s)
\end{equation}</p><p>let&rsquo;s write it as:</p><p>\begin{equation}
U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s) = {\alpha_{\pi}}^{\top} b
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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=HAHAHUGOSHORTCODE73s1HBHB>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=HAHAHUGOSHORTCODE97s0HBHB>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=HAHAHUGOSHORTCODE97s1HBHB>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=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}
\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/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>
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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>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=HAHAHUGOSHORTCODE143s7HBHB>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=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}
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>
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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 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=HAHAHUGOSHORTCODE150s7HBHB>parameter learning</a> in <a href=HAHAHUGOSHORTCODE150s8HBHB>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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