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new_math.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\end_header
\begin_body
\begin_layout Itemize
Lemma 1:
\begin_inset Formula
\begin{align*}
Binom(x|n,p) & ={n \choose x}p^{x}(1-p)^{n-x}\\
Beta(p|x,n-x) & =\frac{p^{x-1}(1-p)^{n-x-1}}{B(x,n-x)}\\
& \therefore\\
Binom(x|n,p) & \propto Beta(p|x+1,n-x+1)\\
\frac{Binom(x|n,p)}{Beta(p|x+1,n-x+1)} & ={n \choose x}B(x+1,n-x+1)
\end{align*}
\end_inset
\end_layout
\begin_layout Itemize
Lemma 2:
\begin_inset Formula $\int_{0}^{U}dxBeta(x|\alpha,\beta)=\frac{B(U;\alpha,\beta)}{B(\alpha,\beta)}$
\end_inset
\end_layout
\begin_layout Itemize
Derivation of joint likelihood:
\begin_inset Formula
\begin{align*}
& p(V_{1},R_{1},V_{2,}R_{2}|M_{i})=\int d\phi_{1}d\phi_{2}p(V_{1},R_{1},V_{2,}R_{2},\phi_{1},\phi_{2}|M_{i})\\
& =\int d\phi_{1}d\phi_{2}p(V_{1},R_{1}|\phi_{1})p(V_{2},R_{2}|\phi_{2})p(\phi_{1},\phi_{2}|M_{i})\\
& =\int d\phi_{1}d\phi_{2}p(V_{1},R_{1}|\phi_{1})p(V_{2},R_{2}|\phi_{2})p(\phi_{2}|\phi_{1},M_{i})p(\phi_{1}|M_{i})\\
& =\int d\phi_{1}p(V_{1},R_{1}|\phi_{1})p(\phi_{1}|M_{i})\int d\phi_{2}p(V_{2},R_{2}|\phi_{2})p(\phi_{2}|\phi_{1},M_{i})
\end{align*}
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\text{If \ensuremath{M_{i}=\text{cocluster}:}}$
\end_inset
\begin_inset Formula
\begin{align*}
p(\phi_{2}|\phi_{1},M_{i})= & I(\phi_{1}=\phi_{2})\\
p(\phi_{1}|M_{i})= & \frac{p(\phi_{1},\phi_{2}|M_{i})}{p(\phi_{2}|\phi_{1},M_{i})}=\\
= & \frac{I(\phi_{1}=\phi_{2})}{I(\phi_{1}=\phi_{2})}\\
= & 1\\
p(V_{1},R_{1},V_{2,}R_{2}|M_{i})= & \int d\phi_{1}p(V_{1},R_{1}|\phi_{1})p(\phi_{1}|M_{i})\int d\phi_{2}p(V_{2},R_{2}|\phi_{2})p(\phi_{2}|\phi_{1},M_{i})\\
= & \int d\phi_{1}p(V_{1},R_{1}|\phi_{1})p(V_{2},R_{2}|\phi_{1})\\
= & \int d\phi_{1}Binom(V_{1}|V_{1}+R_{1},\omega_{v_{1}}\phi_{1})Binom(V_{2}|V_{2}+R_{2},\omega_{v_{2}}\phi_{1})
\end{align*}
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\text{If \ensuremath{M_{i}=\text{garbage}:}}$
\end_inset
\begin_inset Formula
\begin{align*}
p(\phi_{1},\phi_{2}|M_{i}) & =p(\phi_{1}|M_{i})p(\phi_{2}|M_{i})\\
& =1\cdot1\\
p(V_{1},R_{1},V_{2},R_{2}|M_{i}) & =\int d\phi_{1}d\phi_{2}p(V_{1},R_{1}|\phi_{1})p(V_{2},R_{2}|\phi_{2})p(\phi_{1},\phi_{2}|M_{i})\\
& =\int d\phi_{1}p(V_{1},R_{1}|\phi_{1})\int d\phi_{2}p(V_{2},R_{2}|\phi_{2})\\
& =\int d\phi_{1}Binom(V_{1}|V_{1}+R_{1},\omega_{v_{1}}\phi_{1})\int d\phi_{2}Binom(V_{2}|V_{2}+R_{2},\omega_{v_{2}}\phi_{2})\\
& ={V_{1}+R_{1} \choose V_{1}}B(V_{1}+1,R_{1}+1){V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\int_{0}^{1}d\phi_{1}Beta(\omega_{v_{1}}\phi_{1}|V_{1}+1,R_{1}+1)\int_{0}^{1}d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)\\
& ={V_{1}+R_{1} \choose V_{1}}B(V_{1}+1,R_{1}+1){V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\frac{1}{\omega_{v_{1}}\omega_{v_{2}}}\int_{0}^{\omega_{v_{1}}}dP_{1}Beta(P_{1}|V_{1}+1,R_{1}+1)\int_{0}^{\omega_{v_{2}}}dP_{2}Beta(P_{2}|V_{2}+1,R_{2}+1)\\
& ={V_{1}+R_{1} \choose V_{1}}B(V_{1}+1,R_{1}+1){V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\frac{1}{\omega_{v_{1}}\omega_{v_{2}}}\frac{B(\omega_{v_{1}};V_{1}+1,R_{1}+1)}{B(V_{1}+1,R_{1}+1)}\frac{B(\omega_{v_{2}};V_{2}+1,R_{2}+1)}{B(V_{2}+1,R_{2}+1)}\\
& =\frac{1}{\omega_{v_{1}}\omega_{v_{2}}}{V_{1}+R_{1} \choose V_{1}}{V_{2}+R_{2} \choose V_{2}}B(\omega_{v_{1}};V_{1}+1,R_{1}+1)B(\omega_{v_{2}};V_{2}+1,R_{2}+1)
\end{align*}
\end_inset
\end_layout
\begin_layout Itemize
If
\begin_inset Formula $M_{i}\in\{A\rightarrow B,B\rightarrow A,\text{diff\_branches\}:}$
\end_inset
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Formula
\begin{align*}
p(\phi_{2}|\phi_{1},M_{i}) & =\frac{1}{U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i})}I(L_{\phi_{2}}(\phi_{1},M_{i})\leq\phi_{2}\leq U_{\phi_{2}}(\phi_{1},M_{i}))\\
& \text{(\ensuremath{U} and \ensuremath{L} define upper and lower bounds on \ensuremath{\phi_{2})}}\\
p(V_{1},R_{1},V_{2,}R_{2}|M_{i}) & =\int d\phi_{1}p(V_{1},R_{1}|\phi_{1})p(\phi_{1}|M_{i})\frac{1}{U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i})}\int_{L_{\phi_{2}}(\phi_{1},M_{i})}^{U_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}p(V_{2},R_{2}|\phi_{2})\\
\int d\phi_{2}p(V_{2},R_{2}|\phi_{2}) & =\int d\phi_{2}Binom(V_{2}|V_{2}+R_{2},\omega_{v_{2}}\phi_{2})\\
& =\int d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1){V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\\
& ={V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\int d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)\\
& \text{Let \ensuremath{P_{2}=\omega_{v_{2}}\phi_{2}.}Then \ensuremath{\frac{dP_{2}}{d\phi_{2}}=\omega_{v_{2}},}and:}\\
\int d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)= & \frac{1}{\omega_{v_{2}}}\int\Bigg(d\phi_{2}\omega_{v_{2}}\Bigg)Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)\\
= & \frac{1}{\omega_{v_{2}}}\int dP_{2}Beta(P_{2}|V_{2}+1,R_{2}+1)\\
\int_{L_{\phi_{2}}(\phi_{1},M_{i})}^{U_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}p(V_{2},R_{2}|\phi_{2})= & \int_{0}^{U_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}p(V_{2},R_{2}|\phi_{2})-\int_{0}^{L_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}p(V_{2},R_{2}|\phi_{2})\\
= & {V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\\
& \Bigg(\int_{0}^{U_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)-\int_{0}^{L_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}Beta(\omega_{v_{2}}\phi_{2}|V_{2}+1,R_{2}+1)\Bigg)\\
= & {V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\\
& \Bigg(\frac{1}{\omega_{v_{2}}}\int_{0}^{\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i})}dP_{2}Beta(P_{2}|V_{2}+1,R_{2}+1)-2\int_{0}^{\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i})}dP_{2}Beta(P_{2}|V_{2}+1,R_{2}+1)\Bigg)\\
= & {V_{2}+R_{2} \choose V_{2}}B(V_{2}+1,R_{2}+1)\frac{1}{\omega_{v_{2}}B(V_{2}+1,R_{2}+1)}\\
& \Bigg(B(\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)-B(\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)\Bigg)\\
=\frac{1}{\omega_{v_{2}}} & {V_{2}+R_{2} \choose V_{2}}\Bigg(B(\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)-B(\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)\Bigg)\\
p(\phi_{1}|M_{i}) & =\int d\phi_{2}p(\phi_{1}\phi_{2}|M_{i})\\
& =\int_{L_{\phi_{2}(\phi_{1},M_{i})}}^{U_{\phi_{2}(\phi_{1},M_{i})}}d\phi_{2}2\\
& =2\phi_{2}\Big|_{L_{\phi_{2}(\phi_{1},M_{i})}}^{U_{\phi_{2}(\phi_{1},M_{i})}}\\
& =2(U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i}))\\
\text{Alternative derivation:}\\
p(\phi_{1},\phi_{2}|M_{i}) & =2I(0\leq\phi_{2}\leq\phi_{1}\leq1)\\
& =p(\phi_{2}|\phi_{1},M_{i})p(\phi_{1}|M_{i})\\
p(\phi_{1}|M_{i}) & =\frac{p(\phi_{1},\phi_{2}|M_{i})}{p(\phi_{2}|\phi_{1},M_{i})}\\
& =\frac{2I(0\leq\phi_{2}\leq\phi_{1}\leq1)}{\frac{1}{U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i})}I(0\leq\phi_{2}\leq\phi_{1}\leq1)}\\
& =2(U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i}))\\
p(V_{1},R_{1},V_{2,}R_{2}|M_{i})= & \int d\phi_{1}p(V_{1},R_{1}|\phi_{1})p(\phi_{1}|M_{i})\frac{1}{U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i})}\int_{L_{\phi_{2}}(\phi_{1},M_{i})}^{U_{\phi_{2}}(\phi_{1},M_{i})}d\phi_{2}p(V_{2},R_{2}|\phi_{2})\\
= & \int d\phi_{1}Binom(V_{1}|V_{1}+R_{1},\omega_{v_{1}}\phi_{1})p(\phi_{1}|M_{i})\frac{1}{U_{\phi_{2}}(\phi_{1},M_{i})-L_{\phi_{2}}(\phi_{1},M_{i})}\\
& \frac{{V_{2}+R_{2} \choose V_{2}}}{\omega_{v_{2}}}\Bigg(B(\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)-B(\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)\Bigg)\\
= & \int d\phi_{1}Binom(V_{1}|V_{1}+R_{1},\omega_{v_{1}}\phi_{1})2\frac{{V_{2}+R_{2} \choose V_{2}}}{\omega_{v_{2}}}\\
& \Bigg(B(\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)-B(\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)\Bigg)\\
= & 2\frac{{V_{2}+R_{2} \choose V_{2}}}{\omega_{v_{2}}}\int d\phi_{1}Binom(V_{1}|V_{1}+R_{1},\omega_{v_{1}}\phi_{1})\\
& \Bigg(B(\omega_{v_{2}}U_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)-B(\omega_{v_{2}}L_{\phi_{2}}(\phi_{1},M_{i});V_{2}+1,R_{2}+1)\Bigg)
\end{align*}
\end_inset
\end_layout
\end_body
\end_document