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\fancyhead[C]{COMP 136 - 2020s - HW5}
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\begin{document}
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\Large{\bf Student Name: TODO}

\Large{\bf Collaboration Statement:}

Total hours spent: TODO hours

I consulted the following resources:
\begin{itemize}
\item TODO
\end{itemize}

\tableofcontents

\newpage
\officialdirections{
\subsection*{1a: Problem Statement}
Give an expression for the Jacobian $S'(x) = \frac{d}{dx} S(x)$, as a function of $x, \mu, \sigma$, for the specific target-to-source function $S(x) = \frac{1}{\sigma}(x - \mu)$ defined above.
}

\subsection{1a: Solution}

TODO

\newpage
\officialdirections{
\subsection*{1b: Problem Statement}
Show that the pdf of $x = T(u) = \sigma u + \mu$ is given by:
\begin{align}
p(x) = \frac{1}{(2\pi)^{\frac{1}{2}}} \frac{1}{\sigma} e^{-\frac{1}{2} \frac{1}{\sigma^2} (x-\mu)^2}
\end{align}
}


\subsection{1b: Solution}

TODO



\newpage
\officialdirections{
\subsection*{2a: Problem Statement}
 First, give a compact expression for the Jacobian matrix $J_S(x)$ as a function of $x, \mu, L$, for the specific target-to-source function $S(x) = L^{-1} (x - \mu)$ defined above.
}

\subsection{2a: Solution}
TODO




\newpage
\officialdirections{
\subsection*{2b: Problem Statement}
Define $\Sigma = L L^T$. Show the following:
\begin{align}
( L^{-1} (x - \mu))^T ( L^{-1} (x - \mu)) = (x-\mu)^T \Sigma^{-1} (x-mu)
\end{align}
}

\subsection{2b: Solution}
TODO


\newpage
\officialdirections{
\subsection*{2c: Problem Statement}
 Define $\Sigma = L L^T$. Show the following:
\begin{align}
| \det (L^{-1}) | = \frac{1}{(\text{det} \Sigma)^{\frac{1}{2}}}
\end{align}
}

\subsection{2c: Solution}
TODO


 \newpage
\officialdirections{
\subsection*{2d: Problem Statement}
Show that the pdf of $x = T(u) = L u + \mu$ is given by:
\begin{align}
p(x) = \frac{1}{(2\pi)^{\frac{D}{2}}} \frac{1}{\text{det}(\Sigma)^{\frac{1}{2}}} e^{-\frac{1}{2} (x-\mu)^T \Sigma^{-1} (x-\mu)}
\end{align}
where we define $\Sigma = L L^T$. 
}

\subsection{2d: Solution}
TODO

\newpage
\officialdirections{
\subsection*{3a: Problem Statement}
Show the following for any $x > 0, y > 0$:
\begin{align}
\frac
  { \min \left( 1, \frac{x}{y} \right) }
  { \min \left( 1, \frac{y}{x} \right) }
=
\frac{x}{y}
\end{align}
}

\subsection{3a: Solution}
TODO

\newpage
\officialdirections{
\subsection*{3b: Problem Statement}
If $\mathcal{T}$ is the Metropolis-Hastings transition probability distribution, show that it means the detailed balance condition with respect to $p^*$ for any $a,b$ such that $a \neq b$:
\begin{align}
p^*( a) \mathcal{T}( b | a)  = p^*(b) \mathcal{T}( a | b)
\end{align}
}

\subsection{3b: Solution}
TODO

\end{document}