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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
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+% Sample tex file for usage of iidef.sty
+% Homework template for Inference and Information
+% UPDATE: October 12, 2017 by Xiangxiang
+% UPDATE: 22/03/2018 by zhaofeng-shu33
+\documentclass[a4paper]{article}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath, amssymb, amsthm}
+% amsmath: equation*, amssymb: mathbb, amsthm: proof
+\usepackage{moreenum}
+\usepackage{mathtools}
+\usepackage{url}
+\usepackage[outdir=./]{epstopdf}
+\usepackage{graphicx}
+\usepackage{subcaption}
+\usepackage{booktabs} % toprule
+\usepackage[mathcal]{eucal}
+\usepackage{dsfont}
+
+\usepackage[numbered,framed]{matlab-prettifier}
+\lstset{
+ style = Matlab-editor,
+ captionpos =b,
+ basicstyle = \mlttfamily,
+ escapechar = ",
+ mlshowsectionrules = true,
+}
+
+\usepackage[thehwcnt = 6]{iidef}
+\thecourseinstitute{Tsinghua-Berkeley Shenzhen Institute}
+\thecoursename{Information Inference}
+\theterm{Fall 2017}
+\hwname{Coursework}
+\begin{document}
+\courseheader
+\name{YOUR NAME}
+\rule{\textwidth}{1pt}
+\begin{itemize}
+\item {\bf Acknowledgments: \/}
+ This template takes some materials from course CSE 547/Stat 548 of Washington University: \small{\url{https://courses.cs.washington.edu/courses/cse547/17sp/index.html}}.
+
+ If you refer to other materials in your homework, please list here.
+\item {\bf Collaborators: \/}
+ I finish this template by myself. If you finish your homework all by yourself, make a similar statement. If you get help from others in finishing your homework, state like this:
+ \begin{itemize}
+ \item 1.2 (b) was solved with the help from \underline{\hspace{3em}}.
+ \item Discussion with \underline{\hspace{3em}} helped me finishing 1.3.
+ \end{itemize}
+\end{itemize}
+\rule{\textwidth}{1pt}
+
+\vspace{2em}
+
+You may use \texttt{enumerate} to generate answers for each question:
+
+\begin{enumerate}
+ \setlength{\itemsep}{3\parskip}
+
+ \item Type of commonly used notations. Use another \texttt{enumerate} to start generate answers for sub-questions:
+ \begin{enumerate}
+ \item Use \verb|$ $| to get an inline equation: $\Prob(A) = \E[\1_A(\omega)]$.
+ \item Use \texttt{equation} to have equation in display math mode:
+ \begin{equation}
+ \frac{a + b}{2} \geq \sqrt{ab}
+ \label{eq:1}
+ \end{equation}
+
+ \item Use \verb|\eqref| to get reference for equations: \eqref{eq:1} holds when $a\geq 0, b\geq 0$.
+
+ \item Now we would introduce some commonly used notations:
+ \begin{enumerate}
+ \item Use \verb|\mathbb{P}, \mathbb{R}, \mathbb{E}| to type $\mathbb{P}, \mathbb{R}, \mathbb{E}$.
+ \item Use \verb|\mathcal{A}, \mathcal{X}, \mathcal{Y}, \mathcal{N}| to type $\mathcal{A}, \mathcal{X}, \mathcal{Y}, \mathcal{N}$.
+ \item Use \verb|\underline{x}, \underline{y}| to type vectors $\underline{x}, \underline{y}$.
+ \item Use \verb|\mathsf{x}, \mathsf{y}, \mathsf{z}| to type random variables $\rvx, \rvy, \rvz$. For simplicity, I have defined several macros so you could simply type \verb|\rvx, \rvy, \rvz|. Don't forget \verb|$ $|!
+ \item Thanks to these macros, we could have $\reals, \E[\rvx], \Var(\rvy), \Prob(A), \independent, \1$ by typing \verb|\reals, \E[\rvx], \Var(\rvy), \Prob(A),\independent,| \verb|\1|.
+ \item Now you can use \verb|\ux, \uy, \uz| to type vectors $\ux, \uy, \uz$, and use \verb|\urvx, \urvy, \urvz| to type random vectors $\urvx, \urvy, \urvz$.
+ \item Remember that $P_{\rvx|\rvy}(x|y) \defas \Prob(\rvx = x|\rvy = y)$.
+ \begin{enumerate}
+ \item Writing $\Prob(x)$ is wrong. $\Prob$ should only operate on events.
+ \item $\rvx$ is a random variable, while $x$ is a real number.
+ \end{enumerate}
+
+ \end{enumerate}
+ \item You may find \url{https://en.wikibooks.org/wiki/LaTeX} useful.
+ \item Writing \LaTeX\ online may be easier for beginners:
+ \begin{enumerate}
+ \item ShareLaTeX: \url{https://www.sharelatex.com/}.
+ \item Overleaf: \url{https://www.overleaf.com/}.
+ \end{enumerate}
+ \end{enumerate}
+ \item You may need aligned equations for your homework, here are several examples:
+
+ Total propability rule:
+ \begin{equation*}
+ \begin{aligned}
+ \Prob(\rvx = x)
+ &= \sum_{y \in \mathcal{Y}} \Prob(\rvx = x, \rvy = y)\\
+ &= \sum_{y \in \mathcal{Y}} \Prob(\rvx = x| \rvy = y) \Prob(\rvy = y),\\
+ \end{aligned}
+ \end{equation*}
+ or
+ \begin{equation*}
+ \begin{aligned}
+ &\quad~ P_{\rvx}(x)\\
+ &= \sum_{y \in \mathcal{Y}} P_{\rvx\rvy}(x,y)\\
+ &= \sum_{y \in \mathcal{Y}} P_{\rvx|\rvy}(x|y)P_{\rvy}(y).\\
+ \end{aligned}
+ \end{equation*}
+ Indicator function:
+ \begin{equation*}
+ \1_A(\omega)=
+ \left\{
+ \begin{aligned}
+ 1, &\quad\text{if}~ \omega \in A,\\
+ 0, &\quad\text{if}~ \omega \notin A.
+ \end{aligned}
+ \right.
+ \end{equation*}
+
+ \item You may need to add figure and source codes in your homework. Figure \ref{fig:1} is an example that compares the empirical distribution (histogram) and probability density function of the Gaussian random variable.
+ \begin{figure}[htbp]
+ \centering
+ \includegraphics[width = 0.8\textwidth]{pdf_normal.eps}
+ \caption{Gaussian PDF and histogram of samples}
+ \label{fig:1}
+ \end{figure}
+
+ The source code to plot Figure \ref{fig:1} could be found in Appendix \ref{sec:a:code}. Here are the core codes:
+ \lstinputlisting[firstline=4,lastline=4, firstnumber=4]{matlabscript.m}
+ \lstinputlisting[firstline=6,lastline=7, firstnumber=6]{matlabscript.m}
+ To understand line 6, note that if we have $n$ samples of $\rvx$ denoted by $x^{(i)}, i = 1, 2, \cdots, n$, then the probability density function $p_{\rvx}$ could be estimated as
+ \begin{equation*}
+ \begin{aligned}
+ p_{\rvx}(x_0) &= \left.\frac{\mathrm{d}}{\mathrm{d}x} \Prob(\rvx \leq x) \right|_{x = x_0} \\
+ &\approx \frac{\Prob(x_0 - \Delta x < \rvx \leq x_0)}{\Delta x}\\
+ &\approx \frac{1}{n\Delta x} \sum_{i = 1}^n \1_{x^{(i)} \in (x_0 - \Delta x, x_0]}.
+ \end{aligned}
+ \end{equation*}
+
+\item An example of hypothesis testing:
+ \begin{equation*}
+ \log \frac{\Prob(\rvH = H_1|\rvy = y)}{\Prob(\rvH = H_0|\rvy = y)}
+ \mathop{\gtreqless}_{\hat{\rvH} = H_0}^{\hat{\rvH} = H_1} \gamma
+ \end{equation*}
+
+ \end{enumerate}
+
+
+ \newpage
+
+ \appendix
+ \section{Source code}
+ \label{sec:a:code}
+ % \lstlistoflistings
+ Source code for plotting Figure \ref{fig:1} is shown as follows.
+ \lstinputlisting[caption=FigurePlot]{matlabscript.m}
+
+\end{document}
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End: