% 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{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, University of Washington: \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.pdf} \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: