Finally, T1 is in da house

Author: maxbiostat

assignments/CompStats_T1_2022.pdf

@@ -0,0 +1,106 @@
+\documentclass[a4paper,10pt, notitlepage]{report}
+\usepackage[utf8]{inputenc}
+\usepackage{natbib}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage[shortlabels]{enumitem}
+% \usepackage[portuguese]{babel}
+
+
+% Title Page
+\title{O Brother, How Far Art Thou?}
+\author{Computational Statistics \\ Instructor: Luiz Max de Carvalho}
+
+\begin{document}
+\maketitle
+
+\textbf{Hand-in date: 06/10/2021.}
+
+\section*{General guidance}
+\begin{itemize}
+ \item State and prove all non-trivial mathematical results necessary to substantiate your arguments;
+ \item Do not forget to add appropriate scholarly references~\textit{at the end} of the document;
+ \item Mathematical expressions also receive punctuation;
+ \item Please hand in a single PDF file as your final main document.
+ 
+ Code appendices are welcome,~\textit{in addition} to the main PDF document.
+ \end{itemize}
+
+\newpage
+
+\section*{Background}
+
+A large portion of the content of this course is concerned with computing high-dimensional integrals~\textit{via} simulation.
+Today you will be introduced to a simple-looking problem with a complicated closed-form solution and one we can approach using simulation.
+
+Suppose you have a disc $C_R$ of radius $R$. 
+Take $p = (p_x, p_y)$ and $ q = (q_x, q_y) \in C_R$ two points in the disc.  
+Consider the Euclidean distance between $p$  and $q$, $||p-q|| = \sqrt{(p_x-q_x)^2 + (p_y-q_y)^2} = |p-q|$.
+\paragraph{Problem A:} What is the \textit{average} distance between pairs of points in $C_R$ if they are picked uniformly at random?
+
+\section*{Part I: nuts and bolts}
+
+\begin{enumerate}
+ \item To start building intuition, let's solve a related but much simpler problem.
+ Consider an interval $[0, s]$, with $s>0$ and take $x_1,x_2 \in [0, s]$~\textit{uniformly at random}.
+ Show that the average distance between $x_1$ and $x_2$ is $s/3$.
+ \item Show that Problem A is equivalent to computing
+ \begin{equation*}
+  I = \frac{1}{\pi^2 R^4}\int_{0}^{R}\int_{0}^{R}\int_{0}^{2\pi}\int_{0}^{2\pi}\sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos\phi(\theta_1, \theta_2)}r_1r_2\,d\theta_1\,d\theta_2\,dr_1\,dr_2,
+ \end{equation*}
+ where $\phi(\theta_1, \theta_2)$ is the central angle between $r_1$ and $r_2$.
+ 
+ \textit{Hint:} Draw a picture.
+ \item Compute $I$ in closed-form.
+
+ \textit{Hint:} Look up \textit{Crofton's mean value theorem} or \textit{Crofton's formula}. 
+\end{enumerate}
+
+\section*{Part II -- getting your hands dirty}
+
+Now we will move on to implementation.
+
+\paragraph{Problem B:} Employ a simulation algorithm to approximate $I$.
+Provide point and interval estimates and give theoretical guarantees about them (consistency, coverage, etc).
+
+\begin{enumerate}
+ \item You have been (randomly) assigned a simulation method -- see list at the end.
+ Represent $I$ as $\int_{\mathcal{X}} \phi(x)\pi(x)\,dx$ and justify your choice of $\phi$, $\pi$ and $\mathcal{X}$.
+ Recall that these choices are arbitrary up to a point, but they might lead to wildly different empirical performances~\textbf{and} theoretical properties for estimators of $I$.
+ \textbf{Justify} your choices in light of the method you have been given to work with.
+ Choose wisely and be rigorous in your justifications.
+ \item Again, starting from the eventual samples you will obtain with your method, construct a non-empty\footnote{This is a joke.
+ It means you should come up with at least one estimator. But you might, and are even encouraged to, entertain more than one estimator.} family of estimators of $I$ and discuss whether it is (strongly) consistent and whether a central limit theorem can be established.
+ \item Detail a suite of diagnostics that might be employed in your application to detect convergence or performance problems.
+ Extra points for those who design algorithms that exploit the structure of this particular integration problem. 
+ \item For each $R \in \{0.01, 0.1, 1, 10, 100, 1000, 10000\}$, perform $M=500$ runs from your simulation method and compute: (i) variance (ii) bias (iii) standard deviation of the mean (MCSE).
+ \item Can you identify one key quantity missing from the previous item?
+ \textit{Hint:} it bears relevance to the real world application of any computational method.
+estimator.\end{enumerate}
+
+Here we will list a selection of methods that will be randomly assigned to each student, along with some questions that need to be answered for that particular method.
+
+\begin{itemize}
+ \item \textbf{Rejection sampling}
+  \begin{itemize}
+  \item Justify your choice of proposal distribution and show that it conforms to the necessary conditions for the algorithm to work; in particular, try to find a proposal that gives the highest acceptance probability.
+ \end{itemize}
+  \item \textbf{Importance sampling}
+ \begin{itemize}
+  \item Justify your choice of proposal based on the variance of the resulting estimator.
+ \end{itemize}
+  \item \textbf{Gibbs sampling}
+ \begin{itemize}
+  \item Write your full conditionals out and show that they adhere to the Hammersley-Clifford condition.
+ \end{itemize}
+  \item \textbf{Metropolis-Hastings}
+ \begin{itemize}
+  \item Justify your choice of proposal; test different ones if you need to.
+ \end{itemize}
+\end{itemize}
+% 
+% \bibliographystyle{apalike}
+% \bibliography{refs}
+
+\end{document}          
+