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+\newtheorem{remark}[thm]{Remark} +\newtheorem{defn}[thm]{Definition} + +%%%%%%%%%%%%%%%%%%%% Notation stuff +\newcommand{\pr}{\operatorname{Pr}} %% probability +\newcommand{\vr}{\operatorname{Var}} %% variance +\newcommand{\rs}{X_1, X_2, \ldots, X_n} %% random sample +\newcommand{\irs}{X_1, X_2, \ldots} %% infinite random sample +\newcommand{\rsd}{x_1, x_2, \ldots, x_n} %% random sample, realised +\newcommand{\bX}{\boldsymbol{X}} %% random sample, contracted form (bold) +\newcommand{\bx}{\boldsymbol{x}} %% random sample, realised, contracted form (bold) +\newcommand{\bT}{\boldsymbol{T}} %% Statistic, vector form (bold) +\newcommand{\bt}{\boldsymbol{t}} %% Statistic, realised, vector form (bold) +\newcommand{\emv}{\hat{\theta}} +\DeclarePairedDelimiter\ceil{\lceil}{\rceil} +\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} + +% Title Page +\title{Exam 2 (A2)} +\author{Class: Bayesian Statistics \\ Instructor: Luiz Max Carvalho} +\date{02/06/2021} + +\begin{document} +\maketitle + +\textbf{Turn in date: until 16/06/2021 at 23:59h Brasilia Time.} + +\begin{center} +\fbox{\fbox{\parbox{1.0\textwidth}{\textsf{ + \begin{itemize} + \item Please read through the whole exam before starting to answer; + \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 You can write your answer to a question as a point-by-point response or in ``essay'' form, your call; + \item Please hand in a single, \textbf{typeset} ( \LaTeX) PDF file as your final main document. + Code appendices are welcome,~\textit{in addition} to the main PDF document. + \item You may consult any sources, provided you cite \textbf{ALL} of your sources (books, papers, blog posts, videos); + \item You may use symbolic algebra programs such as Sympy or Wolfram Alpha to help you get through the hairier calculations, provided you cite the tools you have used. + \item The exam is worth 100 %$\min\left\{\text{your\:score}, 100\right\}$ + marks. + \end{itemize}} +}}} +\end{center} +% \newpage +% \section*{Hints} +% \begin{itemize} +% \item a +% \item b +% \end{itemize} +% +\newpage + +\section*{Background} + +This exam covers applications, namely estimation, prior sensitivity and prediction. +You will need a working knowledge of basic computing tools, and knowledge of MCMC is highly valuable. +Chapter 6 in \cite{Robert2007} gives an overview of computational techniques for Bayesian statistics. + +\section*{Inferring population sizes -- theory} + +Consider the model +\begin{equation*} + x_i \sim \operatorname{Binomial}(N, \theta), +\end{equation*} +with \textbf{both} $N$ and $\theta$ unknown and suppose one observes $\boldsymbol{x} = \{x_1, x_2, \ldots, x_K\}$. +Here, we will write $\xi = (N, \theta)$. + +\begin{enumerate}[label=\alph*)] + \item (10 marks) Formulate a hierarchical prior ($\pi_1$) for $N$, i.e., elicit $F$ such that $N \mid \alpha \sim F(\alpha)$ and $\alpha \sim \Pi_A$. + Justify your choice; + \item (5 marks) Using the prior from the previous item, write out the full joint posterior kernel for all unknown quantities in the model, $p(\xi \mid \boldsymbol{x})$. \textit{Hint:} do not forget to include the appropriate indicator functions!; + \item (5 marks) Is your model identifiable? + \item (5 marks) Exhibit the marginal posterior density for $N$, $p_1(N \mid \boldsymbol{x})$; + \item (5 marks) Return to point (a) above and consider an alternative, uninformative prior structure for $\xi$, $\pi_2$. + Then, derive $p_2(N \mid \boldsymbol{x})$; + \item (10 marks) Formulate a third prior structure on $\xi$, $\pi_3$, that allows for the closed-form marginalisation over the hyperparameters $\alpha$ -- see (a) -- and write out $p_3(N \mid \boldsymbol{x})$; + \item (10 marks) Show whether each of the marginal posteriors considered is proper. + Then, derive the posterior predictive distribution, $g_i(\tilde{x} \mid \boldsymbol{x})$, for each of the posteriors considered ($i = 1, 2, 3$). + \item (5 marks) Consider the loss function + \begin{equation} + \label{eq:relative_loss} + L(\delta(\boldsymbol{x}), N) = \left(\frac{\delta(\boldsymbol{x})-N}{N} \right)^2. + \end{equation} + Derive the Bayes estimator under this loss. +\end{enumerate} + +\section*{Inferring population sizes -- practice} +Consider the problem of inferring the population sizes of major herbivores~\citep{Carroll1985}. +In the first case, one is interested in estimating the number of impala (\textit{Aepyceros melampus}) herds in the Kruger National Park, in northeastern South Africa. +In an initial survey collected the following numbers of herds: $\boldsymbol{x}_{\text{impala}} = \{15, 20, 21, 23, 26\}$. +Another scientific question is the number of individual waterbuck (\textit{Kobus ellipsiprymnus}) in the same park. +The observed numbers of waterbuck in separate sightings were $\boldsymbol{x}_{\text{waterbuck}} = \{53, 57, 66, 67, 72\}$ and may be regarded (for simplicity) as independent and identically distributed. + +\begin{figure}[H] + \centering + \begin{subfigure}[b]{0.45\textwidth} + \centering + \includegraphics[scale=0.75]{figures/impala.jpeg} + \caption{Impala} + \end{subfigure} + \begin{subfigure}[b]{0.45\textwidth} + \centering + \includegraphics[scale=0.75]{figures/waterbuck.jpeg} + \caption{Waterbuck} + \end{subfigure} + \caption{Two antelope species whose population sizes we want to estimate.} + \label{fig:antelopes} +\end{figure} + + +\begin{enumerate}[label=\alph*)] +\setcounter{enumi}{8} + \item (20 marks) For each data set, sketch the marginal posterior distributions $p_1(N \mid \boldsymbol{x})$, $p_2(N \mid \boldsymbol{x})$ and $p_3(N \mid \boldsymbol{x})$. + Moreover, under each posterior, provide (i) the Bayes estimator under quadratic loss and under the loss in (\ref{eq:relative_loss}) and (ii) a 95\% credibility interval for $N$. + Discuss the differences and similarities between these distributions and estimates: do the prior modelling choices substantially impact the final inferences? If so, how? + \item (25 marks) Let $\bar{x} = K^{-1}\sum_{k =1}^K x_k$ and $s^2 = K^{-1}\sum_{k =1}^K (x_k-\bar{x})^2$. + For this problem, a sample is said to be \textit{stable} if $\bar{x}/s^2 \geq (\sqrt{2} + 1)/\sqrt{2}$ and \textit{unstable} otherwise. + Devise a simple method of moments estimator (MME) for $N$. + Then, using a Monte Carlo simulation, compare the MME to the three Bayes estimators under quadratic loss (\ref{eq:relative_loss}) in terms of relative mean squared error. + How do the Bayes estimators compare to MME in terms of the statibility of the generated samples? + \textit{Hint}: You may want to follow the simulation setup of~\cite{Carroll1985}. +\end{enumerate} + +\bibliographystyle{apalike} +\bibliography{a2} +\end{document} diff --git a/assignments/a2.bib b/assignments/a2.bib new file mode 100644 index 0000000..30b97ee --- /dev/null +++ b/assignments/a2.bib @@ -0,0 +1,16 @@ +@article{Carroll1985, + title={A note on {N} estimators for the binomial distribution}, + author={Carroll, Raymond J and Lombard, F}, + journal={Journal of the American Statistical Association}, + volume={80}, + number={390}, + pages={423--426}, + year={1985}, + publisher={Taylor \& Francis} +} +@book{Robert2007, + title={The {B}ayesian choice: from decision-theoretic foundations to computational implementation}, + author={Robert, Christian}, + year={2007}, + publisher={Springer Science \& Business Media} +} diff --git a/assignments/figures/impala.jpeg b/assignments/figures/impala.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..91050c8ca03135f5ac51b77bb6b4b4cd4aa8536e GIT binary patch literal 9522 zcmZX&cQD*<)HeQEy|enRw%DkNURPVuMJI^fqGX9KdaPcfvj|awC2I8OtQrw5IzggG z7ZH(ozt6lgzcSBt|97tI+~=IR&&)Y<-!0y405sa_TIv7@1OTA_4sf>uL<3~RB#ev< z%uI}o%n)WM>;DCmgOi<=of!&cg+e)4IYsVqvI}xTnR#S*1cgNVVn z|MVsigM&}T8TX9wKxFBA_$SE3xU`(JRcTHOH~zIVW(g^N|broKlgYu~Wx6YG0%`He;j%B8Uk zDNB*IY%Z(A_Wv|7nFmMu^t`^0#}#qE{!$@#t@JH4d4EF9Qok_~kY?ybzJaf*OcqEQ 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