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# 数学代写|凸优化代写Convex Optimization代考|EE364a SEQUENTIAL HYPOTHESIS TESTING

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## 数学代写|凸优化代写Convex Optimization代考|Motivation: Election polls

Let us consider the following “practical” question.
One of $L$ candidates for an office is about to be selected by a populationwide majority vote. Every member of the population votes for exactly one candidate. How do we predict the winner via an opinion poll?
A (naive) model of the situation could be as follows. Let us represent the preference of a particular voter by his preference vector-a basic orth $e$ in $\mathbf{R}^L$ with unit entry in a position $\ell$ meaning that the voter is about to vote for the $\ell$-th candidate. The entries $\mu_{\ell}$ in the average $\mu$, over the population, of these vectors are the fractions of votes in favor of the $\ell$-th candidate, and the elected candidate is the one “indexing” the largest of the $\mu_{\ell}$ ‘s. Now assume that we select at random, from the uniform distribution, a member of the population and observe his preference vector. Our observation $\omega$ is a realization of a discrete random variable taking values in the set $\Omega=\left{e_1, \ldots, e_L\right}$ of basic orths in $\mathbf{R}^L$, and $\mu$ is the distribution of $\omega$ (technically, the density of this distribution w.r.t. the counting measure $\Pi$ on $\Omega$ ). Selecting a small threshold $\delta$ and assuming that the true – unknown to us $-\mu$ is such that the largest entry in $\mu$ is at least by $\delta$ larger than every other entry and that $\mu_{\ell} \geq \frac{1}{N}$ for all $\ell, N$ being the population size, ${ }^{13}$ we can model the population preference for the $\ell$-th candidate with
\begin{aligned} \mu \in M_{\ell} &=\left{\mu \in \mathbf{R}^d: \mu_i \geq \frac{1}{N}, \sum_i \mu_i=1, \mu_{\ell} \geq \mu_i+\delta \forall(i \neq \ell)\right} \ & \subset \mathcal{M}=\left{\mu \in \mathbf{R}^d: \mu>0, \sum_i \mu_i=1\right} \end{aligned}
In an (idealized) poll, we select at random a number $K$ of voters and observe their preferences, thus arriving at a sample $\omega^K=\left(\omega_1, \ldots, \omega_K\right)$ of observations drawn, independently of each other, from an unknown distribution $\mu$ on $\Omega$, with $\mu$ known to belong to $\bigcup_{\ell=1}^L M_{\ell}$. Therefore, to predict the winner is the same as to decide on $L$ convex hypotheses, $H_1, \ldots, H_L$, in the Discrete o.s., with $H_{\ell}$ stating that $\omega_1, \ldots, \omega_K$ are drawn, independently of each other, from a distribution $\mu \in M_{\ell}$. What we end up with, is the problem of deciding on $L$ convex hypotheses in the Discrete o.s. with $L$-element $\Omega$ via stationary $K$-repeated observations.

## 数学代写|凸优化代写Convex Optimization代考|Sequential hypothesis testing

In view of the above analysis, when predicting outcomes of “close run” elections, huge poll sizes are necessary. It, however, does not mean that nothing can be done in order to build more reasonable opinion polls. The classical related statistical idea, going back to Wald [236], is to pass to sequential tests where the observations are processed one by one, and at every instant we either accept some of our hypotheses and terminate, or conclude that the observations obtained so far are insufficient to make a reliable inference and pass to the next observation. The idea is that a properly built sequential test, while still ensuring a desired risk, will be able to make “early decisions” in the case when the distribution underlying observations is “well inside” the true hypothesis and thus is far from the alternatives. Let us show
$$\begin{array}{lll}{[\text { area A] }} & M_1 & \text { dark tetragon + light border strip: candidate A wins with margin } \geq \delta_S \ \text { [area A] } & M_1^s & \text { dark tetragon: candidate A wins with margin } \geq \delta_s>\delta_S \ \text { [area B] } & M_2 & \text { dark tetragon + light border strip: candidate } \mathrm{B} \text { wins with margin } \geq \delta_S \ \text { [area B] } & M_2^s & \text { dark tetragon: candidate B wins with margin } \geq \delta_s>\delta_S \ \text { [area C] } & M_3 & \text { dark tetragon + light border strip: candidate } \mathrm{C} \text { wins with margin } \geq \delta_S \ \text { [area C] } & M_3^s & \text { dark tetragon: candidate } \mathrm{C} \text { wins with margin } \geq \delta_s>\delta_S\end{array} \mathcal{C}s closeness: hypotheses in the tuple \left{G{2 \ell-1}^s: \mu \in M_{\ell}, G_{2 \ell}^s: \mu \in M_{\ell}^s, 1 \leq \ell \leq 3\right} are not \mathcal{C}_s-close to each other if the corresponding M-sets belong to different areas and at least one of the sets is painted dark, like M_1^s and M_2, but not M_1 and M_2. ## 凸优化代写 ## 数学代写|凸优化代写Convex Optimization代考|Motivation: Election polls 让我们考虑以下““实际”问题。 之一L-个职位的候选人即将通过全民多数票选出。人口中的每个成员都投票给一个候选人。我们如何通过民意调查预恻获胜者? 置意味着选民即状投票给 \ell-第一个候选人。参赛作品 \mu_{\ell} 平均而言 \mu ，在总体上，这些向量是支持 \ell-th candidate， and the elected candidate is the one “indexing” the largest of the \mu_{\ell} 的。现在假设我们从均㝏分布中随机选择人口中的一个成员并 交 \mathbf{R}^L ，和 \mu 是分布 \omega (从技术上讲，这个分布的密度 wrt 计数测量 \Pi \Omega 上）。选择一个小的阈值 \delta 并假设真实的一一我们不知道的 -\mu 是这样的，最大的条目 \mu 至少是由 \delta 比其他所有条目都大，并且 \mu_{\ell} \geq \frac{1}{N} 对所有人 \ell, N 作为人口规模， 13 我们可以模拟人口偏 好 \ell-th 候选人 Veft 的分隔符缺朱或无法识别 在 (理想化的) 民意调查中，我们随机选择一个数字 K 选民并观眎他们的偏好，从而得出一个样本 \omega^K=\left(\omega_1, \ldots, \omega_K\right) 来自末 知分布的相互独立的观察 \mu 上 \Omega ，和 \mu 已知属于 \bigcup_{\ell=1}^L M_{\ell}. 因此，预财获胜者与决定获胜者是一样的 L 凸假设, H_1, \ldots, H_L ，在 离散操作系统中，与 H_{\ell} 说明 \omega_1, \ldots, \omega_K 徳此独立地从分布中抽取 \mu \in M_{\ell}. 我们最终得到的是决定的问题 L 离散操作系统中的凸 假设 L- 元责 \Omega 通过固定 K – 反夏观崇。 ## 数学代写|凸优化代写Convex Optimization代考|Sequential hypothesis testing 力。 经典的相关统计思想，可以追淜到 Wald [236]，是传递哈顺序测试，在这些财试中，观究结果被逐个处理，并且在每时 刻，我们要/接受我们的一些假设并紟止，要/得出结论认为观䒺结果是这样获得的far 不足以做出可靠的推断并传递怡下一个观 离替代方案的情况下做出“早期快定”。让我们展示 \ \$$
[ area A] $\quad M_1$ dark tetragon + light border strip: candidate A wins with margin $\geq \delta_S$ [area A] $\quad M_1^s \quad$ dark tetragon: candidate A wins with
$\mathcal{C}$ s接近性: 元组中的假设 $\backslash$ left 的分隔符缺失或无法识别 $\quad$ 不是 $\mathcal{C}_s$-如果对应，则彼此接近 $M$ – 集合属于不同的

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。