凸优化第七章统计估计作业题

最新推荐文章于 2025-10-10 18:20:20 发布

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最新推荐文章于 2025-10-10 18:20:20 发布 · 588 阅读

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本文探讨了极大似然估计在高斯分布中的应用，介绍了最优检测器设计，特别是在二进制假设检验中的误判概率。实验设计部分讨论了解析问题后直接取整得到全局最优实验方案的方法。同时，针对非负单调增加信号的估计，将最大似然估计转化为凸优化问题并解决具体实例。最后，研究了投资回报的最坏情况概率损失，通过离散化处理无限维优化问题并求解最大损失概率。

极大似然估计

Let $x^{(1)}\cdots x^{(N)}$ be independent samples from an N(μ,Σ) distribution, where it is known that $\Sigma ^{min} \leq \Sigma \leq \Sigma^{max}$ , where $\Sigma^{min}$ and $\Sigma^{max}$ are given positive definite matrices. Which of the following is true?

最优检测器设计

Consider a binary detection system with the following confusion matrix

$D = \left[\begin{array}{ll} 0.75 & 0.15 \\ 0.25 & 0.85 \end{array} \right ].$

If both classes are equiprobable, what is the probability of error

错误的概率，即实际是假设1成立，而估计为假设2，以及实际是假设2而估计为假设1。一直每种假设发生的概率均为0.5，故错误的概率= 0.1*-。25+0.15*0.5=0.2

In binary hypothesis testing the likelihood ratio threshold can be interpreted as the cost of misclassification.

实验设计

If you solve the relaxed version of the optimal experiment design problem, and then simply round to get the (integer) number of times to carry out each experiment, you get the globally optimal (integer) experiment design.