Posted on Categories:Statistical inference, 统计代写, 统计代考, 统计推断

# 统计代写|统计推断代考Statistical Inference代写|Best Estimates

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## 统计代写|统计推断代考Statistical Inference代写|Best Estimates

Perhaps surprisingly, there is not a single answer to the best estimate for $\theta$ given the posterier distribution, like the one shown in Figure 6.9. There are several plausible measures, each with their own advantages. Any specific estimate of a parameter (e.g. $\theta$ ) is denoted with a hat (e.g. $\hat{\theta})$ in the descriptions that follow.
The Mode Also known as the maximum a-posteriori probability (MAP) estimate, the mode is the maximum of the posterior probability. In the case of a Beta distribution with $h$ successes in $N$ trials, we have
$$\hat{\theta}{\text {mode }}=\frac{h}{N}$$ The Mean Also known as the expected value or average value, the mean of a distribution of a parameter $\theta$ is defined to be the sum of all of the possible values of $\theta$ times the posterior probability of $\theta$, $$\hat{\theta}{\text {mean }}=\sum_\theta \theta \times P(\theta \mid \text { data })$$
It is one measure of the middle of the distribution. In the special case of a Beta distribution with $h$ successes in $N$ trials, we have
$$\hat{\theta}_{\text {mean }}=\frac{h+1}{N+2}$$

Intuitively this is the same as the MAP of the Beta distribution, with one more success and one more failure than actually observed. Further, for the Beta distribution, the mean value $\hat{\theta}_{\text {mean }}$ represents the predictive probability of a successful event on the next observation.

## 统计代写|统计推断代考Statistical Inference代写|Uncertainty in the Best Estimates

To quantify the uncertainty in the best estimates, we need a value which represents the width of the distribution. Looking at Figure 6.10 we’d like to provide a quick way of saying that the range of probable values lies somewhere between $\theta=0.2$ and $\theta=0.5$ – anything outside of this contributes only a small amount to the probability, or in other words, we are most confident that our best estimate of $\theta$ lies between those 0.2 and 0.5 . Depending on the application, the symmetry of the distribution, and other practical factors one may see a few potential measures of the width of the distribution.

Inter-Quantile Range The Inter-Quantile Range (ICR) is the range between the $25 \%$ and $75 \%$ quartiles, and represents $50 \%$ of the probability.
In Figure 6.10, the Inter-Quantile Range range is [o.29,0.40].
95\% Credible Interval (CI) The $95 \%$ Credible Interval (CI) is the range between the $2.5 \%$ and $97.5 \%$ quantiles, and thus represents $95 \%$ of the probability. According to Table 1.1 on page 51 , it is “very likely” that our best estimate lies in this range.
In Figure 6.10, the $95 \%$ Credible Interval is nearly [0.2,0.5].
Standard Deviation The standard deviation is a measure of the half-width of a distribution, most commonly used specifically with reference to the particular Normal distribution. This will be defined more precisely in Section 7.2 on page 140), and will thus not be defined in general here.
An approximate value for the standard deviation for the Beta distribution is
$$\sigma \approx \sqrt{\hat{\theta}(1-\hat{\theta}) / N}$$
From Figure 6.10, and using the median as the best estimate, $\hat{\theta}$, we get
$$\sigma \approx \sqrt{0.34(1-0.34) / 30}=0.09$$

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Best Estimates

$$\hat{\theta}{\text {mode }}=\frac{h}{N}$$ The Mean Also known as the expected value or average value, the mean of a distribution of a parameter $\theta$ is defined to be the sum of all of the possible values of $\theta$ times the posterior probability of $\theta$, $$\hat{\theta}{\text {mean }}=\sum_\theta \theta \times P(\theta \mid \text { data })$$
It is one measure of the middle of the distribution. In the special case of a Beta distribution with $h$ successes in $N$ trials, we have
$$\hat{\theta}_{\text {mean }}=\frac{h+1}{N+2}$$

## 统计代写|统计推断代考Statistical Inference代写|Uncertainty in the Best Estimates

95％可信区间(CI) $95 \%$可信区间(CI)是$2.5 \%$到$97.5 \%$分位数之间的范围，代表概率的$95 \%$。根据第51页的表1.1，我们的最佳估计“很可能”位于这个范围内。

Beta分布的标准差的近似值为
$$\sigma \approx \sqrt{\hat{\theta}(1-\hat{\theta}) / N}$$

$$\sigma \approx \sqrt{0.34(1-0.34) / 30}=0.09$$

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## 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。