### 统计代写|贝叶斯统计代写Bayesian statistics代考|STA602

statistics-lab™ 为您的留学生涯保驾护航 在代写贝叶斯统计方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写贝叶斯统计代写方面经验极为丰富，各种代写贝叶斯统计相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯统计代写beyesian statistics代考|The Bayesian method

Consider a problem where we wish to make inferences about a parameter $\theta$ given data $x$. In a classical setting the data is treated as if it is random, even after it has been observed, and the parameter is viewed as a fixed unknown constant. Consequently, no probability distribution can be attached to the parameter. Conversely in a Bayesian approach parameters, having not been observed, are treated as random and thus possess a probability distribution whilst the data, having been observed, is treated as being fixed.

Example 1 Suppose that we perform $n$ independent Bernoulli trials in which we observe $x$, the number of times an event occurs. We are interested in making inferences about $\theta$, the probability of the event occurring in a single trial. Let’s consider the classical approach to this problem.
Prior to observing the data, the probability of observing $x$ was
$$P(X=x \mid \theta)=\left(\begin{array}{l} n \ x \end{array}\right) \theta^x(1-\theta)^{n-x}$$

This is a function of the (future) $x$, assuming that $\theta$ is known. If we know $x$ but don’t know $\theta$ we could treat (1) as a function of $\theta, L(\theta)$, the likelihood function. We then choose the value which maximises this likelihood. The maximum likelihood estimate is $\frac{x}{n}$ with corresponding estimator $\frac{X}{n}$.

In the general case, the classical approach uses an estimate $T(x)$ for $\theta$. Justifications for the estimate depend upon the properties of the corresponding estimator $T(X)$ (bias, consistency, …) using its sampling distribution (given $\theta$ ). That is, we treat the data as being random even though it is known! Such an approach can lead to nonsensical answers.

Example 2 Suppose in the Bernoulli trials of Example 1 we wish to estimate $\theta^2$. The maximum likelihood estimator ${ }^1$ is $\left(\frac{X}{n}\right)^2$. However this is a biased estimator as
\begin{aligned} E\left(X^2 \mid \theta\right) & =\operatorname{Var}(X \mid \theta)+E^2(X \mid \theta) \ & =n \theta(1-\theta)+n^2 \theta^2 \ & =n \theta+n(n-1) \theta^2 . \end{aligned}

## 统计代写|贝叶斯统计代写beyesian statistics代考|Bayes’ theorem

Let $X$ and $Y$ be random variables with joint density function $f(x, y)$. The marginal distribution of $Y, f(y)$, is the joint density function averaged over all possible values of $X$,
$$f(y)=\int_X f(x, y) d x .$$
For example, if $Y$ is univariate and $X=\left(X_1, X_2\right)$ where $X_1$ and $X_2$ are univariate then
$$f(y)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\left(x_1, x_2, y\right) d x_1 d x_2 .$$
The conditional distribution of $Y$ given $X=x$ is
$$f(y \mid x)=\frac{f(x, y)}{f(x)}$$
so that by substituting (1.2) into (1.1) we have
$$f(y)=\int_X f(y \mid x) f(x) d x .$$
which is often known as the theory of total probability. $X$ and $Y$ are independent if and only if
$$f(x, y)=f(x) f(y) .$$
Substituting (1.2) into (1.3) we see that an equivalent result is that
$$f(y \mid x)=f(y)$$

so that independence reflects the notion that learning the outcome of $X$ gives us no information about the distribution of $Y$ (and vice versa). If $Z$ is a third random variable then $X$ and $Y$ are conditionally independent given $Z$ if and only if
$$f(x, y \mid z)=f(x \mid z) f(y \mid z) .$$

## 统计代写|贝叶斯统计代写beyesian statistics代考|The Bayesian method

$$P(X=x \mid \theta)=(n x) \theta^x(1-\theta)^{n-x}$$

$$E\left(X^2 \mid \theta\right)=\operatorname{Var}(X \mid \theta)+E^2(X \mid \theta) \quad=n \theta(1-\theta)+n^2 \theta^2=n \theta+n(n-1) \theta^2$$

## 统计代写|贝叶斯统计代写beyesian statistics代考|Bayes’ theorem

$$f(y)=\int_X f(x, y) d x .$$

$$f(y)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f\left(x_1, x_2, y\right) d x_1 d x_2 .$$

$$f(y \mid x)=\frac{f(x, y)}{f(x)}$$

$$f(y)=\int_X f(y \mid x) f(x) d x .$$

$$f(x, y)=f(x) f(y) .$$

$$f(y \mid x)=f(y)$$

$$f(x, y \mid z)=f(x \mid z) f(y \mid z) \text {. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。