### 统计代写|应用随机过程代写Stochastic process代考|Bayesian analysis

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• 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|应用随机过程代写Stochastic process代考|Bayesian analysis

In this chapter, we briefly address the first part of this book’s title, that is, Bayesian Analysis, providing a summary of the key results, methods and tools that are used throughout the rest of the book. Most of the ideas are illustrated through several worked examples showcasing the relevant models. The chapter also sets up the basic notation that we shall follow later on.

In the last few years numerous books dealing with various aspects of Bayesian analysis have been published. Some of the most relevant literature is referenced in the discussion at the end of this chapter. However, in contrast to the majority of these books, and given the emphasis of our later treatment of stochastic processes, we shall here stress two issues that are central to our book, that is, decision-making and computational issues.

The chapter is organized as follows. First, in Section $2.2$ we outline the basics of the Bayesian approach to inference, estimation, hypothesis testing, and prediction. We also consider briefly problems of sensitivity to the prior distribution and the use of noninformative prior distributions. In Section 2.3, we outline Bayesian decision analysis. Then, in Section 2.4, we briefly review Bayesian computational methods. We finish with a discussion in Section 2.5.

## 统计代写|应用随机过程代写Stochastic process代考|Bayesian statistics

The Bayesian framework for inference and prediction is easily described. Indeed, at a conceptual level, one of the major advantages of the Bayesian approach is the ease with which the basic ideas are put into place.

In particular, one of the typical goals in statistics is to learn about one (or more) parameters, say $\theta$, which describe a stochastic phenomenon of interest. To learn about $\boldsymbol{\theta}$, we will observe the phenomenon, collect a sample of data, say $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$

and calculate the conditional density or probability function of the data given $\theta$, which we denote as $f(\mathbf{x} \mid \boldsymbol{\theta})$. This joint density, when thought of as a function of $\theta$, is usually referred to as the likelihood function and will be, in general, denoted as $l(\theta \mid \mathbf{x})$, or $l(\theta \mid$ data) when notation gets cumbersome. Although this will not always be the case in this book, due to the inherent dependence in data generated from stochastic processes, in order to illustrate the main ideas of Bayesian statistics, in this chapter we shall generally assume $\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)$ to be (conditionally) independent and identically distributed (CIID) given $\theta$.

As well as the likelihood function, the Bayesian approach takes into account another source of information about the parameters $\theta$. Often, an analyst will have access to external sources of information such as expert information, possibly based on past experience or previous related studies. This external information is incorporated into a Bayesian analysis as the prior distribution, $f(\theta)$.

The prior and the likelihood can be combined via Bayes’ theorem which provides the posterior distribution $f(\boldsymbol{\theta} \mid \mathbf{x})$, that is the distribution of the parameter $\theta$ given the observed data $\mathbf{x}$,
$$f(\boldsymbol{\theta} \mid \mathbf{x})=\frac{f(\boldsymbol{\theta}) f(\mathbf{x} \mid \boldsymbol{\theta})}{\int f(\boldsymbol{\theta}) f(\mathbf{x} \mid \boldsymbol{\theta}) d \boldsymbol{\theta}} \propto f(\boldsymbol{\theta}) f(\mathbf{x} \mid \boldsymbol{\theta})$$
The posterior distribution summarizes all the information available about the parameters and can be used to solve all standard statistical problems, like point and interval estimation, hypothesis testing or prediction. Throughout this chapter, we shall use the following two examples to illustrate these problems.

## 统计代写|应用随机过程代写Stochastic process代考|Parameter estimation

As an example of usage of the posterior distribution, we may be interested in point estimation. This is typically addressed by summarizing the distribution through, either

Figure 2.1 Prior (dashed line), scaled likelihood (dotted line), and posterior distribution (solid line) for the gambler’s ruin problem.
the posterior mean, that is,
$$E[\boldsymbol{\theta} \mid \mathbf{x}]=\int \boldsymbol{\theta} f(\boldsymbol{\theta} \mid \mathbf{x}) \mathrm{d} \boldsymbol{\theta}$$
or, in the univariate case, through a posterior median, that is,
$$\theta_{\text {med }} \in{y: P(\theta \leq y \mid x)=1 / 2 ; P(\theta \geq y \mid x)=1 / 2}$$
or through a posterior mode, that is
$$\theta_{\text {mode }}=\arg \max f(\theta \mid \mathbf{x})$$

## 统计代写|应用随机过程代写Stochastic process代考|Bayesian statistics

F(θ∣X)=F(θ)F(X∣θ)∫F(θ)F(X∣θ)dθ∝F(θ)F(X∣θ)

## 统计代写|应用随机过程代写Stochastic process代考|Parameter estimation

θ和 ∈是:磷(θ≤是∣X)=1/2;磷(θ≥是∣X)=1/2

θ模式 =参数⁡最大限度F(θ∣X)

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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。