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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|STAT7613 BAYES’ RULE

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|BAYES’ RULE

Bayes’ rule is a basic result in probability that describes a relationship between two conditional distributions $p(X \mid Y)$ and $p(Y \mid X)$ for a pair of random variables (these random variables can also be continuous). More specifically, Bayes’ rule states that for any such pair of random variables, the following identity holds:
$$p(Y=y \mid X=x)=\frac{p(X=x \mid Y=y) p(Y=y)}{p(X=x)} .$$

This result also generally holds true for any two events $A$ and $B$ with the conditional probability $p(X \in A \mid Y \in B)$.

The main advantage that Bayes’ rule offers is inversion of the conditional relationship between two random variables – therefore, if one variable is known, then the other can be calculated as well, assuming the marginal distributions $p(X=x)$ and $p(Y=y)$ are also known.
Bayes’ rule can be proven in several ways. One way to derive it is simply by using the chain rule twice. More specifically, we know that the joint distribution values can be rewritten as follows, using the chain rule, either first separating $X$ or first separating $Y$ :
\begin{aligned} p(X & =x, Y=y) \ & =p(X=x) p(Y=y \mid X=x) \ & =p(Y=y) p(X=x \mid Y=y) . \end{aligned}

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|INDEPENDENT AND CONDITIONALLY INDEPENDENT RANDOM VARIABLES

A pair of random variables $(X, Y)$ is said to be independent if for any $A$ and $B$,
$$p(X \in A \mid Y \in B)=p(X \in A)$$
or alternatively $p(Y \in B \mid X \in A)=p(Y \in B)$ (these two definitions are correct and equivalent under very mild conditions that prevent ill-formed conditioning on an event that has zero probability).

Using the chain rule, it can also be shown that the above two definitions are equivalent to the requirement that $p(X \in A, Y \in B)=p(X \in A) p(Y \in B)$ for all $A$ and $B$.

Independence between random variables implies that the random variables do not provide information about each other. This means that knowing the value of $X$ does not help us infer anything about the value of $Y$-in other words, it does not change the probability of $Y$. (Or vice-versa $-Y$ does not tell us anything about $X$.) While independence is an important concept in probability and statistics, in this book we will more frequently make use of a more refined notion of independence, called “conditional independence”-which is a generalization of the notion of independence described in the beginning of this section. A pair of random variables $(X, Y)$ is conditionally independent given a third random variable $Z$, if for any $A, B$ and $z$, it holds that $p(X \in A \mid Y \in B, Z=z)=p(X \in A \mid Z=z)$.

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|BAYES’ RULE

$$p(Y=y \mid X=x)=\frac{p(X=x \mid Y=y) p(Y=y)}{p(X=x)} .$$

$$p(X=x, Y=y) \quad=p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=y) .$$

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|INDEPENDENT AND CONDITIONALLY INDEPENDENT RANDOMVARIABLES

$$p(X \in A \mid Y \in B)=p(X \in A)$$

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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。