### 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian Basics

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|贝叶斯分析代写Bayesian Analysis代考|Bayes’ rule

The starting point for Bayesian inference is Bayes’ rule. The simplest form of this is
$$P(A \mid B)=\frac{P(A) P(B \mid A)}{P(A) P(B \mid A)+P(\bar{A}) P(B \mid \bar{A})},$$
where $A$ and $B$ are events such that $P(B)>0$. This is easily proven by considering that:
$P(A \mid B)=\frac{P(A B)}{P(B)}$ by the definition of conditional probability $P(A B)=P(A) P(B \mid A) \quad$ by the multiplicative law of probability $P(B)=P(A B)+P(\bar{A} B)=P(A) P(B \mid A)+P(\bar{A}) P(B \mid \bar{A})$
by the law of total probability.
We see that the posterior probability $P(A \mid B)$ is equal to the prior probability $P(A)$ multiplied by a factor, where this factor is given by $P(B \mid A) / P(B)$

As regards terminology, we call $P(A)$ the prior probability of $A$ (meaning the probability of $A$ before $B$ is known to have occurred), and we call $P(A \mid B)$ the posterior probability of $A$ given $B$ (meaning the probability of $A$ after $B$ is known to have occurred). We may also say that $P(A)$ represents our a priori beliefs regarding $A$, and $P(A \mid B)$ represents our a posteriori beliefs regarding $A$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes factors

One way to perform hypothesis testing in the Bayesian framework is via the theory of Bayes factors. Suppose that on the basis of an observed event $D$ (standing for data) we wish to test a null hypothesis
$$H_{0}: E_{0}$$
versus an alternative hypothesis
$$H_{1}: E_{1} \text {, }$$
where $E_{0}$ and $E_{1}$ are two events (which are not necessarily mutually exclusive or even exhaustive of the event space).
Then we calculate:
$\pi_{0}=P\left(E_{0}\right)=$ the prior probability of the null hypothesis
$\pi_{1}=P\left(E_{1}\right)=$ the prior probability of the alternative hypothesis
$P R O=\pi_{0} / \pi_{1}=$ the prior odds in favour of the null hypothesis
$p_{0}=P\left(E_{0} \mid D\right)=$ the posterior probability of the null hypothesis
$p_{1}=P\left(E_{1} \mid D\right)=$ the posterior probability of the alternative hypothesis
$P O O=p_{0} / p_{1}=$ the posterior odds in favour of the null hypothesis.
The Bayes factor is then defined as $B F=P O O / P R O$. This may be interpreted as the factor by which the data have multiplied the odds in favour of the null hypothesis relative to the alternative hypothesis. If $B F>1$ then the data has increased the relative likelihood of the null, and if $B F<1$ then the data has decreased that relative likelihood. The magnitude of $B F$ tells us how much effect the data has had on the relative likelihood.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models

Bayes’ formula extends naturally to statistical models. A Bayesian model is a parametric model in the classical (or frequentist) sense, but with the addition of a prior probability distribution for the model parameter, which is treated as a random variable rather than an unknown constant. The basic components of a Bayesian model may be listed as:

• the data, denoted by $y$
• the parameter, denoted by $\theta$
• the model distribution, given by a specification of $f(y \mid \theta)$ or $F(y \mid \theta)$ or the distribution of $(y \mid \theta)$
• the prior distribution, given by a specification of $f(\theta)$ or $F(\theta)$ or the distribution of $\theta .$
Here, $F$ is a generic symbol which denotes cumulative distribution function (cdf), and $f$ is a generic symbol which denotes probability density function (pdf) (when applied to a continuous random variable) or probability mass function (pmf) (when applied to a discrete random variable). For simplicity, we will avoid the term pmf and use the term pdf or density for all types of random variable, including the mixed type.
Note 1: A mixed distribution is defined by a cdf which exhibits at least one discontinuity (or jump) and is strictly increasing over at least one interval of values.

Note 2: The prior may be specified by writing a statement of the form ‘ $\theta \sim \ldots$, where the symbol ‘ $\sim$ ‘ means ‘is distributed as’, and where ‘…’denotes the relevant distribution. Likewise, the model for the data may be specified by writing a statement of the form ‘ $(y \mid \theta) \sim \ldots$ ‘.

Note 3: At this stage we will not usually distinguish between $y$ as a random variable and $y$ as a value of that random variable; but sometimes we may use $Y$ for the former. Each of $y$ and $\theta$ may be a scalar, vector, matrix or array. Also, each component of $y$ and $\theta$ may have a discrete distribution, a continuous distribution, or a mixed distribution.

In the first few examples below, we will focus on the simplest case where both $y$ and $\theta$ are scalar and discrete.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayes factors

H0:和0

H1:和1,

p0=磷(和0∣D)=原假设的后验概率
p1=磷(和1∣D)=备择假设的后验概率

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Bayesian models

• 数据，表示为是
• 参数，表示为θ
• 模型分布，由规范给出F(是∣θ)或者F(是∣θ)或分布(是∣θ)
• 先验分布，由规范给出F(θ)或者F(θ)或分布θ.
这里，F是表示累积分布函数 (cdf) 的通用符号，并且F是一个通用符号，表示概率密度函数 (pdf)（当应用于连续随机变量时）或概率质量函数 (pmf)（当应用于离散随机变量时）。为简单起见，我们将避免使用术语 pmf 并使用术语 pdf 或密度来表示所有类型的随机变量，包括混合类型。
注 1：混合分布由 cdf 定义，该 cdf 表现出至少一个不连续性（或跳跃）并且在至少一个值区间内严格增加。

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## MATLAB代写

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