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

Bayes theorem is based on the conditional probability law:
$$\mathrm{P}[\mathrm{A} \mid \mathrm{B}]=\mathrm{P}[\mathrm{B} \mid \mathrm{A}] \mathrm{P}[\mathrm{A}] / \mathrm{P}[\mathrm{B}]$$
where $\mathrm{P}[\mathrm{A}]$ is the probability of A before one knows the outcome of the event $B, P[B \mid A]$ is the probability of $B$ assuming what one knows about the event $A$, and $P[A \mid B]$ is the probability of $A$ knowing that event $B$ has occurred. $\mathrm{P}[\mathrm{A}]$ is called the prior probability of $\mathrm{A}$, while $\mathrm{P}[\mathrm{A} \mid \mathrm{B}]$ is called the posterior probability of A.

Another version of Bayes theorem is to suppose $X$ is a continuous observable random vector and $\theta \in \Omega \subset R^{m}$ is an unknown parameter vector, and suppose the conditional density of $X$ given $\theta$ is denoted by $f(X \mid \theta)$. If $X=\left(X_{1}, X_{2}, \ldots, X_{n}\right)$ represents a random sample of size $n$ from a population with density $f(X \mid \theta)$, and $\xi(\theta)$ is the prior density of $\theta$, then Bayes theorem expresses the posterior density as
$$\xi(\theta \mid \mathrm{X})=\mathrm{C} \prod_{i=1}^{i=} f\left(x_{i} \mid \theta\right) \xi(\theta), \quad \mathrm{X}{i} \in \mathrm{R} \text { and } \theta \in \Omega$$ where the proportionality constant is $c$, and the term $\prod{i=1}^{i=n} f\left(x_{i} \mid \theta\right)$ is called the likelihood function. The density $\xi(\theta)$ is the prior density of $\theta$ and represents the knowledge one possesses about the parameter before one observes X. Such prior information is most likely available to the experimenter from other previous related experiments. Note that $\theta$ is considered a random variable and that Bayes theorem transforms one’s prior knowledge of $\theta$, represented by its prior density, to the posterior density, and that the transformation is the combining of the prior information about $\theta$ with the sample information represented by the likelihood function.
‘An essay toward solving a problem in the doctrine of chances’ by the Reverend Thomas Bayes $^{1}$ is the beginning of our subject. He considered a binomial experiment with $n$ trials and assumed the probability $\theta$ of success was uniformly distributed (by constructing a billiard table) and presented a way to calculate $\operatorname{Pr}(a \leq \theta \leq b \mid \mathrm{X}=\mathrm{p})$, where $\mathrm{X}$ is the number of successes in $n$ independent trials. This was a first in the sense that Bayes was making inferences via $\xi(\theta \mid \mathrm{X})$, the conditional density of $\theta$ given $x$. Also, by assuming the parameter as uniformly distributed, he was assuming vague prior information for $\theta$. The type of prior information where very little is known about the parameter is called noninformative or vague information.

It can well be argued that Laplace ${ }^{2}$ is the greatest Bayesian because he made many significant contributions to inverse probability (he did not know of Bayes), beginning in 1774 with ‘Memorie sur la probabilite des causes par la evenemens,’ with his own version of Bayes theorem, and over a period of some 40 years culminating in ‘Theorie analytique des probabilites.’ See Stigler ${ }^{3}$ and Chapters 9-20 of Hald ${ }^{4}$ for the history of Laplace’s contributions to inverse probability.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Binomial Distribution

Where do we begin with prior information, a crucial component of Bayes theorem rules? Bayes assumed the prior distribution of the parameter is uniform, namely
$$\xi(\theta)=1,0 \leq \theta \leq 1 \text {, where }$$
$\theta$ is the common probability of success in $n$ independent trials and

$$f(x \mid \theta)=\left(\begin{array}{l} n \ x \end{array}\right) \theta^{x}(1-\theta)^{n-x}$$
where $x$ is the number of successes $x=0,1,2, \ldots, n$. For the distribution of $X$, the number of successes is binomial and denoted by $X \sim \operatorname{Binomial}(\theta, n)$. The uniform prior was used for many years; however, Lhoste ${ }^{5}$ proposed a different prior, namely
$$\xi(\theta)=\theta^{-1}(1-\theta)^{-1}, 0 \leq \theta \leq 1$$
to represent information which is noninformative and is an improper density function. Lhoste based the prior on certain invariance principals, quite similar to Jeffreys. ${ }^{6}$ Lhoste also derived a noninformative prior for the standard deviation $\sigma$ of a normal population with density
$$f(x \mid \mu, \sigma)=(1 / \sqrt{2 \pi} \sigma) \exp -(1 / 2 \sigma)(x-\mu)^{2}, \mu \in R \text { and } \sigma>0$$
He used invariance as follows: he reasoned that the prior density of $\sigma$ and the prior density of $1 / \sigma$ should be the same, which leads to
$$\xi(\sigma)=1 / \sigma$$
Jeffreys’ approach is similar in that in developing noninformative priors for binomial and normal populations, but he also developed noninformative priors for multi-parameter models, including the mean and standard deviation for the normal density as
$$\xi(\mu, \sigma)=1 / \sigma, \mu \in R \text { and } \sigma>0$$
Noninformative priors were ubiquitous from the 1920 s to the 1980 s and were included in all the textbooks of that period. For example, see Box and Tiao, $^{10}$ Zellner, ${ }^{11}$ and Broemeling. ${ }^{12}$ Looking back, it is somewhat ironic that noninformative priors were almost always used, even though informative prior information was almost always available. This limited the utility of the Bayesian approach, and people saw very little advantage over the conventional way of doing business. The major strength of the Bayesian way is that it a convenient, practical, and logical method of utilizing informative prior information. Surely the investigator knows informative prior information from previous related studies.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Normal Distribution

Of course, the normal density plays an important role as a model for time series. For example, as will be seen in future chapters, the normal distribution will model the observations of certain time series, such as autoregressive and moving average series. How is informative prior information expressed for the parameters $\mu$ and $\sigma$ (the mean and standard deviation)? Suppose a previous study has $m$ observations $x=\left(x_{1} x_{2}, \ldots, x_{m}\right)$, then the density of X given $\mu$ and $\sigma$ is
\begin{aligned} &f(x \mid \mu, \sigma) \propto\left[\sqrt{m} / \sqrt{2 \pi \sigma^{2}}\right] \exp -\left(m / 2 \sigma^{2}\right)(\bar{x}-\mu)^{2} \ &{\left[(2 \pi)^{-(n-1) / 2} \sigma^{-(n-1)}\right] \exp -\left(1 / 2 \sigma^{2}\right) \sum_{i=1}^{i=m}\left(x_{i}-\bar{x}\right)^{2}} \end{aligned}
This is a conjugate density for the two-parameter normal family and is called the normal-gamma density. Note it is the product of two functions, where the first, as a function of $\mu$ and $\sigma$, is the conditional density of $\mu$ given $\sigma$, with mean $\bar{x}$ and variance $\sigma^{2} / m$, while the second is a function of $\sigma$ only and is an inverse gamma density. Or equivalently, if the normal is parameterized with $\mu$ and the precision $\tau=1 / \sigma^{2}$, the conjugate distribution is as follows: (a) the conditional distribution of $\mu$ given $\tau$ is normal with mean $\bar{x}$ and precision $\mathrm{m} \tau$, and (b) the marginal distribution of $\tau$ is gamma with parameters $(m+1) / 2$ and $\sum_{i=1}^{i=m}\left(x_{i}-\bar{x}\right)^{2} / 2=(m-1) S^{2} / 2$, where $S^{2}$ is the sample variance. Thus, if one knows the results of a previous experiment, the likelihood function for $\mu$ and $\tau$ provides informative prior information for the normal population.

For example, the normal serves as the distribution of the observations of a first-order autoregressive process
$$Y(t)=\theta Y(t-1)+W(t), t=1,2, \ldots$$
where
$$W(t), t=1,2, \ldots$$
is a sequence of independent normal random variables with mean zero and precision $\tau$, and $\tau>0$. It is easy to show that the joint distribution of the $n$ observations from the AR(1) process is multivariate normal with mean vector

0 and variance covariance matrix with diagonal entries $1 / \tau\left(1-\theta^{2}\right)$ and $k$-th order covariance $\operatorname{Cov}[Y(t), Y(t+k)]=\theta^{k} / \tau\left(1-\theta^{2}\right),|\theta|<1, k=1,2, \ldots$

Note it is assumed the process is stationary, namely, $|\theta|<1$. Of course, the goal of the Bayesian analysis is to estimate the processes autoregressive parameter $\theta$ and the precision $\tau>0$. For the Bayesian analysis, a prior distribution must be assigned to $\theta$ and $\tau$, which in the conjugate prior case is a normal-gamma. The posterior analysis for the autoregressive time series results in a univariate $t$-distribution for the distribution of $\theta$ as will be shown in Chapter $5 .$

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

X(θ∣X)=C∏一世=1一世=F(X一世∣θ)X(θ),X一世∈R 和 θ∈Ω其中比例常数为C, 和项∏一世=1一世=nF(X一世∣θ)称为似然函数。密度X(θ)是先验密度θ并且表示在观察 X 之前对参数所拥有的知识。这种先验信息最有可能从其他先前的相关实验中提供给实验者。注意θ被认为是一个随机变量，并且贝叶斯定理转换了一个人的先验知识θ，由它的先验密度表示，到后验密度，并且转换是关于先验信息的组合θ用似然函数表示的样本信息。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Binomial Distribution

X(θ)=1,0≤θ≤1， 在哪里
θ是成功的常见概率n独立试验和

F(X∣θ)=(n X)θX(1−θ)n−X

X(θ)=θ−1(1−θ)−1,0≤θ≤1

F(X∣μ,σ)=(1/2圆周率σ)经验−(1/2σ)(X−μ)2,μ∈R 和 σ>0

X(σ)=1/σ
Jeffreys 的方法与为二项式和正态总体开发非信息先验的方法相似，但他也为多参数模型开发了非信息先验，包括正态密度的均值和标准差

X(μ,σ)=1/σ,μ∈R 和 σ>0

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Normal Distribution

F(X∣μ,σ)∝[米/2圆周率σ2]经验−(米/2σ2)(X¯−μ)2 [(2圆周率)−(n−1)/2σ−(n−1)]经验−(1/2σ2)∑一世=1一世=米(X一世−X¯)2

0 和带对角线条目的方差协方差矩阵1/τ(1−θ2)和ķ- 阶协方差这⁡[是(吨),是(吨+ķ)]=θķ/τ(1−θ2),|θ|<1,ķ=1,2,…

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## MATLAB代写

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