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

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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|Introduction

In a statistical context, by inference, one usually means estimation of parameters, tests of hypotheses, and prediction of future observations. With the Bayesian approach, all inferences are based on the posterior distribution of the parameters, which in turn is based on the sample, via the likelihood function and the prior distribution. We have seen the role of the prior density and likelihood function in determining the posterior distribution, and presently will focus on the determination of point and interval estimation of the model parameters, and later will emphasize how the posterior distribution determines a test of hypothesis. Finally, the role of the predictive distribution in testing hypotheses and in goodness of fit will be explained.

When the model has only one parameter, one would estimate that parameter by listing its characteristics, such as the posterior mean, media, and standard deviation and plotting the posterior density. However, if there are several parameters, one would determine the marginal posterior distribution of the relevant parameters and, as above, calculate its characteristics (e.g., mean, median, mode, standard deviation, etc.) and plot the densities. Interval estimates of the parameters are also usually reported and are called credible intervals.

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

Inferences for the normal $(\mu, \tau)$ population are somewhat more demanding, because both parameters are unknown. Assuming the vague prior density $\xi(\mu, \tau) \propto 1 / \tau$, the marginal posterior distribution of the population mean $\mu$ is a $t$-distribution with $n-1$ degrees of freedom, mean $\bar{x}$, and precision $n / s^{2}$; thus, the mean and the median are the same and provide a natural estimator of $\mu$, and because of the symmetry of the $t$-density, a $(1-\alpha)$ credible interval for $\mu$ is $\bar{x} \pm t_{\alpha / 2, n-1} S / \sqrt{n}$, where $t_{\alpha / 2, n-1}$ is the upper 100 $\alpha / 2$ percent point of the $t$-distribution with $n$-1degrees of freedom. To generate values from the $t\left(n-1, \bar{x}, n / S^{2}\right)$ distribution, generate values from Student’s $t$-distribution with $n-1$ degrees of freedom, multiply each by $S / \sqrt{n}$, and then add $\bar{x}$ to each. Suppose $n-30$,

\begin{aligned} &x=(7.8902,4.8343,11.0677,8.7969,4.0391,4.0024,6.6494,8.4788,0.7939,5.0689, \ &6.9175,6.1092,8.2463,10.3179,1.8429,3.0789,2.8470,5.1471,6.3730,5.2907,1.5024, \ &3.8193,9.9831,6.2756,5.3620,5.3297,9.3105,6.5555,0.8189,0.4713) \text {, then } \ &\bar{x}=5.57 \text { and } S=2.92 \end{aligned}
Using the same dataset, the following WinBugs code is used to analyze the problem.
BC $2.1$
Model;
${$ for ( i in 1:30) ${x[i] \sim \operatorname{dnorm}(\mathrm{mu}, \mathrm{tau})}$
mu $\sim$ dnorm $(0.0,0001)$
tau $\sim$ dgamma(. $0001, .0001)$
sigma $<-1 /$ tau $}$
list $(\mathrm{x}=\mathrm{c}(7.8902,4.8343,11.0677,8.7969,4.0391,4.0024,6.6494,8.4788,0.7939,5.0689$, $6.9175,6.1092,8.2463,10.3179,1.8429,3.0789,2.8470,5.1471,6.3730,5.2907,1.5024$, $3.8193,9.9831,6.2756,5.3620,5.3297,9.3105,6.5555,0.8189,0.4713))$
$\operatorname{list}(\mathrm{mu}=0, \mathrm{tau}=1)$
Note, that a somewhat different prior was employed here, compared to previously, in that $\mu$ and $\tau$ are independent and assigned properly, but noninformative distributions. The corresponding analysis gives the following as shown in Table 2.2.

Upper and lower refer to the lower and upper $21 / 2$ percent points of the posterior distribution. Note a $95 \%$ credible interval for mu is $(4.47,6.65)$ and the estimation error is $.003566$. See the Appendix for the details on executing the WinBUGS statements above.

The program generated 30,000 samples from the joint posterior distribution of $\mu$ and $\sigma$ using a Gibbs sampling algorithm, and used 29,000 for the posterior moments and graphs, with a refresh of 100 .

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

An important feature of inference is testing hypotheses. Often in stochastic processes, the scientific hypothesis can be expressed in statistical terms and

a formal test implemented. Suppose $\Omega=\Omega_{0} \cup \Omega_{1}$ is a partition of the parameter space, then the null hypothesis is designated as $H_{0}: \theta \in \Omega_{0}$ and the alternative by $H_{1}: \theta \in \Omega_{1}$, and a test of $H_{0}$ versus $H_{1}$ consists of rejecting $H_{0}$ in favor of $H_{1}$ if the observations $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ belong to a critical region C. In the usual approach, the critical region is based on the probabilities of type I errors, namely $\mathrm{P}{\mathrm{r}}(C \mid \theta)$, where $\theta \in \Omega{0}$ and of type II errors $1-P_{r}(C \mid \theta)$, where $\theta \in \Omega_{1}$. This approach to testing hypothesis was developed formally by Neyman and Pearson and can be found in many of the standard references, such as Lehmann. ${ }^{20}$ Lee ${ }^{21}$ presents a good elementary introduction to testing and estimation in a Bayesian context.
In the Bayesian approach, the posterior probabilities
$$p_{0}=\operatorname{Pr}\left(\theta \in \Omega_{0} \mid \text { data }\right)$$
and
$$p_{1}=\operatorname{Pr}\left(\theta \in \Omega_{1} \mid \text { data }\right)$$
are required, and on the basis of the two, a decision is made whether or not to reject $\mathrm{H}$ in favor of $\mathrm{A}$ or to reject $\mathrm{A}$ in favor of $\mathrm{H}$. Also required are the two corresponding prior probabilities
$$\pi_{0}=\operatorname{Pr}\left(\theta \in \Omega_{0}\right)$$
and
$$\pi_{1}=\operatorname{Pr}\left(\theta \in \Omega_{1}\right) .$$
Now consider the prior odds $\pi_{0} / \pi_{1}$ and posterior odds $p_{0} / p_{1}$. In turn, consider the Bayes factor B in favor of $H_{0}$ relative to $H_{1}$, namely
$$B=\left(p_{0} / p_{1}\right) /\left(\pi_{0} / \pi_{1}\right)$$
Then, the posterior probabilities $p_{0}$ and $p_{1}$ can be expressed in terms of the Bayes factor, thus:
$$p_{0}=1 /\left[1+\left(\pi_{1} / \pi_{1}\right) B^{-1}\right]$$
and the Bayes factor is interpreted as the odds in favor of $H_{0}$ relative to $H_{1}$ as implied by the information from the data.

When the hypotheses are simple, that is, $\Omega_{0}=\left{\theta_{0}\right}$ and $\Omega_{1}=\left{\theta_{1}\right}$, note that the odds ratio can be expressed as the likelihood ratio.
$$B=p\left(x \mid \theta_{0}\right) / p\left(x \mid \theta_{1}\right)$$

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

X=(7.8902,4.8343,11.0677,8.7969,4.0391,4.0024,6.6494,8.4788,0.7939,5.0689, 6.9175,6.1092,8.2463,10.3179,1.8429,3.0789,2.8470,5.1471,6.3730,5.2907,1.5024, 3.8193,9.9831,6.2756,5.3620,5.3297,9.3105,6.5555,0.8189,0.4713)， 然后  X¯=5.57 和 小号=2.92

$F○r(一世一世n1:30)$X[一世]∼规范⁡(米在,吨一个在)$米在$∼$dn○r米$(0.0,0001)$吨一个在$∼$dG一个米米一个(.$0001,.0001)$s一世G米一个$<−1/$吨一个在$

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

p0=公关⁡(θ∈Ω0∣ 数据 )

p1=公关⁡(θ∈Ω1∣ 数据 )

p0=1/[1+(圆周率1/圆周率1)乙−1]

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## MATLAB代写

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