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

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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|Monte-Carlo-Markov Chain

MCMC techniques are especially useful when analyzing data with complex statistical models. For example, when considering a hierarchical model with many levels of parameters, it is more efficient to use a MCMC technique such as Metropolis-Hasting or Gibbs sampling iterative procedure in order to sample from the many posterior distributions. It is very difficult, if not impossible, to use non-iterative direct methods for complex models.

A way to draw samples from a target posterior density $\xi(\theta \mid \mathbf{x})$ is to use Markov chain techniques, where each sample only depends on the last sample drawn. Starting with an approximate target density, the approximations are improved with each step of the sequential procedure. Or in other words, the sequence of samples is converging to samples drawn at random from the target distribution. A random walk from a Markov chain is simulated, where the stationary distribution of the chain is the target density, and the simulated values converge to the stationary distribution or the target density. The main concept in a Markov chain simulation is to devise a Markov process whose stationary distribution is the target density. The simulation must be long enough so that the present samples are close enough to the target. It has been shown that this is possible and that convergence can be accomplished. The general scheme for a Markov chain simulation is to create a sequence $\theta_{t}, t=1,2, \ldots$ by beginning at some value $\theta_{0}$, and at the $t$-th stage, select the present value from a transition function $Q_{t}\left(\theta_{t} \mid \theta_{t-1}\right)$, where the present value $\theta_{t}$ only depends on the previous one, via the transition function. The value of the starting value $\theta_{0}$ is usually based on a good approximation to the target density. In order to converge to the target distribution, the transition function must be selected with care. The account given here is a summary of Gelman et al., 19 , ch. 11 who presents a very complete account of MCMC. Metropolis-Hasting is the general name given to methods of choosing appropriate transition functions, and two special cases of this are the Metropolis algorithm and the other is referred to as Gibbs sampling.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Metropolis Algorithm

Suppose the target density $\xi(\theta \mid x)$ can be computed, then the Metropolis technique generates a sequence $\theta_{t}, t=1,2, \ldots$ with a distribution that converges to a stationary distribution of the chain. Briefly, the steps taken to construct the sequence are:
a. Draw the initial value $\theta_{0}$ from some approximation to the target density,
b. For $t=1,2, \ldots$ generate a sample $\theta_{}$ from the jumping distribution $G_{t}\left(\theta_{} \mid \theta_{t-1}\right)$,
c. Calculate the ratio $\mathrm{s}=\xi\left(\theta_{} \mid \mathrm{X}\right) \xi\left(\theta_{t-1} \mid \mathrm{X}\right)$ and d. Let $\theta_{t}=\theta_{s}$ with probability $\min (s, 1)$ or let $\theta_{t}=\theta_{t-1}$. To summarize the above, if the jump given by b above increases the posterior density, let $\theta_{t}=\theta_{}$; however, if the jump decreases the posterior density, let $\theta_{t}=\theta_{*}$ with probability s, otherwise let $\theta_{t}=\theta_{t-1}$. One must show the sequence generated is a Markov chain with a unique stationary density that converges to the target distribution. For more information, see Gelman et al. 19 , p. 325

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

Another MCMC algorithm is Gibbs sampling that is quite useful for multidimensional problems and is an alternating conditional sampling way to generate samples from the joint posterior distribution. Gibbs sampling can be thought of as a practical way to implement the fact that the joint distribution of two random variables is determined by the two conditional distributions.

The two-variable case is first considered by starting with a pair $\left(\theta_{1}, \theta_{2}\right)$ of random variables. The Gibbs sampler generates a random sample from the joint distribution of $\theta_{1}$ and $\theta_{2}$ by sampling from the conditional distributions of $\theta_{1}$ given $\theta_{2}$ and from $\theta_{2}$ given $\theta_{1}$. The Gibbs sequence of size $k$

$$\theta_{2}^{0}, \theta_{1}^{0} ; \theta_{2}^{1}, \theta_{1}^{1} ; \theta_{2}^{2}, \theta_{1}^{2} ; \ldots ; \theta_{2}^{k}, \theta_{1}^{k}$$
is generated by first choosing the initial values $\theta_{2}^{0}, \theta_{1}^{0}$ while the remaining are obtained iteratively by alternating values from the two conditional distributions. Under quite general conditions, for large enough $k$, the final two values $\theta_{2}^{k}, \theta_{1}^{k}$ are samples from their respective marginal distributions. To generate a random sample of size $n$ from the joint posterior distribution, generate the above Gibbs sequence $n$ times. Having generated values from the marginal distributions with large $k$ and $n$, the sample mean and variance will converge to the corresponding mean and variance of the posterior distribution of $\left(\theta_{1}, \theta_{2}\right)$.

Gibbs sampling is an example of an MCMC because the generated samples are drawn from the limiting distribution of a 2 by 2 Markov chain. See Casella and George ${ }^{25}$ for a proof that the generated values are indeed values from the appropriate marginal distributions. Of course, Gibbs sequences can be generated from the joint distribution of three, four, and more random variables.

The Gibbs sampling scheme is illustrated with a case of three random variables for the common mean of two normal populations.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Monte-Carlo-Markov Chain

MCMC 技术在使用复杂的统计模型分析数据时特别有用。例如，当考虑具有多个参数级别的层次模型时，使用诸如 Metropolis-Hasting 或 Gibbs 采样迭代程序之类的 MCMC 技术来从许多后验分布中进行采样会更有效。对复杂模型使用非迭代直接方法是非常困难的，如果不是不可能的话。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Metropolis Algorithm

a．绘制初始值θ0从某种近似到目标密度，
b。为了吨=1,2,…生成样本θ从跳跃分布G吨(θ∣θ吨−1)，
c。计算比率s=X(θ∣X)X(θ吨−1∣X)和 d。让θ吨=θs有概率分钟(s,1)或让θ吨=θ吨−1. 综上所述，如果上面 b 给出的跳跃增加了后验密度，让θ吨=θ; 然而，如果跳跃降低后验密度，让θ吨=θ∗概率为 s，否则让θ吨=θ吨−1. 必须证明生成的序列是一个马尔可夫链，它具有收敛到目标分布的独特平稳密度。有关详细信息，请参阅 Gelman 等人。19，页。325

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

θ20,θ10;θ21,θ11;θ22,θ12;…;θ2ķ,θ1ķ

Gibbs 抽样是 MCMC 的一个例子，因为生成的样本是从 2×2 马尔可夫链的极限分布中抽取的。见卡塞拉和乔治25证明生成的值确实是来自适当边际分布的值。当然，吉布斯序列可以从三个、四个和更多随机变量的联合分布中生成。

Gibbs 抽样方案以两个正态总体的共同平均值的三个随机变量为例进行说明。

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## MATLAB代写

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