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

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

Suppose that we wish to sample from a univariate distribution with pdf $f(x)$ for which rejection sampling and the other techniques described previously are problematic (say). Then another way to proceed is via the Metropolis algorithm. This is an example of Markov chain Monte Carlo (MCMC) methods. The Metropolis algorithm may be described as follows.

As with the Newton-Raphson algorithm, we begin by specifying an initial value of $x$, call it $x_0$. We then also need to specify a suitable driver distribution which is easy to sample from, defined by a pdf,
$$g(t \mid x)$$
For now, we will assume the driver to be symmetric, in the sense that
$$g(t \mid x)=g(x \mid t)$$
or more precisely,
$$g(t=a \mid \theta=b)=g(t=b \mid \theta=a) \quad \forall a, b \in \mathfrak{R} .$$
Note: The driver distribution may also be non-symmetric, but this case will be discussed later.

We then do the following iteratively for each $j=1,2,3, \ldots, K$ (where $K$ is ‘large’):
(a) Generate a candidate value of $x$ by sampling $x_j^{\prime} \sim g\left(t \mid x_{j-1}\right)$. We call $x_j^{\prime}$ the proposed value and $g\left(t \mid x_{j-1}\right)$ the proposal density.
(b) Calculate the acceptance probability as $p=\frac{f\left(x_j^{\prime}\right)}{f\left(x_{j-1}\right)}$.
Note: If $p>1$ then we take $p=1$. Also, if $x_j^{\prime}$ is outside the range of possible values for the random variable $x$, then $f\left(x_j^{\prime}\right)=0$ and so $p=0$.
(c) Accept the proposed value $x_j^{\prime}$ with probability $p$.
To determine if $x_j^{\prime}$ is accepted, generate $u \sim U(0,1)$ (independently). If $u<p$ then accept $x_j^{\prime}$, and otherwise reject $x_j^{\prime}$.
(d) If $x_j^{\prime}$ has been accepted then let $x_j=x_j^{\prime}$, and otherwise let $x_j=x_{j-1}$ (i.e. repeat the last value $x_{j-1}$ in the case of a rejection).
This procedure results in the realisation of a Markov chain, $x_0, x_1, x_2, \ldots, x_K$. The last value of this chain, $x_K$, may be taken as an observation from $f(x)$, at least approximately. The approximation will be extremely good if $K$ is sufficiently large.

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Changing the tuning parameter

What happens if we make the tuning parameter $c=0.15$ larger? Figures 6.3 and 6.4 are a repeat of Figures 6.1 and 6.2, respectively, but using simulated values from a run of the Metropolis algorithm with $c=0.65$.
In this case the acceptance rate is only $20.8 \%$ and the histogram is a poorer estimate of the true density (to which it would however converge as $J \rightarrow \infty)$. We say that the algorithm is now displaying poor mixing compared to results in the first run of 500 where $c=0.15$.

What happens if we make $c=0.15$ smaller? Figures 6.5 and 6.6 are a repeat of Figures 6.1 and 6.2, respectively, but using simulated values from a run of the Metropolis algorithm with $c=0.05$.

In this case the acceptance rate is higher at $83 \%$, there is greater autocorrelation, and the histogram is again a poorer estimate of the true density (to which it would however still converge as $J \rightarrow \infty$ ). We again say that the algorithm is mixing poorly.

It is important to stress that even if the algorithm is mixing poorly (whether this be due to the tuning constant being too large or too small), it will eventually (with a sufficiently large value of $J$ ) yield a sample that is useful for inference to the desired degree of precision. Tweaking the tuning constant is merely a device for optimising computational efficiency.

# 贝叶斯分析代写

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

$$g(t \mid x)$$

$$g(t \mid x)=g(x \mid t)$$

$$g(t=a \mid \theta=b)=g(t=b \mid \theta=a) \quad \forall a, b \in \mathfrak{R} .$$

(a)对$x_j^{\prime} \sim g\left(t \mid x_{j-1}\right)$进行抽样，产生一个候选值$x$。我们称$x_j^{\prime}$为建议值，称$g\left(t \mid x_{j-1}\right)$为建议密度。
(b)计算接受概率为$p=\frac{f\left(x_j^{\prime}\right)}{f\left(x_{j-1}\right)}$。

(c)以$p$的概率接受建议值$x_j^{\prime}$。

(d)如果$x_j^{\prime}$已被接受，则设$x_j=x_j^{\prime}$，否则设$x_j=x_{j-1}$(即在被拒绝的情况下重复最后一个值$x_{j-1}$)。

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Changing the tuning parameter

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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。