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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|Random number generation

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Random number generation

So far we have assumed the availability of the sample required for Monte Carlo estimation, such as $x_1, \ldots, x_J \sim$ iid $f(x)$. The issue was skipped over by making use of ready made functions in $\mathrm{R}$ such as runif(), $\operatorname{rbeta()}$ and rgamma(). However, many applications involve dealing with complicated distributions from which sampling is not straightforward.

So we will next discuss some basic techniques that can be used to generate the required Monte Carlo sample from a given distribution. More advanced techniques will be treated later. We will first treat the discrete case, which is the simplest, and then the continuous case. It will be assumed throughout that we can at least sample easily from the standard uniform distribution, i.e. that we can readily generate $u \sim U(0,1)$.

Note: This sampling is easily achieved using the runif() function in R. Alternatively, it can be done physically by using a hat with 10 cards in it, where these have the numbers $0,1,2, \ldots ., 9$ written on them. Three cards (say) are drawn out of the hat, randomly and with replacement. The three numbers thereby selected are written down in a row, and a decimal point is placed in front of them. The resulting number (e.g. $0.472,0.000$ or 0.970 ) is an approximate draw from the standard uniform distribution. Repeating the entire procedure several times results in a random sample from that distribution. Increasing ‘three’ above (to ‘five’, say) improves the approximation (e.g. yielding $0.47207,0.00029$ or 0.97010 ).

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Sampling from an arbitrary discrete distribution

Suppose we wish to sample a value $x \sim f(x)$ where $f(x)$ is a discrete pdf defined over the possible values $x=x_1, \ldots, x_K$. First define
$$f_k=f\left(x_k\right)$$
and
$$F_k=f_1+\ldots+f_k(k=1, \ldots, K),$$
noting that $F_K=1$.
Then sample $u \sim U(0,1)$, and finally return:
\begin{aligned} & x=x_1 \quad \text { if } 0 \leq u \leq F_1 \ & x=x_2 \quad \text { if } F_1<u \leq F_2 \ & x=x_K \quad \text { if } F_{K-1}<u \leq F_K(=1) . \ & \end{aligned}
One way to implement the above is to set $k=1$, to repeatedly increment $k$ by 1 until $F_{k-1}<u \leq F_k$, and then, using the final value of $k$ thereby obtained, to return $x=x_k$.

Note 1: We see that this procedure will work also in the case where $K$ is infinite. In that case a practical alternative is to redefine $K$ as a value $k$ for which $F_k$ is very close to 1 (e.g. 0.9999) and then approximate $f(x)$ by zero for all $x>x_K$.

Note 2: In R, an alternative to using $u \sim U(0,1)$ is to apply the function sample() with appropriate specifications of $x_1, \ldots, x_K$ and $f_1, \ldots, f_K$ (as illustrated in an exercise below).

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|Sampling from an arbitrary discrete distribution

$$f_k=f\left(x_k\right)$$

$$F_k=f_1+\ldots+f_k(k=1, \ldots, K),$$

\begin{aligned} & x=x_1 \quad \text { if } 0 \leq u \leq F_1 \ & x=x_2 \quad \text { if } F_1<u \leq F_2 \ & x=x_K \quad \text { if } F_{K-1}<u \leq F_K(=1) . \ & \end{aligned}

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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。