### 统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS 3023

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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|Solving equations

In most of the Bayesian models so far examined, the calculations required could be done analytically. For example, the model given by:
$$(Y \mid \theta) \sim \operatorname{Binomial}(5, \theta)$$
$$\theta \sim U(0,1) \text {, }$$
together with data $y=5$, implies the posterior $(\theta \mid y) \sim \operatorname{Beta}(6,1)$. So $\theta$ has posterior pdf $f(\theta \mid y)=6 \theta^{5}$ and posterior cdf $F(\theta \mid y)=\theta^{6}$. Then, setting $F(\theta \mid y)=1 / 2$ yields the posterior median, $\theta=1 / 2^{1 / 6}=0.8909$.
But what if the equation $F(\theta \mid y)=1 / 2$ were not so easy to solve? In that case we could employ a number of strategies. One of these is trial and error, and another is via special functions in software packages, for example using the qbeta () function in $R$. This yields the correct answer. Yet another method is the Newton-Raphson algorithm, our next topic.
R Code for Section $4.1$
qbeta(0.5,6,1) #0.8908987

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Newton-Raphson algorithm

The Newton-Raphson (NR) algorithm is a useful technique for solving equations of the form $g(x)=0$.

This algorithm involves choosing a suitable starting value $x_{0}$ and iteratively applying the equation
$$x_{j+1}=x_{j}-g^{\prime}\left(x_{j}\right)^{-1} g\left(x_{j}\right)$$
until convergence had been achieved to a desired degree of precision.
How does the NR algorithm work? Figure 4.1 illustrates the idea.

Here, $a$ is the desired solution of the equation $g(x)=0, c$ is a guess at that solution, and $b$ is a better estimate of $a$. Observe that the slope of the tangent at point $Q$ is equal to both $g^{\prime}(c)$ and $g(c) /(c-b)$. Equating these two expressions we get $b=c-g(c) / g^{\prime}(c)$.

Note: Sometimes the NR algorithm takes a long time to converge, and sometimes it converges to the wrong or even impossible value or gets ‘stuck’ and fails to converge at all. This is a general problem with the NR algorithm, namely its instability and the need to start it off with an initial guess that is sufficiently close to the desired solution.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The multivariate Newton-Raphson algorithm

The Newton-Raphson algorithm can also be used to solve several equations simultaneously, say
$$g_{k}\left(x_{1}, \ldots, x_{K}\right)=0, k=1, \ldots, K \text {. }$$

Let: $x=\left(\begin{array}{c}x_{1} \ \vdots \ x_{K}\end{array}\right), g(x)=\left(\begin{array}{c}g_{1}(x) \ \vdots \ g_{K}(x)\end{array}\right), 0=\left(\begin{array}{c}0 \ \vdots \ 0\end{array}\right)$ (a column vector of length $K$ ).
Then the system of $K$ equations may be expressed as
$$g(x)=0,$$
and the NR algorithm involves iterating according to
$$x^{(j+1)}=x^{(j)}-g^{\prime}\left(x^{(j)}\right)^{-1} g\left(x^{(j)}\right),$$
where: $x^{(j)}=\left(\begin{array}{c}x_{1}^{(j)} \ \vdots \ x_{K}^{(j)}\end{array}\right)$ is the value of $x$ at the $j$ th iteration
$$x^{(j+1)}=\left(\begin{array}{c} x_{1}^{(j+1)} \ \vdots \ x_{K}^{(j+1)} \end{array}\right), \quad g\left(x^{(j)}\right)=\left(\begin{array}{c} g_{1}\left(x^{(j)}\right) \ \vdots \ g_{K}\left(x^{(j)}\right) \end{array}\right)=\left[\left.\begin{array}{c} g_{1}(x) \ \vdots \ \left.g_{K}(x)\right) \end{array}\right|{x{x} x^{(j)}}\right]$$
\begin{aligned} &g^{\prime}\left(x^{(j)}\right)=\left[\left.g^{\prime}(x)\right|{x=x^{\prime \prime \prime}}\right] \ &g^{\prime}(x)=\left(\begin{array}{ccc} \partial g{1}(x) / \partial x^{T} \ \vdots \ \partial g_{K}(x) / \partial x^{T} \end{array}\right)=\left(\begin{array}{ccc} \partial g_{1}(x) / \partial x_{1} & \cdots & \partial g_{1}(x) / \partial x_{K} \ \vdots & \ddots & \vdots \ \partial g_{K}(x) / \partial x_{1} & \cdots & \partial g_{K}(x) / \partial x_{K} \end{array}\right) . \end{aligned}

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

(是∣θ)∼二项式⁡(5,θ)

θ∼在(0,1),

qbeta (0.5,6,1) #0.8908987

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The Newton-Raphson algorithm

Newton-Raphson (NR) 算法是求解以下方程的有用技术G(X)=0.

Xj+1=Xj−G′(Xj)−1G(Xj)

NR 算法是如何工作的？图 4.1 说明了这个想法。

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|The multivariate Newton-Raphson algorithm

Newton-Raphson 算法也可用于同时求解多个方程，例如

Gķ(X1,…,Xķ)=0,ķ=1,…,ķ.

G(X)=0,
NR算法涉及根据

X(j+1)=X(j)−G′(X(j))−1G(X(j)),

X(j+1)=(X1(j+1) ⋮ Xķ(j+1)),G(X(j))=(G1(X(j)) ⋮ Gķ(X(j)))=[G1(X) ⋮ Gķ(X))|XXX(j)]

G′(X(j))=[G′(X)|X=X′′′] G′(X)=(∂G1(X)/∂X吨 ⋮ ∂Gķ(X)/∂X吨)=(∂G1(X)/∂X1⋯∂G1(X)/∂Xķ ⋮⋱⋮ ∂Gķ(X)/∂X1⋯∂Gķ(X)/∂Xķ).

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## MATLAB代写

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