### 数学代写|概率论代写Probability theory代考|STAT4061

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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

The adaptive MH (AMH) algorithm was proposed by Beck and Au [22] to settle the difficulty in sampling according to the posterior PDF directly. The algorithm introduces a sequence of intermediate PDFs that converge to the posterior PDF for sample generation, and finally achieves the samples from the posterior PDF.

Let $\left{p^{(1)}, p^{(2)}, \ldots, p^{\left(s_{0}\right)}\right}$ denotes a sequence of PDFs, and $p^{(s)}$ is chosen as updated PDF from Bayes’ theorem based on an increasing amount of data:
$$p^{(s)}=p\left(\theta \mid D^{(s)}, C\right)$$
where $D^{(1)} \subset D^{(2)} \subset \ldots \subset D^{\left(s_{0}\right)}=D$. If the updated PDF with data $D$ has the following form:
$$p(\theta \mid D, C)=c \exp \left(-\frac{J(\theta)}{2 \varphi^{2}}\right)$$
where $J(\theta)$ is the goodness-of-fit function, and $\varphi$ is a measure of the size of the prediction error. Therefore, the sequence $\left{p^{(s)}\right}$ can be constructed as:

$$p^{(s)}=c^{(s)} \exp \left(-\frac{J(\theta)}{2\left(\varphi^{(s)}\right)^{2}}\right)$$
where $\left(\varphi^{(s)}\right)^{2}=2^{s_{0}-s} \varphi^{2}$ with $2^{s_{n}} \approx \varphi^{-2}$ if the length scale of the prior PDF is of order one. The following content shows the main steps of AMH algorithm:
(1) Starting with the prior PDF as the proposal PDF and $p^{(1)}$ as the target PDF, simulate the samples $\left{\theta_{1}^{(1)}, \theta_{2}^{(1)}, \ldots, \theta_{N}^{(1)}\right}$ by using the MH algorithm;
(2) At the sth level, simulate the samples using the procedure similar to level 1 except that the proposal PDF in MH algorithm is constructed by the kernel density. The kernel density or proposal PDF at the sth level can be constructed by using the samples generated at the $(s-1)$ th level:
$$p^{(s-1)}(\theta)=\sum_{n-1}^{N} w_{n} g\left(\theta ; \theta_{n}^{(s-1)}, \sum_{n}\right)$$
where $g\left(\theta ; \boldsymbol{\theta}{n}^{(s-1)}, \Sigma{n}\right)$ is the multidimensional Gaussian PDF evaluated at $\boldsymbol{\theta}$ with mean $\boldsymbol{\theta}{n}^{(s-1)}$ and covariance matrix $\Sigma{n} ; w_{n}$ is the weight with $\sum_{n=1}^{N} w_{n}=1$.
(3) Repeat step (2) until the $s_{0}$ th simulation level is finished, and obtain the samples from the target updated PDF $p(\theta \mid D, C)=p^{\left(5_{0}\right)}$.

With the help of AMH algorithm, the samples can be simulated from the posterior PDF that is very peaked or multimodal. However, this algorithm is still inefficient for the high-dimensional problems due to the inefficiency of kernel density estimations for high-dimensional problems. Based on experience, problem may arise when the number of uncertainty parameters is more than $4 .$

## 数学代写|概率论代写Probability theory代考|Transitional Markov Chain Monte Carlo Method

The transitional Markov chain Monte Carlo (TMCMC) method was proposed by Ching and Chen [47] on the basis of AMH algorithm. The TMCMC algorithm uses the same idea of “intermediate PDFs” as proposed by Beck and Au [22], but it does not require kernel density estimation. An additional advantage is that the intermediate PDFs can be automatically selected. The sequence of intermediate PDFs will ultimately converge to the posterior PDF and is defined by:
$$f_{s}(\theta) \propto f(\theta \mid C) \cdot f(D \mid \theta, C)^{p_{x}}\left(s=0,1, \ldots, N_{s} ; 0=p_{0}<p_{1}<\cdots<p_{N_{x}}=1\right)$$
where the subscript $s$ denotes the stage number and $N_{S}$ is the total number. The PDF of the initial stage with $p_{0}=0$ is proportional to the prior PDF, while the PDF of the

final stage with $p_{N_{x}}=1$ is proportional to the posterior PDF. The used intermediate PDFs can enable a good means of transitioning and obtaining the samples between the adjacent intermediate PDFs. Besides, the TMCMC algorithm can also estimate the evidence without solving any integral problem. The said algorithm can be enacted by following these steps:
(i) At the initial stage $(s=0)$, draw the samples $\left{\theta_{0, k}: k=1,2, \ldots, N_{0}\right}$ according to the prior $\operatorname{PDF} f_{0}(\theta)$;
(ii) Choose $p_{s+1}$ such that the COV of $\left{f\left(D \mid \theta_{s, k}, C\right)^{p_{s+1}-p_{s}}: k=1,2, \ldots, N_{s}\right}$ is equal to a prescribed threshold. Then, compute the plausibility weights $w\left(\boldsymbol{\theta}{s, k}\right)=f\left(D \mid \boldsymbol{\theta}{s, k}, C\right)^{p_{x+1}-p_{s}}$ for $k=1,2, \ldots, N_{s}$ and $S_{s}=\sum_{k=1}^{N_{x}} w\left(\boldsymbol{\theta}{s, k}\right) / N{s} .$ Here, $\theta_{s, k}$ denotes the $k$ th sample that belongs to level $s$ and the factor will be used to compute the evidence $f(D \mid C)$;
(iii) Apply the $\mathrm{MH}$ algorithm to generate the samples $\left{\theta_{s+1, k}: k=1,2, \ldots, N_{s+1}\right}$ from $f_{s+1}(\theta)$. The candidate sample $\boldsymbol{\theta}^{c}$ of the sth sample in the Markov chain is generated from $N\left(\theta_{s, l}, \Sigma_{s}\right)$, in which $\theta_{s, l}$ is one of the current samples $\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}$. The lth initial sample $\boldsymbol{\theta}{s, l}$ is chosen with probability $w\left(\boldsymbol{\theta}{s, l}\right) / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right) . \Sigma{s}$ is the covariance matrix of the Gaussian proposal PDF centered at the current sample $\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}$ and can be estimated using the following equation:
\begin{aligned} \sum_{s}=\beta^{2} \sum_{k=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, k}\right) &\left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right)\right]\right} \ & \times\left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}_{s, l}\right)\right]\right}^{T} \end{aligned}

## 数学代写|概率论代写Probability theory代考|Soil Suction

According to the level of the water contained in the soils, the soils can be divided into two broad categories: saturated soils and unsaturated soils. The unsaturated soil is commonly defined as having three phases: solid, water and air. However, the air-water interface or contractile skin must be considered as an independent phase when considering the stress state of unsaturated soil [48]. Since the 1930 s, the geotechnical scholars began to study the issue of unsaturated soils. The suction is an important factor for studying the properties of unsaturated soils. It exhibits the intensity of interaction between soil-water and soil particles and the curvature state of the air-water interface in unsaturated soil [48].

The total suction consists of two components: matric suction (matric or capillary component of free energy) and osmotic suction (osmotic or solute component of free energy). Aitchison [49] gave the total suction as follows:
$$\phi=\left(u_{a}-u_{w}\right)+\pi$$
where $\phi$ denotes total suction; $\left(u_{a}-u_{w}\right.$ ) denotes matric suction, in which $u_{a}$ and $u_{w}$ represent pore-air and pore-water pressure, respectively; and $\pi$ denotes the osmotic suction.

As shown in Fig. 1.1, water rises in a capillary tube immersed in water, similar to the pores of the soil with small radius acting as capillary tubes that cause the soil water to rise above the water table. The pressure of capillary water is negative relative to the air pressure, which is generally atmospheric (i.e., $u_{a}=0$ ). Therefore, the capillary pressure, that is, matric suction, can be expressed as follows:
$$u_{a}-u_{w}=\frac{2 T_{s} \cos \alpha}{r}$$

where $T_{s}$ is the surface tension of water; $\alpha$ is the contact angle; $r$ is the radius of the capillary tube, that is, the radius of the pore in the soil. It can be seen from Eq. (1.23) that the matric suction increases as the pore radius decreases.

Some researchers have studied the effect of suction on the shear strength, permeability and compressibility of soils. Most experimental evidences show the shear strength increases nonlinearly with soil suction, and the brittleness and dilatancy of an unsaturated soil also increase [50-53]. Fredlund and Rahardjo [54] stated that the permeability of an unsaturated soil depends on degree of saturation, and it can be estimated using the relationship between degree of saturation and suction. Alonso et al. [55] pointed that the compressibility of an unsaturated soil generally increases with the decrease of suction. Wheeler and Sivakumar [56] and Chiu and Ng [57] found that the compressibility under unsaturated conditions is larger than that under saturated conditions through the experiments. Estabragh et al. [58] found that the compressibility of a compacted silty soil does not change monotonically with suction, and it reaches maximum at certain suction. On the other hand, the osmotic suction is insensitive to the changes of soil-water content [54].

## 概率论代考

Beck和Au [22]提出了自适应MH（AMH）算法，以解决直接根据后验PDF进行采样的困难。该算法引入了一系列中间PDF，这些中间PDF会收敛到后验PDF进行样本生成，最终从后验PDF获得样本。

p(s)=p(θ∣D(s),C)

p(θ∣D,C)=C经验⁡(−Ĵ(θ)2披2)

p(s)=C(s)经验⁡(−Ĵ(θ)2(披(s))2)

(1) 从先验 PDF 作为提案 PDF 开始，p(1)作为目标PDF，模拟样本\left{\theta_{1}^{(1)}, \theta_{2}^{(1)}, \ldots, \theta_{N}^{(1)}\right}\left{\theta_{1}^{(1)}, \theta_{2}^{(1)}, \ldots, \theta_{N}^{(1)}\right}通过使用MH算法；
(2) 在第 s 层，使用类似于第 1 层的过程模拟样本，不同之处在于 MH 算法中的提议 PDF 是由核密度构造的。第s层的核密度或proposal PDF可以通过使用在第s层生成的样本来构建(s−1)级别：

p(s−1)(θ)=∑n−1ñ在nG(θ;θn(s−1),∑n)

(3) 重复步骤 (2) 直到s0仿真关卡完成，从目标更新的PDF中获取样本p(θ∣D,C)=p(50).

## 数学代写|概率论代写Probability theory代考|Transitional Markov Chain Monte Carlo Method

Fs(θ)∝F(θ∣C)⋅F(D∣θ,C)pX(s=0,1,…,ñs;0=p0<p1<⋯<pñX=1)

(i) 在初始阶段(s=0), 抽取样本\left{\theta_{0, k}: k=1,2, \ldots, N_{0}\right}\left{\theta_{0, k}: k=1,2, \ldots, N_{0}\right}根据之前的PDF格式⁡F0(θ);
(ii) 选择ps+1这样的 COV\left{f\left(D \mid \theta_{s, k}, C\right)^{p_{s+1}-p_{s}}: k=1,2, \ldots, N_{s} \正确的}\left{f\left(D \mid \theta_{s, k}, C\right)^{p_{s+1}-p_{s}}: k=1,2, \ldots, N_{s} \正确的}等于规定的阈值。然后，计算合理性权重在(θs,ķ)=F(D∣θs,ķ,C)pX+1−ps为了ķ=1,2,…,ñs和小号s=∑ķ=1ñX在(θs,ķ)/ñs.这里，θs,ķ表示ķ属于级别的第 th 个样本s并且该因子将用于计算证据F(D∣C);
(iii) 应用米H生成样本的算法\left{\theta_{s+1, k}: k=1,2, \ldots, N_{s+1}\right}\left{\theta_{s+1, k}: k=1,2, \ldots, N_{s+1}\right}从Fs+1(θ). 候选样本θC马尔可夫链中第 s 个样本的ñ(θs,l,Σs), 其中θs,l是当前样本之一\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}. 第 l 个初始样本θs,l被概率选中在(θs,l)/∑l=1ñs在(θs,l).Σs是以当前样本为中心的高斯提议 PDF 的协方差矩阵\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}\left{\theta_{s, l}: l=1,2, \ldots, N_{s}\right}并且可以使用以下等式进行估计：

\begin{aligned} \sum_{s}=\beta^{2} \sum_{k=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, k}\right) & \left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right)\right]\right} \ & \times\left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l }\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}_{s, l}\right)\右]\right}^{T} \end{对齐}\begin{aligned} \sum_{s}=\beta^{2} \sum_{k=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, k}\right) & \left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l}\right)\right]\right} \ & \times\left{\boldsymbol{\theta}{s, k}-\left[\sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}{s, l }\right) \boldsymbol{\theta}{s, l} / \sum_{l=1}^{N_{s}} w\left(\boldsymbol{\theta}_{s, l}\right)\右]\right}^{T} \end{对齐}

φ=(在一个−在在)+圆周率

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## MATLAB代写

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