### 数学代写|概率论代写Probability theory代考|STAT4528

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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

## 数学代写|概率论代写Probability theory代考|Parametric Identification

Consider a linear or nonlinear relationship (model class $C$ ) to describe the quantity of concern $y:$
$$y=y\left(\mathbf{x} ; \theta_{\mathrm{m}}, C\right)$$
where $\theta_{m}$ is the uncertain parameters contained in the model $C ; \mathbf{x}$ is the measured variable vector. Use $Y$ to denote the measurement of $y$ and it is assumed that the difference can be adequately modeled as Gaussian random variable:
$$Y=y+\varepsilon$$
where $\varepsilon$ is a Gaussian random variable with zero mean and variance $\sigma_{\varepsilon}^{2}$ referred hereafter as prediction-error variance, and it is adopted to represent the measurement noise and modeling error. The uncertain parameter vector $\theta=\left[\theta_{\mathrm{m}}^{T}, \sigma_{\varepsilon}^{2}\right]^{T}$ includes the model parameter vector $\theta_{\mathrm{m}}$ and the prediction-error variance $\sigma_{\varepsilon}^{2}$. Their uncertainties can be represented by using the posterior PDFs, and the probability expression of posterior PDF given the data $D$ is written as:
$$p(\boldsymbol{\theta} \mid D, C)=c_{0} p(\boldsymbol{\theta} \mid C) p(D \mid \boldsymbol{\theta}, C)$$
where $c_{0}=1 / p(D \mid C)$ is a normalizing constant; $p(D \mid C)$ is the evidence of model class $C ; p(\theta \mid C)$ is the prior PDF of the uncertain parameters in $\theta$, which is based on the previous knowledge or user’s judgment; $p(D \mid \theta, C)$ is the likelihood function expressing the level of data fitting. If the prediction errors in different measured data are statistically independent, the likelihood function can be computed by:
$$p(D \mid \theta, C)=\left(2 \pi \sigma_{\varepsilon}^{2}\right)^{-\frac{N}{2}} \exp \left[-\frac{N}{2 \sigma_{\varepsilon}^{2}} J_{g}\left(\theta_{\mathrm{m}} ; D, C\right)\right]$$
where $N$ is the number of measured data; $J_{g}\left(\theta_{\mathrm{m}} ; D, C\right)$ is the goodness-of-fit function, and is given by:
$$J_{g}\left(\theta_{\mathrm{m}} ; D, C\right)=\frac{1}{N} \sum_{n=1}^{N}[Y(n)-y(n)]^{2}$$

## 数学代写|概率论代写Probability theory代考|Model Class Selection

The probability/plausibility of a model class conditional on a database $D$ can be obtained by using the Bayes’ theorem as follows [41]:
$$P\left(C_{j} \mid D\right)=\frac{p\left(D \mid C_{j}\right) \cdot P\left(C_{j}\right)}{p(D)}, j=1,2, \ldots, N_{c}$$

where $N_{C}$ is the number of model classes to be considered; $p(D)=$ $\sum_{j=1}^{N_{c}} p\left(D \mid C_{j}\right) P\left(C_{j}\right)$ is a normalizing constant and $P\left(C_{j}\right)$ expresses the user’s judgment on the initial plausibility of the model classes with $\sum_{j=1}^{N_{c}} P\left(C_{j}\right)=1$. In general, uniform prior plausibility can be assumed, that is, $P\left(C_{j}\right)=1 / N_{c}$. The evidence of the model class $C_{j} p\left(D \mid C_{j}\right)$ can be obtained using the theorem of total probability:
$$p\left(D \mid C_{j}\right)=\int_{\Theta} p\left(D \mid \theta, C_{j}\right) p\left(\theta \mid C_{j}\right) d \theta, j=1,2, \ldots, N_{c}$$
where the parameter vector $\theta$ belongs to the parameter space $\Theta_{j} \subset \Re^{N_{j}}$ and it is defined in each model class $\left(C_{j}\right)$. In general, Eq. $(1.10)$ involves a high-dimensional integral and it is computationally expensive for direct computation. In the globally identifiable cases, the posterior PDF given large volume of data can be approximated by a Gaussian distribution, so the evidence $p\left(D \mid C_{j}\right)$ can be approximated by using Laplace’s method for asymptotic expansion [42]:
$$p\left(D \mid C_{j}\right) \approx p\left(D \mid \theta^{}, C_{j}\right) p\left(\theta^{} \mid C_{j}\right)(2 \pi)^{\frac{x_{j}}{2}}\left|\mathbf{H}{j}\left(\theta^{}\right)\right|^{-\frac{1}{2}}, j=1,2, \ldots, N{c}$$
where $N_{j}$ is the number of uncertain parameters for model class $C_{j}, \boldsymbol{\theta}^{}$ is the optimal parameter vector that maximizes $p\left(\theta \mid D, C_{j}\right)$. The maximum likelihood value $p\left(D \mid \theta^{}, C_{j}\right)$ is larger for the model class that can better fit the data. Thus, the model class with more uncertain parameters is more likely to have greater maximum likelihood value. The remaining terms in Eq. (1.11) are called Ockham factor [43]: $$O_{j}=p\left(\theta^{} \mid C_{j}\right)(2 \pi)^{\frac{N_{j}}{2}}\left|\mathbf{H}_{j}\left(\theta^{*}\right)\right|^{-\frac{1}{2}}$$
The Ockham factor provides a measurement for the robustness of the model class and its value decreases exponentially with the number of uncertain parameters in the model class.

On the other hand, the exact expression for $p\left(D \mid C_{j}\right)$ can be obtained based on the Bayes’ theorem:
$$p\left(D \mid C_{j}\right)=\frac{p\left(D \mid \theta^{}, C_{j}\right) p\left(\theta^{} \mid C_{j}\right)}{p\left(\theta^{*} \mid D, C_{j}\right)}$$

## 数学代写|概率论代写Probability theory代考|Metropolis-Hastings Algorithm

The Metropolis algorithm [44] is the first sampling method for iterative simulations, and it can be conducted through simple steps called the proposal-and-decision steps. Starting with the initial point, the algorithm proceeds by transferring the current state to the new state generated from a symmetric proposal distribution with a probability value. This algorithm was later generalized by Hastings [45] and the asymmetric proposal distribution is allowed to be used in generating the new state. This generalized algorithm is called the Metropolis-Hastings (MH) algorithm. In the MH method, the simulated Markov chain samples tend to follow the target PDF as the sample number approaches to infinite. With respect to the problem involved in Bayesian approach, the posterior PDF in Eq. (1.3) is the target PDF, and it is expressed as the product of a constant and a function, that is, $p(\theta \mid D, C)=c_{0} q(\theta)$.
(1) Select an appropriate proposal PDF $T(\xi \mid \theta)$, which is a PDF for $\xi$ that depends on $\theta$;
(2) At stage $k=1$, choose an arbitrary point to be the initial sample $\theta_{1}$;
(3) At the stage $k$ ( $k$ starts from 2), a new point $\theta_{c}$ randomly generates from the proposal distribution $T\left(\theta_{c} \mid \theta_{k-1}\right)$, and compute the ratio of densities $r$ :
$$r=\frac{q\left(\boldsymbol{\theta}{c}\right) T\left(\boldsymbol{\theta}{k-1} \mid \boldsymbol{\theta}{c}\right)}{q\left(\boldsymbol{\theta}{k-1}\right) T\left(\boldsymbol{\theta}{c} \mid \boldsymbol{\theta}{k-1}\right)}$$
(4) Generate a random number $u$ from a uniform distribution $U(0,1)$;
(5) Determine whether $\theta_{c}$ is acceptable with the following acceptance rule: if $u \leq r$, $\theta_{c}$ is accepted and set $\theta_{k}=\theta_{c}$; otherwise, $\operatorname{set} \theta_{k}=\theta_{k-1}$. Then go back to step (3);

(6) Repeat steps (3)-(5) until the target number of samples $(N)$ is reached, and a group of Markov chain samples $\left[\theta_{k}: k=1,2, \ldots, N\right]$ can be obtained. Under the assumption of ergodicity, the Markov chain will converge to the stationary state as the sample number reaches to infinite, that is, the sample tends to $p(\theta \mid D, C)$. Before the Markov chain reaches the stationary state, the generated samples as burn-in samples should be discarded. However, when the sample size is limited, the ergodicity of Markov chain is still an issue.

In the MH algorithm, the proposal PDF plays an important role, and its determination has significant influence on the efficiency of algorithm. In particular, it is not easy to conduct MH algorithm efficiently for these two cases: (a) If the scale of proposal PDF is too large, most of the candidate samples will be rejected and the accepted ratio of sample will be very low. The Markov chain will contain many repeated samples, and the correlation among samples will be large. (b) If its scale is too small, the accepted ratio will be very high and the Markov chain needs longer time to run over important regions of the posterior PDF. The Markov chain samples will be highly dependent. In order to enhance the efficiency of $\mathrm{MH}$ algorithm, the guidelines of the proposal PDF, which control the accepted ratio in a certain range $(20-40 \%)$, were suggested [46]. However, the MH algorithm still may not be efficient in many cases, for example, the posterior PDF is very peaked, high-dimensional or multimodal, and so on [47]. Thus, it is not feasible to simulate samples according to the updated PDF directly using the MH algorithm.

## 数学代写|概率论代写Probability theory代考|Parametric Identification

p(θ∣D,C)=C0p(θ∣C)p(D∣θ,C)

p(D∣θ,C)=(2圆周率σe2)−ñ2经验⁡[−ñ2σe2ĴG(θ米;D,C)]

ĴG(θ米;D,C)=1ñ∑n=1ñ[是(n)−是(n)]2

## 数学代写|概率论代写Probability theory代考|Model Class Selection

p(D∣Cj)=∫θp(D∣θ,Cj)p(θ∣Cj)dθ,j=1,2,…,ñC

p(D∣Cj)≈p(D∣θ,Cj)p(θ∣Cj)(2圆周率)Xj2|Hj(θ)|−12,j=1,2,…,ñC

○j=p(θ∣Cj)(2圆周率)ñj2|Hj(θ∗)|−12

p(D∣Cj)=p(D∣θ,Cj)p(θ∣Cj)p(θ∗∣D,Cj)

## 数学代写|概率论代写Probability theory代考|Metropolis-Hastings Algorithm

Metropolis 算法 [44] 是迭代模拟的第一个采样方法，它可以通过称为建议和决策步骤的简单步骤来进行。从初始点开始，算法通过将当前状态转移到具有概率值的对称提议分布生成的新状态来继续进行。该算法后来被 Hastings [45] 推广，并且允许使用非对称提议分布来生成新状态。这种广义算法称为 Metropolis-Hastings (MH) 算法。在 MH 方法中，随着样本数接近无穷大，模拟的马尔可夫链样本倾向于跟随目标 PDF。关于贝叶斯方法中涉及的问题，方程式中的后验概率密度函数。(1.3) 是目标 PDF，p(θ∣D,C)=C0q(θ).
(1) 选择合适的提案 PDF吨(X∣θ), 这是一个 PDFX这取决于θ;
(2) 阶段ķ=1，选择任意一个点作为初始样本θ1;
(3) 舞台上ķ ( ķ从 2) 开始，一个新的点θC从提案分布中随机生成吨(θC∣θķ−1)，并计算密度比r :

r=q(θC)吨(θķ−1∣θC)q(θķ−1)吨(θC∣θķ−1)
(4) 产生一个随机数在从均匀分布在(0,1);
(5) 判断是否θC符合以下接受规则是可以接受的：如果在≤r, θC被接受并设置θķ=θC; 否则，放⁡θķ=θķ−1. 然后返回步骤（3）；

(6)重复步骤(3)-(5)，直到达到目标样本数(ñ)达到，一组马尔可夫链样本[θķ:ķ=1,2,…,ñ]可以获得。在遍历性假设下，随着样本数达到无穷大，马尔可夫链会收敛到平稳状态，即样本趋向于p(θ∣D,C). 在马尔可夫链达到静止状态之前，应丢弃生成的作为老化样本的样本。然而，当样本量有限时，马尔可夫链的遍历性仍然是一个问题。

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## MATLAB代写

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