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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|MAP Estimation with Latent Variable

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|MAP Estimation with Latent Variable

When latent variables are introduced to the estimation problem (such as in the case of unsupervised learning), MAP estimation can become much more cumbersome. Assume a joint distribution that factorizes as follows:
$$p\left(x^{(1)}, \ldots, x^{(n)}, z^{(1)}, \ldots, z^{(n)}, \theta \mid \alpha\right)=p(\theta \mid \alpha) \prod_{i=1}^n p\left(z^{(i)} \mid \theta, \alpha\right) p\left(x^{(i)} \mid z^{(i)}, \theta, \alpha\right)$$
The latent structures are denoted by the random variables $Z^{(i)}$, and the observations are denoted by the random variables $X^{(i)}$. The posterior has the form:
$$p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$
The most comprehensive way to get a point estimate from this posterior through MAP estimation is to marginalize $Z^{(i)}$ and then find $\theta^$ as following: $$\theta^=\arg \max \theta \sum{z^{(1)}, \ldots, z^{(n)}} p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$
However, such an estimate often does not have an analytic form, and even computing it numerically can be inefficient. One possible way to avoid this challenge is to change the optimization problem in Equation 4.9 to:
$$\theta^*=\arg \max \theta \max {z^{(1)}, \ldots, z^{(n)}} p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|POSTERIOR APPROXIMATIONS BASED ON THE MAP SOLUTION

In cases where $\Theta \subset \mathbb{R}^K$, the mode of the posterior can be used to obtain an approximate distribution of the posterior. This approximation assumes that the posterior behaves similarly to a multivariate normal distribution with a mean at the posterior mode (note that the mode and the mean of the multivariate normal distribution are identical).

Let $x$ be an observation from the likelihood $p(x \mid \theta)$ with a prior $p(\theta)$. This normal approximation to the posterior (also called “Laplace approximation”) assumes that:
$$p(\theta \mid x) \approx f\left(\theta \mid \theta^, \Sigma^\right)$$

where
$$f\left(\theta \mid \theta^, \Sigma^\right)=\frac{1}{(2 \pi)^{-K / 2} \sqrt{\left|\operatorname{det}\left(\Sigma^\right)\right|}} \exp \left(-\frac{1}{2}\left(\theta-\theta^\right)^{\top}\left(\Sigma^\right)^{-1}\left(\theta-\theta^\right)\right),$$
is the density of the multivariate normal distribution with mean $\theta^$ (the mode of the posterior) and covariance matrix $\Sigma^$ defined as inverse of the Hessian of the negated log-posterior at point $\theta^:$ $$\left(\Sigma^\right)_{i, j}^{-1}=\frac{\partial^2 h}{\partial \theta_i \partial \theta_j}\left(\theta^*\right)$$
with $h(\theta)=-\log p(\theta \mid X=x)$. Note that the Hessian must be a positive definite matrix to serve as the covariance matrix of the distribution in Equation 4.11. This means that the Hessian has to be a symmetric matrix. A necessary condition is that the second derivatives of the logposterior next to the mode are continuous.

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|MAP Estimation with Latent Variable

$$p\left(x^{(1)}, \ldots, x^{(n)}, z^{(1)}, \ldots, z^{(n)}, \theta \mid \alpha\right)=p(\theta \mid \alpha) \prod_{i=1}^n p\left(z^{(i)} \mid \theta, \alpha\right) p\left(x^{(i)} \mid z^{(i)}, \theta, \alpha\right)$$

$$p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$

$$\theta^{=} \arg \max \theta \sum z^{(1)}, \ldots, z^{(n)} p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$

$$\theta^*=\arg \max \theta \max z^{(1)}, \ldots, z^{(n)} p\left(\theta, z^{(1)}, \ldots, z^{(n)} \mid x^{(1)}, \ldots, x^{(n)}, \alpha\right)$$

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|POSTERIOR APPROXIMATIONS BASED ON THE MAP SOLUTION

$\begin{array}{ll}\text { 是具有均值的多元正态分布的密度缺少上标或下标参数 } & \text { (后验模式) 和协方差矩 }\end{array}$ 阵枿少上标或下标参数 定义为点处取反对数后验的 Hessian 矩阵的倒数 $\theta$ :

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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。