assignmentutor-lab™ 为您的留学生涯保驾护航 在代写贝叶斯分析Bayesian Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写贝叶斯分析Bayesian Analysis代写方面经验极为丰富，各种代写贝叶斯分析Bayesian Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
assignmentutor™您的专属作业导师

统计代写|贝叶斯分析代写Bayesian Analysis代考|THE USE OF CONJUGATE PRIORS WITH LATENT VARIABLE

Earlier in this section, it was demonstrated that conjugate priors make Bayesian inference tractable when complete data is available. Example $3.1$ demonstrated this by showing how the posterior distribution can easily be identified when assuming a conjugate prior. Explicit computation of the evidence normalization constant with conjugate priors is often unnecessary, because the product of the likelihood together with the prior lead to an algebraic form of a well-known distribution.

As mentioned earlier, the calculation of the posterior normalization constant is the main obstacle in performing posterior inference. If this is the case, we can ask: do conjugate priors help in the case of latent variables being present in the model? With latent variables, the normalization constant is more complex, because it involves the marginalization of both the parameters and the latent variables. Assume a full distribution over the parameters $\theta$, latent variables $z$ and observed variables $x$ (both being discrete), which factorize as follows:
$$p(\theta, z, x \mid \alpha)=p(\theta \mid \alpha) p(z \mid \theta) p(x \mid z, \theta)$$
The posterior over the latent variables and parameters has the form (see Section $2.2 .2$ for a more detailed example of such posterior):
$$p(\theta, z \mid x, \alpha)=\frac{p(\theta \mid \alpha) p(z \mid \theta) p(x \mid z, \theta)}{p(x \mid \alpha)}$$
and therefore, the normalization constant $p(x \mid \alpha)$ equals:
$$p(x \mid \alpha)=\sum_{z}\left(\int_{\theta} p(\theta) p(z \mid \theta) p(x \mid z, \theta) d \theta\right)=\sum_{z} D(z)$$
where $D(z)$ is defined to be the term inside the sum above. Equation $3.6$ demonstrates that conjugate priors are useful even when the normalization constant requires summing over latent variables. If the prior family is conjugate to the distribution $p(X, Z \mid \theta)$, then the function $D(z)$ will be mathematically easy to compute for any $z$. However, it is not true that $\sum_{z} D(z)$ is always tractable, since the form of $D(z)$ can be quite complex.

统计代写|贝叶斯分析代写Bayesian Analysis代考|MIXTURE OF CONJUGATE PRIORS

Mixture models are a simple way to extend a family of distributions into a more expressive family. If we have a set of distributions $p_{1}(X), \ldots, p_{M}(X)$, then a mixture model over this set of distributions is parametrized by an $M$ dimensional probability vector $\left(\lambda_{1}, \ldots, \lambda_{M}\right)\left(\lambda_{i} \geq 0\right.$, $\left.\sum_{i} \lambda_{i}=1\right)$ and defines distributions over $X$ such that:
$$p(X \mid \lambda)=\sum_{i=1}^{M} \lambda_{i} p_{i}(X)$$
Section 1.5.3 gives an example of a mixture-of-Gaussians model. The idea of mixture models can also be used for prior families. Let $p(\theta \mid \alpha)$ be a prior from a prior family with $\alpha \in A$. Then, it is possible to define a prior of the form:
$$p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}, \lambda_{1}, \ldots, \lambda_{M}\right)=\sum_{i=1}^{M} \lambda_{i} p\left(\theta \mid \alpha^{i}\right)$$
where $\lambda_{i} \geq 0$ and $\sum_{i=1}^{M} \lambda_{i}=1$ (i.e., $\lambda$ is a point in the $M-1$ dimensional probability simplex). This new prior family, which is hyperparametrized by $\alpha^{i} \in A$ and $\lambda_{i}$ for $i \in{1, \ldots M}$ will actually be conjugate to a likelihood $p(x \mid \theta)$ if the original prior family $p(\theta \mid \alpha)$ for $\alpha \in A$ is also conjugate to this likelihood.
To see this, consider that when using a mixture prior, the posterior has the form:
\begin{aligned} p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right) &=\frac{p(x \mid \theta) p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}\right.}{\int_{\theta} p(x \mid \theta) p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}\right.} \ &=\frac{\sum_{i=1}^{\cdot M} \lambda_{i} p(x \mid \theta) p\left(\theta \mid \alpha^{I}\right)}{\sum_{i=1}^{M} \lambda_{i} Z_{i}} \end{aligned}
where
$$Z_{i}=\int_{\theta} p(x \mid \theta) p\left(\theta \mid \alpha^{i}\right) d \theta$$
Therefore, it holds that:
$$p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right)=\frac{\sum_{i=1}^{M}\left(\lambda_{i} Z_{i}\right) p\left(\theta \mid x, \alpha^{i}\right)}{\sum_{i=1}^{M} \lambda_{i} Z_{i}}$$

because $p(x \mid \theta) p\left(\theta \mid \alpha^{i}\right)=Z_{i} p\left(\theta \mid x, \alpha^{i}\right)$. Because of conjugacy, each $p\left(\theta \mid x, \alpha^{i}\right)$ is equal to $p\left(\theta \mid \beta^{i}\right)$ for some $\beta^{i} \in A(i \in{1, \ldots, M})$. The hyperparameters $\beta^{i}$ are the updated hyperparameters following posterior inference. Therefore, it holds:
$$p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right)=\sum_{i=1}^{M} \lambda_{i}^{\prime} p\left(\theta \mid \beta^{i}\right)$$
for $\lambda_{i}^{\prime}=\lambda_{i} Z_{i} /\left(\sum_{i=1}^{M} \lambda_{i} Z_{i}\right)$.

统计代写|贝叶斯分析代写Bayesian Analysis代考|RENORMALIZED CONJUGATE DISTRIBUTIONS

In the previous section, we saw that one could derive a more expressive prior family by using a basic prior distribution in a mixture model. Renormalizing a conjugate prior is another way to change the properties of a prior family while still retaining conjugacy.

Let us assume that a prior $p(\theta \mid \alpha)$ is defined over some parameter space $\Theta$. It is sometimes the case that we want to further constrain $\Theta$ into a smaller subspace, and define $p(\theta \mid \alpha)$ such that its support is some $\Theta_{0} \subset \Theta$. One way to do so would be to define the following distribution $p^{\prime}$ over $\Theta_{0}$ :

$$p^{\prime}(\theta \mid \alpha)=\frac{p(\theta \mid \alpha)}{\int_{\theta^{\prime} \in \Theta_{0}} p\left(\theta^{\prime} \mid \alpha\right) d \theta^{\prime}} .$$
This new distribution retains the same ratio between probabilities of elements in $\Theta_{0}$ as $p$, but essentially allocates probability 0 to any element in $\Theta \backslash \Theta_{0}$.

It can be shown that if $p$ is a conjugate family to some likelihood, then $p^{\prime}$ is conjugate to the same likelihood as well. This example actually demonstrates that conjugacy, in its pure form does not necessitate tractability by using the conjugate prior together with the corresponding likelihood. More specifically, the integral over $\Theta_{0}$ in the denominator of Equation $3.7$ can often be difficult to compute, and approximate inference is required.

The renormalization of conjugate distributions arises when considering probabilistic context-free grammars with Dirichlet priors on the parameters. In this case, in order for the prior to allocate zero probability to parameters that define non-tight PCFGs, certain multinomial distributions need to be removed from the prior. Here, tightness refers to a desirable property of a PCFG so that the total measure of all finite parse trees generated by the underlying context-free grammar is 1 . For a thorough discussion of this issue, see Cohen and Johnson (2013).

统计代写|贝叶斯分析代写Bayesian Analysis代考|THE USE OF CONJUGATE PRIORS WITH LATENT VARIABLE

$$p(\theta, z, x \mid \alpha)=p(\theta \mid \alpha) p(z \mid \theta) p(x \mid z, \theta)$$

$$p(\theta, z \mid x, \alpha)=\frac{p(\theta \mid \alpha) p(z \mid \theta) p(x \mid z, \theta)}{p(x \mid \alpha)}$$

$$p(x \mid \alpha)=\sum_{z}\left(\int_{\theta} p(\theta) p(z \mid \theta) p(x \mid z, \theta) d \theta\right)=\sum_{z} D(z)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|MIXTURE OF CONJUGATE PRIORS

$$p(X \mid \lambda)=\sum_{i=1}^{M} \lambda_{i} p_{i}(X)$$
1.5.3 节给出了一个混合高斯模型的例子。混合模型的思想也可以用于先验族。让 $p(\theta \mid \alpha)$ 来自以前的家庭 $\alpha \in A$. 然后，可以定义形式的 先验:
$$p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}, \lambda_{1}, \ldots, \lambda_{M}\right)=\sum_{i=1}^{M} \lambda_{i} p\left(\theta \mid \alpha^{i}\right)$$

$$p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right)=\frac{p(x \mid \theta) p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}\right.}{\int_{\theta} p(x \mid \theta) p\left(\theta \mid \alpha^{1}, \ldots, \alpha^{M}\right.} \quad=\frac{\sum_{i=1}^{M} \lambda_{i} p(x \mid \theta) p\left(\theta \mid \alpha^{I}\right)}{\sum_{i=1}^{M} \lambda_{i} Z_{i}}$$

$$Z_{i}=\int_{\theta} p(x \mid \theta) p\left(\theta \mid \alpha^{i}\right) d \theta$$

$$p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right)=\frac{\sum_{i=1}^{M}\left(\lambda_{i} Z_{i}\right) p\left(\theta \mid x, \alpha^{i}\right)}{\sum_{i=1}^{M} \lambda_{i} Z_{i}}$$

$$p\left(\theta \mid x, \alpha^{1}, \ldots, \alpha^{M}, \lambda\right)=\sum_{i=1}^{M} \lambda_{i}^{\prime} p\left(\theta \mid \beta^{i}\right)$$

统计代写|贝叶斯分析代写Bayesian Analysis代考|RENORMALIZED CONJUGATE DISTRIBUTIONS

$$p^{\prime}(\theta \mid \alpha)=\frac{p(\theta \mid \alpha)}{\int_{\theta^{\prime} \in \Theta_{0}} p\left(\theta^{\prime} \mid \alpha\right) d \theta^{\prime}}$$

有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。