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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|PARTITIONING THE LATENT VARIABLES

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|PARTITIONING THE LATENT VARIABLES

Perhaps the most important choice when designing an MCMC sampler for a Bayesian NLP model is about the way in which the (latent) random variables of interest are partitioned. In their most general form, MCMC methods do not require the partitioning of the latent variables in the model. In fact, most of the introductory text to MCMC methods describes a single global state of all of the latent variables in the model (the state is a tuple with assignments for all $Z^{(i)}$ for $i \in{1, \ldots, n}$ and $\theta)$, and a chain that moves between states in this space. This state is often represented by a single random variable.

However, in NLP problems, treating the sample space with a single random variable without further refining it yields challenging inference problems, since this single random variable would potentially represent a complex combinatorial structure, such as a set of trees or graphs. Therefore, the set of latent variables is carved up into smaller subsets.

As mentioned earlier, NLP models are usually defined over discrete structures, and therefore the variables $Z^{(i)}$ commonly denote a structure such as a parse tree, an alignment or a sequence. We assume this type of structure for $Z^{(i)}$ for the rest of this section-i.e., a discrete compositional structure.

There are two common choices for partitioning the latent variables $Z^{(i)}$ in order to sample from the posterior:

Keep each variable $Z^{(i)}$ as a single atomic unit. When moving between states, re-sample a whole $Z^{(i)}$ structure for some $i$, possibly more than one structure at a time. This is one type of blocked sampling where the atomic unit is a whole predicted structure. It is often the case that each $Z^{(i)}$ is sampled using a dynamic programming algorithm. See Section 5.3 for more details.

Refine the predicted structure into a set of random variables, and sample each of them separately. This means, for example, that if $Z^{(i)}$ denotes a dependency tree, it will be refined to a set of random variables denoting the existence of edges in the tree. When moving between states, only one edge (or a small number of edges) is changed at a time. This is also often called pointwise sampling. See Section 5.3 for more details.

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|GIBBS SAM

The Gibbs sampling algorithm (Geman and Geman, 1984) is one of the most common MCMC algorithms used in the context of Bayesian NLP. In this setting, Gibbs sampling explores the state space, sampling $u_i$ each time for some $i \in{1, \ldots, p}$. These $u_i$ are drawn from the conditional distributions, $p\left(U_i \mid U_{-i}, \boldsymbol{X}\right)$. The full algorithm is given in Algorithm 5.1.

Note that at each step, the “state” of the algorithm is a set of values for $U$. At each iteration, the distributions $p\left(U_i \mid U_{-i}\right)$ condition on values $u_{-i}$ from the current state, and modify the current state-by setting a new value for one of the $U_i$. The update to the current state of the algorithm is immediate when a new value is drawn for one of the variables. The Gibbs algorithm does not delay the global state update, and each new draw of a random variable is immediately followed by a global state update. (However, see Section 5.4 for information about using a “stale” state for parallelizing the Gibbs sampler.)

Algorithm 5.1 returns a single sample once the Markov chain has converged. However, once the Gibbs sampling has converged, a stream of samples can be produced repeatedly by changing the state according to the conditional distributions and traversing the search space, collecting a set of samples. All of these samples are produced from the target distribution $p(U \mid \boldsymbol{X})$. While these samples are not going to be independent of each other, the farther a pair of samples are from each other, the less correlated they are.

To follow the discussion in Section 5.2.1, the Gibbs sampler can be pointwise or blocked (Gao and Johnson, 2008). Pointwise sampling implies that the Gibbs sampler alternates between steps that make very local changes to the state, such as sampling a single part-of-speech tag, while blocked sampling implies that larger pieces of the structure are sampled at each step.

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|GIBBS SAM

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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。