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# 统计代写|贝叶斯分析代考Bayesian Analysis代写|ST308 INDEPENDENCE ASSUMPTIONS IN MODELS

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## 统计代写|贝叶斯分析代考Bayesian Analysis代写|INDEPENDENCE ASSUMPTIONS IN MODELS

How do we actually construct a generative model given a phenomeon we want to model? We first have to decide which random variables compose the model. Clearly, the observed data need to be associated with a random variable, and so do the predicted values. We can add latent variables as we wish, if we believe there are hidden factors that link between the observed and predicted values.

Often, the next step is deciding exactly how these random variables relate to each other with respect to their conditional independence. At this point, we are not yet assigning distributions to the various random variables, but just hypothesizing about the information flow between them. These independence assumptions need to balance various trade-offs. On one hand, the weaker they are (i.e., the more dependence there is), the more expressive the model family isin other words, the model family includes more distributions. On the other hand, if they are too expressive, we might run into problems such as overfitting with small amounts of data, or technical problems such as computationally expensive inference.

Looking at various models in NLP, we see that the independence assumptions we make are rather strong-it is usually the case that a given random variable depends on a small number of other random variables. In that sense, the model has “local factors” in its joint distribution.

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|DIRECTED GRAPHICAL MODELS

As mentioned above, detailing the full generative story or joint distribution is necessary to fully comprehend the inner workings of a given statistical model. However, in cases where we are just interested in describing the independence assumptions that exist in the model, graphical representations can help to elucidate these assumptions.

Given that Bayesian models are often generative, the most important type of graphical representation for them is “directed graphical models” (or “Bayesian networks”). See Murphy (2012) for an introduction to other types of graphical models (GMs), such as undirected graphical models. In a Bayesian network, each random variable in the joint distribution is represented as a vertex in a graph. There are incoming edges to each vertex $X$ from all random variables that $X$ conditions on, when writing down the joint distribution and inspecting the factor that describes the distribution over $X$.

The basic type of independence assumption a Bayesian network describes is the following. A random variable $X$ is conditionally independent of all its ancestors when conditioned on its immediate parents. This type of property leads to an extensive calculus and a set of graphtheoretic decision rules that can assist in determining whether one set of random variables in the model is independent of another set when conditioned on a third set (i.e., the random variables in the third set are assumed to be observed for that conditional independence test). The calculus includes a few logical relations, including symmetry: if $X$ and $Y$ are conditionally independent given $Z$, then $Y$ and $X$ are conditionally independent given $Z$; decomposition: if $X$ and $Y \cup W$ are conditionally independent given $Z$, then $X$ and $Y$ are conditionally independent given $Z$; contraction: if $X$ and $Y$ are conditionally independent given $Z$, and $X$ and $Y$ are conditionally independent given $Y \cup Z$, then $X$ is conditionally independent of $W \cup Y$ given $Z$; weak union:

if $X$ and $Y \cup Z$ are conditionally independent given $W$, then $X$ and $Y$ are conditionally independent given $Z \cup W$. Here, $X, Y, Z$ and $W$ are subsets of random variables in a probability distribution. See Pearl (1988) for more information.

Bayesian networks also include a graphical mechanism to describe a variable or unfixed number of random variables, using so-called “plate notation.” With plate notation, a set of random variables is placed inside plates. A plate represents a set of random variables with some count. For example, a plate could be used to describe a set of words in a document. Figure 1.1 provides an example of a use of the graphical plate language. Random variables, denoted by circles, are the basic building blocks in this graphical language; the plates are composed of such random variables (or other plates) and denote a “larger object.” For example, the random variable $W$ stands for a word in a document, and the random variable $Z$ is a topic variable associated with that word. As such, the plate as a whole denotes a document, which is indeed a larger object composed from $N$ random variables from the type of $Z$ and $W$.

# 贝叶斯分析代写

## 统计代写|贝叶斯分析代考Bayesian Analysis代写|DIRECTED GRAPHICAL MODELS

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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。