### 统计代写 | Statistical Learning and Decision Making代考|Conditional Distributions

statistics-lab™ 为您的留学生涯保驾护航 在代写Statistical Learning and Decision Making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Statistical Learning and Decision Making代写方面经验极为丰富，各种代写Statistical Learning and Decision Making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 高等楖率论
• 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

## 统计代写 | Statistical Learning and Decision Making代考|Conditional Distributions

The previous section introduced the idea of independence, which can help reduce the number of parameters used to define a joint distribution. However, as was mentioned, independence can be too strong of an assumption. This section will introduce the idea of conditional independence, which can help reduce the number of independent parameters without making assumptions that are as strong. Before discussing conditional independence, we will first introduce the notion of a conditional distribution, which is a distribution over a variable given the value of one or more others.

The definition of conditional probability states that
$$P(x \mid y)=\frac{P(x, y)}{P(y)}$$
where $P(x \mid y)$ is read as “probability of $x$ given $y$.” In some contexts, it is common to refer to $y$ as evidence.

Since a conditional probability distribution is a probability distribution over one or more variables given some evidence, we know that
$$\sum_{x} P(x \mid y)=1$$
for a discrete $X$. If $X$ is continuous, it integrates to 1 .
We can incorporate the definition of conditional probability into equation (2.18) to obtain a slightly different form of the law of total probability:
$$P(x)=\sum_{y} P(x \mid y) P(y)$$
for a discrete distribution.
Another useful relationship that follows from the definition of conditional probability is Bayes’ rule: ${ }^{12}$
$$P(x \mid y)=\frac{P(y \mid x) P(x)}{P(y)}$$
If we have a representation of a conditional distribution $P(y \mid x)$, we can apply Bayes’s rule to swap the $y$ and $x$ to obtain the conditional distribution $P(x \mid y)$.
We will now discuss a variety of ways to represent conditional probability distributions over discrete and continuous variables.

## 统计代写 | Statistical Learning and Decision Making代考|Discrete Conditional Models

A conditional probability distribution over discrete variables can be represented using a table. In fact, we can use the same discrete factor representation that we used in section $2.3 .1$ for joint distributions. Table $2.3$ shows an example of a table representing $P(X \mid Y, Z)$ with all binary variables. In contrast with a joint table (e.g, table $2.1$ ), the column containing the probabilities need not sum to 1. However, if we sum over the probabilities that are consistent with what we are conditioning on, we must get 1 . For example, conditioning on $y^{0}$ and $z^{0}$ (the evidence), we have
$$P\left(x^{0} \mid y^{0}, z^{0}\right)+P\left(x^{1} \mid y^{0}, z^{0}\right)=0.08+0.92=1$$ Conditional probability tables can become quite large. If we were to create a table like table $2.3$ where all variables can take on $m$ values and we are conditioning on $n$ variables, there would be $m^{n+1}$ rows. However, since the $m$ values of the variable we are not conditioning on must sum to 1 , there are only $(m-1) m^{n}$ independent parameters. There is still an exponential growth in the number of variables on which we condition. When there are many repeated values in the conditional probability table, a decision tree (introduced in section 2.3.1) may be a more efficient representation.

## 统计代写 | Statistical Learning and Decision Making代考|Conditional Linear Gaussian Models

The conditional linear Gaussian model combines the ideas of conditional Gaussian and linear Gaussian models to be able to handle conditioning a continuous variable on both discrete and continuous variables. Suppose we want to represent $p(X \mid Y, Z)$, where $X$ and $Y$ are continuous and $Z$ is discrete with values $1: n$. The conditional density function is then
$$p(x \mid y, z)= \begin{cases}\mathcal{N}\left(x \mid m_{1} y+b_{1}, \sigma_{1}^{2}\right) & \text { if } z^{1} \ \vdots \ \mathcal{N}\left(x \mid m_{n} y+b_{n}, \sigma_{n}^{2}\right) & \text { if } z^{n}\end{cases}$$
Above, the parameter vector $\theta=\left[m_{1: n}, b_{1: n}, \sigma_{1: n}\right]$ has $3 n$ components.

We can use a sigmoid ${ }^{13}$ model to represent a distribution over a binary variable conditioned on a continuous variable. For example, we may want to represent $P\left(x^{1} \mid y\right)$, where $x$ is binary and $y$ is continuous. Of course, we could just set a threshold $\theta$ and say $P\left(x^{1} \mid y\right)=0$ if $y<\theta$ and $P\left(x^{1} \mid y\right)=1$ otherwise. However, in many applications, we may not want to have such a hard threshold that results in assigning zero probability to $x^{1}$ for certain values of $y$.

Instead of a hard threshold, we could use a soft threshold that assigns low probabilities when below a threshold and high probabilities when above a threshold. One way to represent a soft threshold is to use a logit model, which produces a sigmoid curve:
$$P\left(x^{1} \mid y\right)=\frac{1}{1+\exp \left(-2 \frac{y-\theta_{1}}{\theta_{2}}\right)}$$
The parameter $\theta_{1}$ governs the location of the threshold, and $\theta_{2}$ controls the “softness” or spread of the probabilities. Figure $2.10$ shows an example plot of $P\left(x^{1} \mid y\right)$ with a logit model.

∑X磷(X∣是)=1

## 统计代写 | Statistical Learning and Decision Making代考|Conditional Linear Gaussian Models

p(X∣是,和)={ñ(X∣米1是+b1,σ12) 如果 和1 ⋮ ñ(X∣米n是+bn,σn2) 如果 和n

## 有限元方法代写

tatistics-lab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。