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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Joint Distributions

A joint distribution is a probability distribution over multiple variables. A distribution over a single variable is called a univariate distribution, and a distribution over multiple variables is called a multivariate distribution. If we have a joint distribution over two discrete variables $X$ and $Y$, then $P(x, y)$ denotes the probability that both $X=x$ and $Y=y$.

From a joint distribution, we can compute a marginal distribution of a variable or a set of variables by summing out all other variables using what is known as the law of total probability:?
$$P(x)=\sum_{y} P(x, y)$$
This property is used throughout this book.
Real-world decision making often requires reasoning about joint distributions involving many variables. Sometimes there are complex relationships between the variables that are important to represent. We may use different strategies to represent joint distributions depending on whether the variables involve discrete or continuous values.

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

If the variables are discrete, the joint distribution can be represented by a table like the one shown in table 2.1. That table lists all possible assignments of values to three binary variables. Each variable can only be 0 or 1 , resulting in $2^{3}=8$ possible assignments. As with other discrete distributions, the probabilities in the table must sum to 1 . It follows that although there are eight entries in the table, only seven of them are independent. If $\theta_{i}$ represents the probability in the $i$ th row in the table, then we only need the parameters $\theta_{1}, \ldots, \theta_{7}$ to represent the distribution because we know $\theta_{8}=1-\left(\theta_{1}+\ldots+\theta_{7}\right)$.

If we have $n$ binary variables, then we need as many as $2^{n}-1$ independent parameters to specify the joint distribution. This exponential growth in the number of parameters makes storing the distribution in memory difficult. In some cases, we can assume that our variables are independent, which means that the realization of one does not affect the probability distribution of the other. If $X$ and $Y$ are independent, which is sometimes written $X \perp Y$, then we know $P(x, y)=P(x) P(y)$

for all $x$ and $y$. Suppose we have binary variables $X_{1}, \ldots, X_{n}$ that are all independent of each other, resulting in $P\left(x_{1: n}\right)=\prod_{i} P\left(x_{i}\right)$. This factorization allows us to represent this joint distribution with only $n$ independent parameters instead of the $2^{n}-1$ required when we cannot assume independence (see table $2.2$ ). Independence can result in an enormous savings in terms of representational complexity, but it is often a poor assumption.

We can represent joint distributions in terms of factors. A factor $\phi$ over a set of variables is a function from assignments of those variables to the real numbers. In order to represent a probability distribution, the real numbers in the factor must be non-negative. A factor with non-negative values can be normalized such that it represents a probability distribution. Algorithm 2.1 provides an implementation for discrete factors, and example $2.3$ demonstrates how they work.

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

We can also define joint distributions over continuous variables. A rather simple distribution is the multivariate uniform distribution, which assigns a constant probability density everywhere there is support. We can use $\mathcal{U}(\mathbf{a}, \mathbf{b})$ to represent a uniform distribution over a box, which is a Cartesian product of intervals with the $i$ th interval being $\left[a_{i}, b_{i}\right]$. This family of uniform distributions is a special type of multivariate product distribution, which is a distribution defined in terms of the product of univariate distributions. In this case,
$$\mathcal{U}(\mathbf{x} \mid \mathbf{a}, \mathbf{b})=\prod_{i} \mathcal{U}\left(x_{i} \mid a_{i}, b_{i}\right)$$
We can create a mixture model from a weighted collection of multivariate uniform distributions, just as we can with univariate distributions. If we have a joint distribution over $n$ variables and $k$ mixture components, we need to define $k(2 n+1)-1$ independent parameters. For each of the $k$ components, we need to define the upper and lower bounds for each of the variables in addition to their weights. We can subtract 1 because the weights must sum to 1 . Figure $2.6$ shows an example that can be represented by five components.

It is also common to represent piecewise constant density functions by discretizing each of the variables independently. The discretization is represented by a set of bin edges for each variable. These bin edges define a grid over the variables. We then associate a constant probability density with each grid cell. The bin edges do not have to be uniformly separated. In some cases, it may be desirable to have increased resolution around certain values. Different variables might have different bin edges associated with them. If there are $n$ variables and $m$ bins for each variable, then we need $m^{n}-1$ independent parameters to define the distribution-in addition to the values that define the bin edges.

In some cases, it may be more memory efficient to represent a continuous joint distribution as a decision tree in a manner similar to what we discussed for discrete joint distributions. The internal nodes compare variables against thresholds and the leaf nodes are density values. Figure $2.7$ shows a decision tree that represents the density function in figure 2.6.

Another useful distribution is the multivariate Gaussian distribution with the density function
$$\mathcal{N}(\mathbf{x} \mid \mu, \Sigma)=\frac{1}{(2 \pi)^{n / 2}|\Sigma|^{1 / 2}} \exp \left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right)$$ where $\mathbf{x}$ is in $\mathbb{R}^{n}, \boldsymbol{\mu}$ is the mean vector, and $\boldsymbol{\Sigma}$ is the coovariance matrix. The density function above requires that $\Sigma$ be positive definite ${ }^{i \omega}$. The number of independent parameters is equal to $n+(n+1) n / 2$, the number of components in $\mu$ added to the number of components in the upper triangle of matrix $\Sigma^{11}$ Appendix B shows plots of different multivariate Gaussian density functions. We can also define multivariate Gaussian mixture models. Figure $2.8$ shows an example of one with three components.

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

ñ(X∣μ,Σ)=1(2圆周率)n/2|Σ|1/2经验⁡(−12(X−μ)⊤Σ−1(X−μ))在哪里X在Rn,μ是平均向量，并且Σ是协方差矩阵。上面的密度函数要求Σ是肯定的一世ω. 独立参数的数量等于n+(n+1)n/2, 中的组件数μ添加到矩阵的上三角形中的组件数Σ11附录 B 显示了不同的多元高斯密度函数图。我们还可以定义多元高斯混合模型。数字2.8显示了一个具有三个组件的示例。

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## MATLAB代写

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