### 统计代写|贝叶斯分析代写Bayesian Analysis代考|MSH3

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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|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

It is possible to define several random variables on the same sample space. For example, for a discrete sample space, such as a set of words, we can define two random variables $X$ and $Y$ that take integer values-one could measure word length and the other could measure the count of vowels in a word. Given two such random variables, the joint distribution $P(X, Y)$ is a function that maps pairs of events $(A, B)$ as follows:
$$p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)$$
It is often the case that we take several sets $\left{\Omega_{1}, \ldots, \Omega_{m}\right}$ and combine them into a single sample space $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. Each of the $\Omega_{i}$ is associated with a random variable. Based on this, a joint probability distribution can be defined for all of these random variables together. For example, consider $\Omega=V \times P$ where $V$ is a vocabulary of words and $P$ is a part-of-speech tag. This sample space enables us to define probabilities $p(x, y)$ where $X$ denotes a word associated with a part of speech $Y$. In this case, $x \in V$ and $y \in P$.

With any joint distribution, we can marginalize some of the random variables to get a distribution which is defined over a subset of the original random variables (so it could still be a joint distribution, only over a subset of the random variables). Marginalization is done using integration (for continuous variables) or summing (for discrete random variables). This operation of summation or integration eliminates the random variable from the joint distribution. The result is a joint distribution over the non-marginalized random variables.

For the simple part-of-speech example above, we could either get the marginal $p(x)=$ $\sum_{y \in P} p(x, y)$ or $p(y)=\sum_{x \in V} p(x, y)$. The marginals $p(X)$ and $p(Y)$ do not uniquely determine the joint distribution value $p(X, Y)$. Only the reverse is true. However, whenever $X$ and $Y$ are independent then the joint distribution can be determined using the marginals. More about this in Section 1.3.2.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

Joint probability distributions provide an answer to questions about the probability of several random variables to obtain specific values. Conditional distributions provide an answer to a different, but related question. They help to determine the values that a random variable can obtain, when other variables in the joint distribution are restricted to specific values (or when they are “clamped”).

Conditional distributions are derivable from joint distributions over the same set of random variables. Consider a pair of random variables $X$ and $Y$ (either continuous or discrete). If $A$ is an event from the sample space of $X$ and $y$ is a value in the sample space of $Y$, then:
$$p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}$$
is to be interpreted as a conditional distribution that determines the probability of $X \in A$ conditioned on $Y$ obtaining the value $y$. The bar denotes that we are clamping $Y$ to the value $y$ and identifying the distribution induced on $X$ in the restricted sample space. Informally, the conditional distribution takes the part of the sample space where $Y=y$ and re-normalizes the joint distribution such that the result is a probability distribution defined only over that part of the sample space.

When we consider the joint distribution in Equation $1.1$ to be a function that maps events to probabilities in the space of $X$, with $y$ being fixed, we note that the value of $p(Y=y)$ is actually a normalization constant that can be determined from the numerator $p(X \in A, Y=y)$. For example, if $X$ is discrete when using a PMF, then:
$$p(Y=y)=\sum_{x} p(X=x, Y=y) .$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

Bayes’ rule is a basic result in probability that describes a relationship between two conditional distributions $p(X \mid Y)$ and $p(Y \mid X)$ for a pair of random variables (these random variables can also be continuous). More specifically, Bayes’ rule states that for any such pair of random variables, the following identity holds:
$$p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}$$

This result also generally holds true for any two events $A$ and $B$ with the conditional probability $p(X \in A \mid Y \in B)$.

The main advantage that Bayes’ rule offers is inversion of the conditional relationship between two random variables – therefore, if one variable is known, then the other can be calculated as well, assuming the marginal distributions $p(X=x)$ and $p(Y=y)$ are also known.
Bayes’ rule can be proven in several ways. One way to derive it is simply by using the chain rule twice. More specifically, we know that the joint distribution values can be rewritten as follows, using the chain rule, either first separating $X$ or first separating $Y$ :
\begin{aligned} p(X&=x, Y=y) \ &=p(X=x) p(Y=y \mid X=x) \ &=p(Y=y) p(X=x \mid Y=y) \end{aligned}
Taking the last equality above, $p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=$ $y)$, and dividing both sides by $p(X=x)$ results in Bayes’ rule as described in Equation 1.2.
Bayes’ rule is the main pillar in Bayesian statistics for reasoning and learning from data. Bayes’ rule can invert the relationship between “observations” (the data) and the random variables we are interested in predicting. This makes it possible to infer target predictions from such observations. A more detailed description of these ideas is provided in Section 1.5, where statistical modeling is discussed.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|JOINT DISTRIBUTION OVER MULTIPLE RANDOM VARIABLES

$$p(X \in A, Y \in B)=p\left(X^{-1}(A) \cap Y^{-1}(B)\right)$$
$\mathrm{~ 我 们 经 常 会 采 取 几 组 ~ V e f t { 1 O m e g a _ { 1 } , ~ \ d o t s , ~ I O m e g a _ { m }}$ $\Omega=\Omega_{1} \times \ldots \times \Omega_{m}$. 每个 $\Omega_{i}$ 与随机变量相关联。基于此，可以为所有这些随机变量一起定义联合概率分布。例 如，考虑 $\Omega=V \times P$ 在哪里 $V$ 是一个词汇表和 $P$ 是词性标签。这个样本空间使我们能够定义概率 $p(x, y)$ 在哪里 $X$ 表示与词性相关的词 $Y$. 在这种情况下， $x \in V$ 和 $y \in P$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|CONDITIONAL DISTRIBUTIONS

$$p(X \in A \mid Y=y)=\frac{p(X \in A, Y=y)}{p(Y=y)}$$

$$p(Y=y)=\sum_{x} p(X=x, Y=y)$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|BAYES’ RULE

$$p(Y=y \mid X=x)=\frac{p(X-x \mid Y-y) p(Y-y)}{p(X=x)}$$

$$p(X=x, Y=y) \quad=p(X=x) p(Y=y \mid X=x)=p(Y=y) p(X=x \mid Y=y)$$

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## MATLAB代写

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