### 统计代写|贝叶斯分析代写Bayesian Analysis代考|MAST90125

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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|INDEPENDENT AND CONDITIONALLY INDEPENDENT

A pair of random variables $(X, Y)$ is said to be independent if for any $A$ and $B$,
$$p(X \in A \mid Y \in B)=p(X \in A),$$
or alternatively $p(Y \in B \mid X \in A)=p(Y \in B)$ (these two definitions are correct and equivalent under very mild conditions that prevent ill-formed conditioning on an event that has zero probability).

Using the chain rule, it can also be shown that the above two definitions are equivalent to the requirement that $p(X \in A, Y \in B)=p(X \in A) p(Y \in B)$ for all $A$ and $B$.

Independence between random variables implies that the random variables do not provide information about each other. This means that knowing the value of $X$ does not help us infer anything about the value of $Y$-in other words, it does not change the probability of $Y$. (Or vice-versa $-Y$ does not tell us anything about $X$.) While independence is an important concept in probability and statistics, in this book we will more frequently make use of a more refined notion of independence, called “conditional independence”-which is a generalization of the notion of independence described in the beginning of this section. A pair of random variables $(X, Y)$ is conditionally independent given a third random variable $Z$, if for any $A, B$ and $z$, it holds that $p(X \in A \mid Y \in B, Z=z)=p(X \in A \mid Z=z)$.

Conditional independence between two random variables (given a third one) implies that the two variables are not informative about each other, if the value of the third one is known. 3
Conditional independence (and independence) can be generalized to multiple random variables as well. We say that a set of random variables $X_{1}, \ldots, X_{n}$, are mutually conditionally independent given another set of random variables $Z_{1}, \ldots, Z_{m}$ if the following applies for any $A_{1}, \ldots, A_{n}$ and $z_{1}, \ldots, z_{m}:$
$$\begin{gathered} p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)= \ \prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) . \end{gathered}$$
This type of independence is weaker than pairwise independence for a set of random variables, in which only pairs of random variables are required to be independent. (Also see exercises.)

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

Another type of relationship that can be present between random variables is that of exchangeability. A sequence of random variables $X_{1}, X_{2}, \ldots$ over $\Omega$ is said to be exchangeable, if for any finite subset, permuting the random variables in this finite subset, does not change their joint distribution. More formally, for any $S=\left{a_{1}, \ldots, a_{m}\right}$ where $a_{i} \geq 1$ is an integer, and for any permutation $\pi$ on ${1, \ldots, m}$, it holds that: ${ }^{4}$
$$p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .$$
Due to a theorem by de Finetti (Finetti, 1980), exchangeability can be thought of as meaning “conditionally independent and identically distributed” in the following sense. De Finetti showed that if a sequence of random variables $X_{1}, X_{2}, \ldots$ is exchangeable, then under some regularity conditions, there exists a sample space $\Theta$ and a distribution over $\Theta, p(\theta)$, such that:
$$p\left(X_{a_{1}}, \ldots, X_{a_{m}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta,$$
for any set of $m$ integers, $\left{a_{1}, \ldots, a_{m}\right}$. The interpretation of this is that exchangeable random variables can be represented as a (potentially infinite) mixture distribution. This theorem is also called the “representation theorem.”

The frequentist approach assumes the existence of a fixed set of parameters from which the data were generated, while the Bayesian approach assumes that there is some prior distribution over the set of parameters that generated the data. (This will hecome clearer as the hook progresses.) De Finetti’s theorem provides another connection between the Bayesian approach and the frequentist one. The standard “independent and identically distributed” (i.i.d.) assumption in the frequentist setup can be asserted as a setup of exchangeability where $p(\theta)$ is a point-mass distribution over the unknown (but single) parameter from which the data are sampled. This leads to the observations being unconditionally independent and identically distributed. In the Bayesian setup, however, the observations are correlated, because $p(\theta)$ is not a point-mass distribution. The prior distribution plays the role of $p(\theta)$. For a detailed discussion of this similarity, see O’Neill (2009).

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

If we consider again the naive definition of random variables, as functions that map the sample space to real values, then it is also useful to consider various ways in which we can summarize these random variables. One way to get a summary of a random variable is by computing its expectation, which is its weighted mean value according to the underlying probability model.
It is easiest to first consider the expectation of a continuous random variable with a density function. Say $p(\theta)$ defines a distribution over the random variable $\theta$, then the expectation of $\theta$, denoted $E[\theta]$ would be defined as:
$$E[\theta]=\int_{\theta} p(\theta) \theta d \theta .$$
For the discrete random variables that we consider in this book, we usually consider expectations of functions over these random variables. As mentioned in Section 1.2, discrete random variable values often range over a set which is not numeric. In these cases, there is no “mean value” for the values that these random variables accept. Instead, we will compute the mean value of a real-function of these random variables.
With $f$ being such a function, the expectation $E[f(X)]$ is defined as:
$$E[f(X)]=\sum_{x} p(x) f(x)$$ For the linguistic structures that are used in this book, we will often use a function $f$ that indicates whether a certain property holds for the structure. For example, if the sample space of $X$ is a set of sentences, $f(x)$ can be an indicator function that states whether the word “spring” appears in the sentence $x$ or not; $f(x)=1$ if the word “spring” appears in $x$ and 0 , otherwise. In that case, $f(X)$ itself can be thought of as a Bernoulli random variable, i.e., a binary random variable that has a certain probability $\theta$ to be 1 , and probability $1-\theta$ to be 0 . The expectation $E[f(X)]$ gives the probability that this random variable is 1 . Alternatively, $f(x)$ can count how many times the word “spring” appears in the sentence $x$. In that case, it can be viewed as a sum of Bernoulli variables, each indicating whether a certain word in the sentence $x$ is “spring” or not.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|INDEPENDENT AND CONDITIONALLY INDEPENDENT

$$p(X \in A \mid Y \in B)=p(X \in A),$$

$$p\left(X_{1} \in A_{1}, \ldots, X_{n} \in A_{n} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right)=\prod_{i=1}^{n} p\left(X_{i} \in A_{i} \mid Z_{1}=z_{1}, \ldots, Z_{m}=z_{m}\right) \text {. }$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXCHANGEABLE RANDOM VARIABLES

$\mathrm{S}=|$ left{a_{1}，Vdots, a_{m}|right} 在哪里 $a_{i} \geq 1$ 是一个整数，并且对于任何排列 $\pi$ 上 $1, \ldots, m$ ，它认为: ${ }^{4}$
$$p\left(x_{a_{1}}, \ldots, x_{a_{m}}\right)=p\left(x_{a_{\pi(1)}}, \ldots, x_{\left.a_{\pi(m)}\right)}\right) .$$

$$p\left(X_{a_{1}}, \ldots, X_{a_{m i}}\right)=\int_{\theta} \prod_{i=1}^{m} p\left(X_{a_{i}} \mid \theta\right) p(\theta) d \theta$$

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|EXPECTATIONS OF RANDOM VARIABLES

$$E[\theta]=\int_{\theta} p(\theta) \theta d \theta$$

$$E[f(X)]=\sum_{x} p(x) f(x)$$

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