### 统计代写|贝叶斯网络代写Bayesian network代考|CMPT310

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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|贝叶斯网络代写Bayesian network代考|Representation of the Joint Probability Distributions

In general, BNs are represented by joint probability distributions. Let’s consider a BN containing $n$ number of nodes: $X_{1}$ to $X_{n}$. A particular value in the joint distribution can be represented by $P\left(X_{1}=x_{1}, X_{2}=x_{2}, X_{3}=x 3, \ldots, X_{n}=x_{n}\right)$ or, simply, $P\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$. Using the chain rule of probability theory, the joint probabilities can be factorized as:
\begin{aligned} P\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right) &=P\left(x_{1}\right) \times P\left(x_{2} / x_{1}\right) \cdots \times P\left(x_{n} / x_{1}, x_{2}, \ldots, x_{n-1}\right) \ &=\prod_{i} P\left(x_{i} \mid x_{1}, \ldots, x_{i-1}\right) \end{aligned}
Now, the structure of a BN implies that the value of a particular node is conditional only on the values of its parent nodes, simplifying the joint probability expression to
$$P\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i} P\left(X_{i} \mid \operatorname{Parents}\left(x_{i}\right)\right)$$
provided Parents $\left(X_{i}\right) \subseteq\left{X_{1}, X_{2}, \ldots, X_{i-1}\right}$. For example, in Fig.2.1 the joint probability expression $P(M=T, W=F, S=T, H=F, C=T)$ can be written as: \begin{aligned} P(M=T, W=F, S=T, H=F, C=T) & \ &=P(M=T) P(W=F) \ & \times P(S=T \mid M=T, W=F) \ & \times P(H=F \mid W=F) \ & \times P(C=T \mid S=T, H=F) \end{aligned}

## 统计代写|贝叶斯网络代写Bayesian network代考|Conditional Independence

One of the crucial things in understanding the working principle of Bayesian network is to know the relationship between the conditional probabilities and the network.

• Causal Chain: A causal chain of three nodes has been depicted in Fig. 2.3a, where the variable A causes variable B which in turn causes variable C. Causal chains lead to conditional independence, such as for the Fig. 2.3a:
$$P(C \mid A, B)=P(C \mid B)$$
This indicates that the probability of $\mathrm{C}$, given $\mathrm{B}$ is the same as the probability of $C$, given both $B$ and $A$. In other words, knowing that $A$ has occurred does not provide any added information to change our beliefs about $\mathrm{C}$ if we already know that B has occurred. In Fig. 2.1, the probability that there is a System Crash depends directly on whether there is any $O S$ Failure. If we do not know whether there is $O S$ Failure, but we find out the presence of Malware, that would increase our belief that there is $O S$ Failure and there is System Crash. However, if we already knew that there is OS Failure, then the presence of Malware would not make any difference to the probability of System Crash. That is, System Crash is conditionally independent of Malware given there is $O S$ Failure.
• Common Causes: Two variables A and C having a common influencing variable or cause B is represented in Fig. $2.3$ b Common causes (also called common ancestors) give rise to similar conditional independence structure as that of chains:
$$P(C \mid A, B)=P(C \mid B)$$
For example, if there is no evidence or information about Power Failure, then learning that there is an occurrence of $O S$ Failure or Hardware Failure will increase

the chances of Power Failure, which in turn will increase the probability of the occurrence of OS or Hardware Failure, and ultimately the System Crash. However, if we already know about Power Failure, then an additional occurrence of $O S$ Failure would not tell us anything new about the chances of Hardware Failure.

• Common Effects: A common effect is indicated by a network v-structure, as shown in Fig. 2.3c. This represents the situation where a variable/node has two causes (influencing variables). Common effects generate exactly the opposite conditional independence structure as that produced by chains and common causes. More specifically, in this case, the parents are marginally independent but become dependent when the information about the common effect are given (i.e., they are conditionally dependent):
$$P(A \mid C, B) \neq P(A \mid B)$$
For instance, with reference to the Fig. 2.1, if we know the effect (e.g., OS Failure), and then we find out that one of the causes is absent (e.g., there is no Power Failure), this raises the probability of the other cause (e.g., presence of Malware)-which is just the inverse of the previous one.

## 统计代写|贝叶斯网络代写Bayesian network代考|Bayesian Network and Decision Making

One of the important characteristics of Bayesian network remains in its capability to generate inference i.e. to the compute the posterior probability for a query variable given an observed event. The variables having assignment of values are called evidence variables whereas the other variables without having the assigned values are called hidden variables. The inference in a Bayesian Network can formulated as follows:
Let $E$ represents a set of evidence variables
$Y=\left{y_{1}, y_{2}, y_{3}, \ldots, y_{n}\right}=$ Set of non-evidence variables
$X=$ The query variable
In this context, the Bayesian network can be represented using joint probability as $P(X, E, Y)$. Now, the posterior probability of $X$, given the observed evidence $E$ can be written as follows:
\begin{aligned} P(X \mid E) &=\alpha P(X, E) \ &=\alpha \sum_{Y} P(Y) \cdot P(X, E \mid Y) \ &=\alpha \sum_{Y} P(X, E, Y) \end{aligned}
where $\alpha$ is a normalization constant. Using the above procedure of inference generation from Bayesian Network, necessary decision can be undertaken.

## 统计代写|贝叶斯网络代写Bayesian network代考|Conditional Independence

• 因果链：图 2.3a 中描绘了三个节点的因果链，其中变量 A 导致变量 B，而变量 B 又导致变量 C。因果链导致条件独立，例如图 2.3a：
磷(C∣一个,乙)=磷(C∣乙)
这表明概率C, 给定乙与概率相同C, 给定两者乙和一个. 换句话说，知道一个已发生不提供任何附加信息来改变我们的信念C如果我们已经知道 B 已经发生。在图 2.1 中，发生系统崩溃的概率直接取决于是否存在任何○小号失败。如果我们不知道是否有○小号失败，但我们发现了恶意软件的存在，这将增加我们的信念，即存在○小号失败并出现系统崩溃。但是，如果我们已经知道存在操作系统故障，那么恶意软件的存在不会对系统崩溃的可能性产生任何影响。也就是说，系统崩溃有条件地独立于恶意软件，因为存在○小号失败。
• 共同原因：具有共同影响变量或原因 B 的两个变量 A 和 C 如图所示。2.3b 共同原因（也称为共同祖先）产生与链类似的条件独立结构：
磷(C∣一个,乙)=磷(C∣乙)
例如，如果没有关于停电的证据或信息，则得知发生了停电○小号故障或硬件故障将增加

• 共同效应：共同效应由网络 v 结构表示，如图 2.3c 所示。这表示变量/节点有两个原因（影响变量）的情况。共同影响产生的条件独立结构与链和共同原因产生的条件独立结构完全相反。更具体地说，在这种情况下，父母是勉强独立的，但在给出关于共同效应的信息时变得依赖（即，他们是有条件的依赖）：
磷(一个∣C,乙)≠磷(一个∣乙)
例如，参考图 2.1，如果我们知道结果（例如，OS 故障），然后我们发现其中一个原因不存在（例如，没有电源故障），这提高了发生故障的概率。另一个原因（例如，存在恶意软件）——这与前一个原因正好相反。

## 统计代写|贝叶斯网络代写Bayesian network代考|Bayesian Network and Decision Making

Y=\left{y_{1}, y_{2}, y_{3}, \ldots, y_{n}\right}=Y=\left{y_{1}, y_{2}, y_{3}, \ldots, y_{n}\right}=非证据变量集
X=查询变量

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