### 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Formalism for System Dependability

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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|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Probabilistic graphical models: BN

In this section, the BN formalism is introduced as a probabilistic graphical model [PEA 88]. Mathematical objects are based on graph theory and probability theory. A BN represents a factorized model of a joint probability distribution of several discrete random variables. Graph theory provides the algorithms required to analyze graphical property. Probability theory brings a formalism to quantify the dependencies between variables by introducing conditional probability laws.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|a formalism to model dependability

The probabilistic graphical model (PGM) considered here is a directed acyclic graph (DAG). A DAG comprises nodes and directed edges. Nodes can be classified into two classes: parent nodes and child nodes. A parent node is a node with outgoing edges while a child node is a node with incoming edges. A parent node is called a root node if it has no incoming edges. A child node is a leaf node if it has no outgoing edge. Each parent node $x$ in a graphical probabilistic model is assigned a marginal probability distribution $P(x)$ and each child node $E$ is associated with a conditional probability $P(E \mid p a(E))$, where $p a(E)$ is the set of all parent nodes of $E$. For instance, in Figure $1.1$ $p a\left(E_{2}\right)=x_{2}, x_{3}$ and $p a\left(E_{1}\right)=x_{1}$.

PGM is defined by the structure of the graph and the probabilistic parameters. According to the graph structure shown in Figure 1.1, the a priori probability laws are: $P\left(x_{1}\right), P\left(x_{2}\right)$ and $P\left(x_{3}\right)$; while the conditional probabilities are $P\left(E_{1} \mid x_{1}\right), \quad P\left(E_{2} \mid x_{2}, x_{3}\right)$ and$P\left(y \mid E_{1}, E_{2}\right)$. The conditional probabilities are defined by a conditional orobability table (CPT) as a matrix giving the probability distribution of the variable with respect to the Cartesian product of its parent variable states. For instance, the conditional probability $P\left(y \mid E_{1}, E_{2}\right)$ is given in Table $1.1$ for the $\left{h_{1}^{y}, \ldots, h_{n}^{y}\right}$ states of $y$, according to the $\left{h_{1}^{E 1}, \ldots, h_{n}^{E 1}\right}$ states of $E_{1}$ and the $\left{h_{1}^{E 2}, \ldots, h_{n}^{E 2}\right}$ states of $E_{2}$.

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Inference mechanism

Like several other modeling tools, BN are interesting for their graphical aspect. However, the probabilistic inference mechanism is more interesting and is the actual strength of the tool. Thanks to this inference, a $\mathrm{BN}$ is able to compute the marginal probability distribution of any variable according to:

• the realizations or measurements of observed variables (evidence);
• the likelihood regarding the state of certain variables;
• an a priori knowledge about the probability distribution of unobserved variables;
• the conditional probability distribution between variables.
The inference mechanisms are explained in [JEN 96, PEA 88] and are outside the scope of this book. Nevertheless, several inference mechanisms exist to compute the exact probabilities or the approximate probabilities for very complex systems. The inference algorithms are used to integrate new information in the model as soft or hard evidence. This information modeled as new observations on some variable states is a way to compute the impacts of situations on target variables. In maintenance or risk management, it is interesting to integrate specific situations or compute the impacts of some scenarios or maintenance actions. In all inference mechanisms, Bayes theorem is used to propagate the probabilities on the variables and to update the probabilities of all the variables given the observations of states or likelihoods of states.

In computer science, current research focuses mainly on inference efficiency to handle increasingly complex models and to increase the number of variables handled. For the exact inference, efficient algorithms use the BN structure to solve the non-deterministic polynomial-time-hard (NP-hard) problem to compute an a posteriori probability distribution of random variables [PEA 88, PEO 91, JEN 90 , SHA 96, MAD 99, FAY 00, ALL 03]. The best known algorithms are based on the junction tree. For a detailed explanation, refer to [JEN 96 , pp.76]. The newest algorithms attempt to reduce the memory requirements and to increase the computing speed to deal with larger models [JAE 02, WUI 12]. In dependability analyses, these abilities help model industrial-scale systems.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|a formalism to model dependability

PGM 由图的结构和概率参数定义。根据图 1.1 所示的图结构，先验概率规律为：磷(X1),磷(X2)和磷(X3); 而条件概率是磷(和1∣X1),磷(和2∣X2,X3)和磷(是∣和1,和2). 条件概率由条件概率表 (CPT) 定义为一个矩阵，该矩阵给出变量相对于其父变量状态的笛卡尔积的概率分布。例如，条件概率磷(是∣和1,和2)在表中给出1.1为了\left{h_{1}^{y}, \ldots, h_{n}^{y}\right}\left{h_{1}^{y}, \ldots, h_{n}^{y}\right}的状态是，根据\left{h_{1}^{E 1}, \ldots, h_{n}^{E 1}\right}\left{h_{1}^{E 1}, \ldots, h_{n}^{E 1}\right}的状态和1和\left{h_{1}^{E 2}, \ldots, h_{n}^{E 2}\right}\left{h_{1}^{E 2}, \ldots, h_{n}^{E 2}\right}的状态和2.

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Inference mechanism

• 观测变量的实现或测量（证据）；
• 关于某些变量状态的可能性；
• 关于未观察到的变量的概率分布的先验知识；
• 变量之间的条件概率分布。
推理机制在 [JEN 96, PEA 88] 中进行了解释，超出了本书的范围。然而，存在几种推理机制来计算非常复杂系统的精确概率或近似概率。推理算法用于将模型中的新信息整合为软证据或硬证据。这种信息建模为对某些变量状态的新观察，是一种计算情况对目标变量的影响的方法。在维护或风险管理中，整合特定情况或计算某些场景或维护操作的影响是很有趣的。在所有推理机制中，贝叶斯定理用于传播变量的概率，并在给定状态观察或状态可能性的情况下更新所有变量的概率。

## 广义线性模型代考

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