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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Joint distribution

For any system, a probability is defined for each state corresponding to the Cartesian product between the states of each component and the system states to define the joint probability [SHA 96, p. 2]. The advantage of this representation is to show all the possible situations (working or failure). The main drawback is the size of the Cartesian product that increases rapidly and becomes excessive for the analyst, particularly in industrial-scale systems.

Tables $1.3$ and $1.4$ provide the application of the multi-state system with three valves. The joint probability is defined by $P\left(y, x_{1}, x_{2}, x_{3}\right)$, where $y$ represents the system states and $x_{i}$ represents the states of the components in the three-valve system. If the system is functioning, then $y=O k$, otherwise $y=H s$.

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

The system reliability depends on the components’ reliability $\left(x_{1}\right.$, $x_{2}$ and $\left.x_{3}\right)$ and the relation between the system reliability, $y$, and the component states. This relation is the structure function. The joint probability distribution $P\left(y, x_{1}, x_{2}, x_{3}\right)$ can be computed on any structure function. In the case of the three-valve system, the reliability can be computed from the joint probability distribution. The reliability is then given by marginalization $P(y=O k)=0.345721859$, which is the sum of all state combinations, where the system state is $O k$. Note that it is possible to compute all conditional probabilities from the joint probability distribution.From the Cartesian product of states, a factorized version of the joint probability distribution can be computed by introducing conditional independence. Components $x_{1}, x_{2}$ and $x_{3}$ are state independent. Thus, the expression becomes $P\left(x_{1}, x_{2}, x_{3}\right)=$ $P\left(x_{1}\right) . P\left(x_{2}\right) \cdot P\left(x_{3}\right)$. Nevertheless, the functioning state of the system$(y=O k)$ depends on the state of components $x_{1}, x_{2}$ and $x_{3}$. The joint probability distribution can be rewritten in the following factorized form: Equation [1.1] is the factorized form of the joint probability distribution $P\left(y, x_{1}, x_{2}, x_{3}\right)$. The conditional probability distribution $P\left(y \mid x_{1}, x_{2}, x_{3}\right)$, which is deterministic, remains unwieldy. Nevertheless, the conditional probability distribution may be factorized again by introducing intermediate variables, as done in fault trees. For instance, the system can be divided into two stages, as shown in Figure 1.3. Two variables $E_{1}$ and $E_{2}$ are introduced that characterize the states of the system stage. $E_{1}$ characterizes the possibility of controlling the flow in stage 1 and $E_{2}$ in stage 2 .

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

The tables that define the conditional probabilities model the structure function of the system. This structure function is an equation that describes the relation between the component states and the system states. If the structure function is constant, then it implies that the conditional probability distribution is time independent. Defining the conditional probability using a table allows the modeling of any relations between the system states and the component states. If the relation is based on Boolean operation (AND, OR, etc.), then the CPT is deterministic, but more complex relations can be modeled. The reliability of the system is well modeled if the structure function is correctly modeled by the BN and if all scenarios are described. A CPT contains all the knowledge about the relation between the input states and the output states requested by the analysis.

In the classical case of binary state hypothesis, i.e. the system and its components can have two states ${O k, H s}$, the structure function is similar to a Boolean function. The CPT translates this Boolean relation. In this case, there is an exact correspondence between the BN model and a RBD when considering the working case or a fault tree

when considering the failure case. Note that for our illustration, a non-binary function with three state components is deliberately chosen, to go beyond usual cases with RBD and fault tree and to exhibit part of the advantages of the BN model.

In our illustration as in all binary cases, there is no uncertainty between the combination of component states and system states. The probabilities of $P\left(y \mid x_{1}, x_{2}, x_{3}\right)$ are equal to 0 or 1 . Therefore, CPT is deterministic. Note that this is not necessarily the case, for example, in a non-deterministic model, $P\left(y \mid x_{1}, x_{2}, x_{3}\right) \in[0,1]$. This case models some situations where there is an uncertainty about the consequence of a component state combination, an uncertain function due to a human factor, an uncertain context, etc.

In this chapter, some of the main advantages of $\mathrm{BN}$ techniques have been discussed in an academic and industrial context. It is not necessary to know the joint probability of the system to find the BN model. The analyst can build the model gradually, but he should conduct his analysis with a semantic guide.

## 广义线性模型代考

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