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统计代写|贝叶斯分析代写Bayesian Analysis代考|Multiobject Bayesian Network Models

There are two types of situations in which it becomes inefficient or impractical to model a problem using a single $\mathrm{BN}$ :

  1. When the model contains so many nodes that it becomes conceptually too difficult to understand. One of the key benefits of $\mathrm{BNs}$ is that they are such a powerful visual aid. But, unless there is some especially simple structure, as soon as a model contains more than, say, 30 nodes its visual representation becomes hard to follow. It needs to be broken up into smaller understandable chunks.
  2. When the model contains many similar repeated fragments, such as the case when there are repeated instances of variable names that differ only because they represent a different point in time. For example, the model in Figure $8.48$ (which represents a sequence of days in which we assess the risk of flood) contains several such repeated variables that differ only by the day they are recorded.

In both cases we seek to decompose the model into smaller component models. For example, the large model shown in Figure $8.49$ ought to be decomposable somehow into three smaller models as indicated by the grouping of the nodes, whereas the flood model ought to be decomposable into a sequence of identical models of the type shown in Figure 8.50.

The component models we need are called object-oriented BNs (OOBNs), because they have some of the properties associated with object-oriented modeling.

An OOBN is simply a BN with certain additional features that make it reusable as part of a larger $\mathrm{BN}$ model. The most important feature of an OOBN is that it will generally have input and/or output nodes. These represent the “external interface” of the BN and enable us to link OOBNs in a well-defined way.

For example, the model shown in Figure $8.49$ can be decomposed into three OOBNs with input and output nodes as shown in Figure 8.51.
In a tool like AgenaRisk the individual OOBNs can be embedded and linked into a higher-level model as shown in Figure 8.52. You need to specify which nodes in an OOBN are input and output nodes; at the higher level only the input and output nodes are shown (so the OOBN “Test Quality” has a single output node called test quality and the OOBN “Reliability During Test” has a single output node called reliability during test). We can then simply link relevant pairs of input and output nodes in two different OOBNs as shown in Figure 8.52.

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Missing Variable Fallacy

One of the advantages of using Bayesian networks is that it encourages domain experts to articulate and make visible all their assumptions and then ensures they are carefully modeled. One way to determine whether your model has really articulated all of the necessary assumptions is to build and run the model and determine whether the conclusions make sense.

The missing variable fallacy is explained by the following example. Suppose it is known that, on average, $50 \%$ of the students who start a course pass it. Is it correct to conclude the following?
a. A course that starts with 100 students will end up, on average, with 50 passes.
b. A course that ends with 50 passes will, on average, have started with 100 students.
In fact, although the first conclusion is normally correct you may be very surprised to learn that the second conclusion is normally not true. This has everything to do with the way we reason with prior assumptions, which, as we have seen, lies at the heart of the Bayesian approach to probability.

To explain how this fallacy results in the wrong BN model (and how to fix it) we will add some more detail to the example. The crucial prior assumption in this case is the probability distribution of student numbers who start courses. Let’s suppose that these are courses in a particular college where the average number of students per course is 180 . We know that some courses will have more than 180 and some less. Let’s suppose the distribution of student numbers looks like that in Figure 8.57. This is a Normal distribution whose mean is 180. As we have already seen in Chapter 2, Normal distributions are characterized not just by the mean but also by the standard deviation, which is how spread out the distribution is. In this example the standard deviation is 20 .
Because the number of students who pass is obviously influenced by the number who start, we represent this relationship by the $\mathrm{BN}$ shown in Figure 8.58(a).

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

统计代写|贝叶斯分析代写Bayesian Analysis代考|Multiobject Bayesian Network Models

有两种情况会导致使用单个问题建模问题变得低效或不切实际乙ñ :

  1. 当模型包含如此多的节点时,它在概念上变得难以理解。的主要好处之一乙ñs是它们是如此强大的视觉辅助工具。但是,除非有一些特别简单的结构,否则一旦模型包含超过 30 个节点,它的视觉表示就会变得难以理解。它需要分解成更小的可理解的块。
  2. 当模型包含许多相似的重复片段时,例如变量名称的重复实例仅因为它们代表不同的时间点而不同。例如图中的模型8.48(代表我们评估洪水风险的一系列日子)包含几个这样的重复变量,它们仅在记录的日期不同。

在这两种情况下,我们都试图将模型分解为更小的组件模型。例如图所示的大模型8.49正如节点分组所表明的那样,应该可以以某种方式分解为三个更小的模型,而洪水模型应该可以分解为图 8.50 所示类型的一系列相同模型。

我们需要的组件模型称为面向对象的 BN (OOBN),因为它们具有一些与面向对象建模相关的属性。

OOBN 只是一个具有某些附加功能的 BN,使其可作为更大的一部分重复使用乙ñ模型。OOBN 最重要的特征是它通常具有输入和/或输出节点。这些代表 BN 的“外部接口”,使我们能够以明确定义的方式链接 OOBN。

例如,如图所示的模型8.49可以分解为三个具有输入和输出节点的 OOBN,如图 8.51 所示。
在像 AgenaRisk 这样的工具中,各个 OOBN 可以嵌入并链接到更高级别的模型中,如图 8.52 所示。您需要指定 OOBN 中的哪些节点是输入和输出节点;在较高级别仅显示输入和输出节点(因此 OOBN“测试质量”有一个称为测试质量的单个输出节点,而 OOBN“测试期间的可靠性”有一个称为测试期间可靠性的单个输出节点)。然后,我们可以简单地链接两个不同 OOBN 中的相关输入和输出节点对,如图 8.52 所示。

统计代写|贝叶斯分析代写Bayesian Analysis代考|The Missing Variable Fallacy

使用贝叶斯网络的优点之一是它鼓励领域专家清晰表达他们的所有假设并使其可见,然后确保对它们进行仔细建模。确定您的模型是否真正阐明了所有必要假设的一种方法是构建和运行模型并确定结论是否有意义。

下面的例子解释了缺失变量谬误。假设已知,平均而言,50%开始课程的学生通过了它。以下结论是否正确?
一个。一门有 100 名学生的课程最终平均会通过 50 次通过。
湾。以 50 次通过结束的课程平均开始时有 100 名学生。
事实上,虽然第一个结论通常是正确的,但您可能会非常惊讶地发现第二个结论通常不正确。这与我们对先验假设进行推理的方式有关,正如我们所见,这是贝叶斯概率方法的核心。

为了解释这个谬误如何导致错误的 BN 模型(以及如何修复它),我们将在示例中添加更多细节。在这种情况下,关键的先验假设是开始课程的学生人数的概率分布。假设这些是特定大学的课程,每门课程的平均学生人数为 180 。我们知道有些课程会超过 180,有些则更少。假设学生人数的分布如图 8.57 所示。这是一个均值为 180 的正态分布。正如我们在第 2 章中已经看到的,正态分布的特征不仅在于均值,还在于标准差,即分布的分散程度。在此示例中,标准偏差为 20 。
因为通过的学生人数明显受开始人数的影响,我们用乙ñ如图 8.58(a) 所示。

统计代写|贝叶斯分析代写Bayesian Analysis代考|

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