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统计代写|贝叶斯分析代写Bayesian Analysis代考|Mutually Exclusive Events and Pathways

In the mountain pass example traveling by car and traveling by train were mutually exclusive options. Only by modeling these options as mutually exclusive states within the same node (mode of transport) were we able to “solve” the mutually exclusive paths problems. However, in certain situations this neat solution will not work (we actually already saw one such situation in the “slips” problem whereby it was infeasible to capture mutually exclusive outcomes simply by declaring them as states of a single outcome node).
In particular, we are concerned with the situation where there are two or more mutually exclusive states, which each belong to a separate causal pathway. Merging the causal pathways into a single node may detract from the semantics of the model and make elicitation and communication difficult.
Consider, for example, an inquest into the death of Joe Smith; there are three possible mutually exclusive causes: “natural,” “unlawful” and “suicide.” There is evidence Joe may have had a serious illness and there is also evidence he may have suffered depression which could have caused suicide. A suicide note was found by Joe’s body, but it is not known if it was written by Joe and there are possible suspicious wounds found on his body.

In the BN model of Figure 8.40a we have used a single node whose states correspond to the possible causes of death to ensure mutual exclusivity. However, this solution requires us to complete NPTs which (in realistic examples with multiple causes and alternatives) are infeasibly large and for which the vast majority of entries are either redundant or meaningless. For example:

  • The NPT for the cause of death node: Although each parent node influences only one possible outcome we are forced to give (redundant) separate probabilities conditioned on every combination of all the other causal factor states. For example, although “Illness” only influences whether or not Joe died of natural causes, we have to specify the probability of death from natural causes conditioned on illness and every possible combination of values for the other parent states-none of which is relevant.
  • The NPT for child nodes of the cause of death node: Since some of these are also only relevant for at most a single cause of death, we again have to unnecessarily specify separate probabilities conditioned on each of the different alternative causes of death.

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

The mutual exclusivity problem is especially pertinent when we need to introduce classifications into a model.

With a taxonomic hierarchy we aim to classify hidden attributes of an object using direct or inferred knowledge. There are many reasons why we might wish to model such a thing using a BN. The key feature of a taxonomic hierarchy is that mutual exclusivity expresses the logical existence constraints that exist at a given level in a taxonomy; so an object can be classified as {Mammal, Dog, Alsatian} but not as a {Mammal, Cat, Abyssinian $}$ at the same time.

The taxonomic hierarchy is therefore governed by values denoting membership rather than probability assignments. Taxonomic relations are examples of the definitional/synthesis idiom.

Figure $8.43$ shows a taxonomic hierarchy for military asset types. Here we present a simple classification hierarchy where the more abstract class is refined into more particular classes. This could continue to any particular depth as determined by the problem domain. A class is denoted by a rectangle and an edge denotes class membership between parent class and child class.

At the top level we have three mutually exclusive classes: {Land, Sea, and Air]. If an asset is a Land unit it cannot be a Sea or Air unit or vice versa. We then further refine the classes of Land unit into {Armored, Armored Reconnaissance, Mechanized Infantry, Infantry, and Supply}. We can have similar refinements for Air and Sea units and the key thing in the taxonomy is that the mutual exclusivity is maintained as we progress through the levels in the hierarchy.

We can use this information to help build a $\mathrm{BN}$ and then supplement this BN with additional nodes that reflect measurement idioms and so on. Each value in the parent class becomes a child node in the $\mathrm{BN}$ and this is further refined as necessary. A taxonomic decomposition like this is very simple. However, when translated into a BN, as the size and depth of the classification grows the need to manage the complexity increases. The problem is that, because of the mutually exclusive states, we find that very quickly we have a model with a very large state space; meaning large, complex NPTs.

We can recast a classification model by treating each subclass as a separate “variable,” but to ensure that the subclasses remain mutually exclusive we must introduce a new state-NA, for not applicable-to each child class. Whenever the parent class takes a value inconsistent with the child class under consideration NA is set to true. These NA states must then be maintained through all child nodes of each class node to ensure that the mutual exclusivity constraints are respected. The necessary NPTs are then set up as shown by the example in Table 8.9.

统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS3023

统计代写|贝叶斯分析代写Bayesian Analysis代考|Mutually Exclusive Events and Pathways

在山口示例中,汽车旅行和火车旅行是相互排斥的选择。只有通过将这些选项建模为同一节点(运输方式)内的互斥状态,我们才能“解决”互斥路径问题。然而,在某些情况下,这种简洁的解决方案将不起作用(我们实际上已经在“滑倒”问题中看到了一种这样的情况,即仅通过将相互排斥的结果声明为单个结果节点的状态来捕获它们是不可行的)。
特别是,我们关注存在两个或多个互斥状态的情况,每个状态都属于单独的因果路径。将因果路径合并到单个节点中可能会减损模型的语义并使启发和交流变得困难。
例如,考虑对乔·史密斯之死的调查;存在三种可能的相互排斥的原因:“自然”、“非法”和“自杀”。有证据表明乔可能患有严重疾病,也有证据表明他可能患有可能导致自杀的抑郁症。在乔的尸体上发现了一张遗书,但不知道是不是乔写的,他的身上可能发现了可疑的伤口。

在图 8.40a 的 BN 模型中,我们使用了单个节点,其状态对应于可能的死亡原因,以确保互斥性。但是,该解决方案要求我们完成 NPT,这些 NPT(在具有多种原因和替代方案的现实示例中)非常大,并且绝大多数条目要么是多余的,要么是无意义的。例如:

  • 死因节点的 NPT:虽然每个父节点只影响一个可能的结果,但我们被迫给出(冗余)单独的概率,条件是所有其他因果因素状态的每种组合。例如,虽然“疾病”仅影响乔是否死于自然原因,但我们必须指定以疾病为条件的自然原因死亡的概率以及其他父状态的每个可能的值组合——这些都不相关。
  • 死因节点的子节点的 NPT:由于其中一些也最多仅与单个死因相关,因此我们再次不必要地指定以每个不同的替代死因为条件的单独概率。

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

当我们需要将分类引入模型时,互斥性问题尤其相关。

通过分类层次,我们旨在使用直接或推断的知识对对象的隐藏属性进行分类。我们可能希望使用 BN 对这样的事物进行建模的原因有很多。分类层次的关键特征是互斥性表达了分类中给定级别存在的逻辑存在约束;所以一个物体可以被归类为{Mammal, Dog, Alsatian},但不能被归类为{Mammal, Cat, Abyssinian}}同时。

因此,分类层次由表示成员资格的值而不是概率分配来控制。分类关系是定义/综合习语的例子。

数字8.43显示了军事资产类型的分类层次结构。在这里,我们提出了一个简单的分类层次结构,其中更抽象的类被细化为更具体的类。这可以继续到由问题域确定的任何特定深度。一个类由一个矩形表示,一条边表示父类和子类之间的类成员关系。

在顶层,我们有三个互斥的类:{Land、Sea 和 Air]。如果资产是陆地单位,则它不能是海上或空中单位,反之亦然。然后,我们将陆地单位的类别进一步细化为{装甲、装甲侦察、机械化步兵、步兵和补给}。我们可以对空中和海上单位进行类似的改进,分类中的关键是在我们通过层次结构的各个级别时保持互斥性。

我们可以使用这些信息来帮助建立一个乙ñ然后用反映测量习语等的附加节点来补充这个BN。父类中的每个值都成为乙ñ并根据需要进一步细化。像这样的分类分解非常简单。然而,当翻译成 BN 时,随着分类规模和深度的增长,管理复杂性的需求也会增加。问题是,由于相互排斥的状态,我们很快发现我们有一个状态空间非常大的模型。意味着大型、复杂的 NPT。

我们可以通过将每个子类视为一个单独的“变量”来重新构建分类模型,但是为了确保子类保持互斥,我们必须引入一个新的状态 NA,因为不适用于每个子类。每当父类采用与所考虑的子类不一致的值时,NA 设置为 true。然后必须通过每个类节点的所有子节点维护这些 NA 状态,以确保尊重互斥性约束。然后设置必要的 NPT,如表 8.9 中的示例所示。

统计代写|贝叶斯分析代写Bayesian Analysis代考|

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