# 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

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|Mutually Exclusive Events and Pathways

• 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

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