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统计代写|贝叶斯分析代写Bayesian Analysis代考|Labeled Nodes and Comparative Expressions

A labeled node is one whose set of states is simply a set of labels. A number of examples of such labeled nodes are shown in Table 9.1. Irrespective of what the names of the labels are, they have no underlying semantic meaning. So, for example, in the node Injury there is no underlying assumption that, say, “broken bone” is better or worse than “damaged ligament” or that “broken bone” is better or worse than “other.” All that is assumed by the model is that the states are different. This makes sense for the nodes Student, Injury, Sport, Program language, Die roll, and Bank loss in the table. It makes less sense for the other nodes because implicitly in those cases we would like the states of the nodes to carry some other meaning. By declaring a node like Water level to be labeled (which, as we will see, is possible but not recommended), there is no implicit knowledge that the state “medium” represents a higher level than the state “low.” Any meaning has to be provided manually by the modeler in the way the NPT is defined. That was what we did in the way we specified the NPT in the example in Box 9.1. For example, in the NPT, if the water level at the start of the day was “medium,” then the probability the water level at the end of the day was “high,” given 2-5 mm of rainfall, is greater than the probability the water level is “high” if it started the day “low.”

In most cases when you create a labeled node there is little alternative but to define the NPT manually cell by cell.

统计代写|贝叶斯分析代写Bayesian Analysis代考|NoisyAND Function for Boolean Nodes

Just as NoisyOR provides a highly flexible “noisy” version of the OR function there is an analogous NoisyAND for the AND function (see Box $9.8$ for the formal definition). Again we assume there is a set of factors (represented as Boolean variables $X_1, \ldots, X_n$ ) that can be considered as “causes” independently of each other in terms of their effect on some Boolean variable $Y$. Consider the following example:

A soccer team generally only wins a match if all of the team “components”-the keeper, defence, midfield and attack-all play well. However, there is a “luck” factor which means that (rarely) they might win if some, or even all, components do not play well. Similarly because of “luck” there is a chance that they will not win even if all of them play well. In addition to luck some components are more critical than others. For example, although it is rare to win if even one component does not play well it is especially critical that attack plays well in order to win.

Assuming that each of the team components are independent of each other in terms of their effect on the match result (Figure 9.22), this type of problem is easily modelled as a noisyAND function. As with noisyOR we must specify a weight value (between 0 and 1 ) associated with each parent representing its “criticality” and a leak value (between 0 and 1) representing the “luck” factor.

For example, using the AgenaRisk syntax, we can define the NPT for “team wins” as:
noisyand (Keeper, 0.7, Defence, 0.9, Midfield, $0.8$, Attack, 1, 0.05)
If you run the model you will find that, when all of the components are true there is still a $0.05$ probability (equal to the leak value) that the team will not win. To see the impact of the weights, note that when all the components are true except for “Attack plays well” which is false, then the probability of winning is 0 because the “Attack” component has weight 1 (i.e. maximum criticality). However, because “Midfield” is not as critical there is still a small probability of winning when “Midfield plays well” is false and the other components are true.

统计代写|贝叶斯分析代写Bayesian Analysis代考|STAT4102

统计代写|贝叶斯分析代写Bayesian Analysis代考|Labeled Nodes and Comparative Expressions

标记节点是其状态集只是一组标签的节点。表 9.1 中显示了此类标记节点的一些示例。无论标签的名称是什么,它们都没有潜在的语义含义。因此,例如,在节点损伤中,没有潜在的假设,即“骨折”比“韧带损伤”更好或更差,或者“骨折”比“其他”更好或更差。该模型所假设的只是状态不同。这对于表中的节点 Student、Injury、Sport、Program language、Die roll 和 Bank loss 是有意义的。这对其他节点来说意义不大,因为在这些情况下,我们隐含地希望节点的状态具有其他含义。通过声明要标记的节点(如水位)(正如我们将看到的那样,这是可能的,但不推荐),没有隐含的知识表明状态“中”比状态“低”代表更高的水平。任何含义都必须由建模者以定义 NPT 的方式手动提供。这就是我们在框 9.1 的示例中指定 NPT 的方式所做的。例如,在 NPT 中,如果一天开始时的水位为“中等”,那么在降雨量为 2-5 毫米的情况下,一天结束时水位为“高”的概率大于如果一天开始时水位“低”,则水位“高”的概率。

在大多数情况下,当您创建标记节点时,除了手动逐个单元格地定义 NPT 外,别无选择。

统计代写|贝叶斯分析代写Bayesian Analysis代考|NoisyAND Function for Boolean Nodes

正如 NoisyOR 提供了一个高度灵活的“嘈杂”版本的 OR 函数一样,AND 函数也有一个类似的 NoisyAND(见方框9.8为正式定义)。我们再次假设有一组因素(表示为布尔变量X1,…,Xn) 就其对某些布尔变量的影响而言,可以被视为彼此独立的“原因”是. 考虑以下示例:

一支足球队通常只有在球队的所有“组成部分”——守门员、防守、中场和进攻——都发挥出色的情况下才能赢得比赛。但是,有一个“运气”因素,这意味着(很少)如果某些甚至所有组件表现不佳,他们可能会获胜。同样,由于“运气”,即使他们都打得很好,他们也有可能不会赢。除了运气之外,一些因素比其他因素更重要。例如,虽然即使一个组件打得不好也很难取胜,但要想取胜,打好进攻尤为关键。

假设每个团队组件在对匹配结果的影响方面彼此独立(图 9.22),这种类型的问题很容易建模为噪声与函数。与noisyOR 一样,我们必须指定与每个父级关联的权重值(0 到1 之间),表示其“关键性”,以及表示“运气”因素的泄漏值(0 到1 之间)。

例如,使用 AgenaRisk 语法,我们可以将“团队获胜”的 NPT 定义为:
noisyand (Keeper, 0.7, Defence, 0.9, Midfield,0.8, Attack, 1, 0.05)
如果你运行模型你会发现,当所有的组件都为真时,仍然有一个0.05球队不会获胜的概率(等于泄漏值)。要查看权重的影响,请注意,当所有组件都为真时,除了“Attack play well”是错​​误的,那么获胜的概率为 0,因为“Attack”组件的权重为 1(即最大关键性)。但是,由于“中场”并不那么关键,所以当“中场打得好”为假而其他成分为真时,获胜的可能性仍然很小。

统计代写|贝叶斯分析代写Bayesian Analysis代考|

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