# 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|Decision Trees

Decision trees (DTs) have traditionally been used to choose an optimal decision from a finite set of choices, which are sometimes called policies. Typically, the value being optimized is some utility function expressed for each possible outcome of the decision. A DT represents the structure of a decision problem by modeling all possible combinations of decisions and observations, usually in the particular sequence in which one would expect observations and decisions to be made. DTs are composed of three types of nodes: chance nodes, decision nodes, and utility nodes. Each outgoing arc from a chance node represents an outcome and is labeled with the name and probability of this outcome. Each outgoing arc from a decision node is labeled with a decision alternative.

Now that we have introduced HIDs and represented our decision analysis problem in HID form, we can translate HIDs into a DT in order to do the computation and identify the optimal decisions.

Computing this model using AgenaRisk, as a $\mathrm{BN}$, requires the use of the hybrid influence diagram analysis function. The decision nodes, observable chance nodes, ultimate utility node, and evaluation policy are identified in the analyzer and then calculated. The analyzer determines the decision/information sequence between the nodes in the model. Providing there are no ambiguities in the model (AgenaRisk provides warnings if there are), it computes the optimal decision policy and presents the results as a decision tree, as shown in Figure $11.2$ (for the Example $11.1$ problem).

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Advanced Hybrid Influence Diagrams

Traditionally, influence diagrams and decision trees have only been applied to decision problems where the chance and utility nodes are discrete. Likewise, they have also only been applied when the expectation of the utility is being optimized. This is because the previous available algorithms were not able to conveniently and accurately handle nodes with continuous distribution functions. The HID analyzer in AgenaRisk can, however, handle continuous chance node types as well as utility nodes.

Let’s consider an example where we have a nonlinear utility function modeling a trade-off between expected payoff and the variance in the payoff, such as when a decision maker has an aversion to risk (in the form of uncertain returns). In this case, the decision maker would prefer a lower expected utility if it were less volatile to one that had a higher expected utility but was more volatile. This might be motivated by a fear of an extreme negative utility outcome, which might cause financial ruin.

that modeled the trade-off between expected utility and the variance of the expected utility. So, a given increase in variance of the expected utility would be offset by an increase in the mean (average) utility. What might this function be? Well, if we examine the Wildcatter $\mathrm{BN}$ and inspect the marginal probability distribution for the ultimate utility node, we find that the expected utility, $\mu$, is 5 and the variance, $\sigma^2$, is $5.525$. Given the ratio between these values is $1000: 1$, we could “normalize” the trade-off between them by dividing the variance by 1000 . In this way, a change of 1000 in variance is equivalent to a change of 1 in the expectation. Next, we might want to represent the risk averseness of the decision maker and apply a risk aversion factor-the higher the value, the more averse to high variance they might be. Let’s denote this as $\lambda$.

## 统计代写|贝叶斯分析代写贝叶斯分析代考|高级混合影响图

，它模拟了期望效用和期望效用方差之间的权衡。因此，预期效用方差的增加会被平均效用的增加所抵消。这个函数是什么呢?如果我们检验Wildcatter模型$\mathrm{BN}$并检验最终效用节点的边际概率分布，我们会发现期望效用$\mu$是5，方差$\sigma^2$是$5.525$。假设这些值之间的比率是$1000: 1$，我们可以通过将方差除以1000来“规范化”它们之间的权衡。这样，方差1000的变化等价于期望1的变化。接下来，我们可能想要表示决策者的风险厌恶程度，并应用风险厌恶因素——该值越高，表示他们对高方差的厌恶程度越高。我们将其表示为$\lambda$ .

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