cs188-note13

Decision Networks

Nodes:

• Chance: 类比贝叶斯网络
• Action: 我们能控制并做出选择的节点
• Utility: 前两种的children，基于前两种输出一个效能

在决策网络中，我们的目标还是选择能产生maximum expected utility(MEU) 的行动:

• calculate the posterior probabilities of all chance node parents of the utility node into which the action node feeds
• compute the expected utility of taking that action given the posterior probabilities computed in the previous step. The expected utility of taking an action a given evidence e and n chance nodes is computed with the following formula:

$$EU(a|e)=\sum _{x_1,\dots ,x_n}P(x_1,\dots ,x_n|e)U(a,x_1,\dots ,x_n)$$

• Finally, select the action which yielded the highest utility to get the MEU.

能产生最大预期效用的行动就是 “采取”，因此这也是决策网络向我们推荐的行动。更正式地说，可以通过求预期效用的 argmax 来确定能产生 MEU 的行动

Outcome Trees

就是把上面的内容拆成了图，下面是图

The Value of Perfect Information

General Formula

$$\begin{array}{rl}MEU(e)& =\underset{a}{max}\sum _{s}P(s|e)U(s,a)\\ MEU(e,e^{\prime })& =\underset{a}{max}\sum _{s}P(s|e,e^{\prime })U(s,a)\\ MEU(e,E^{\prime })& =\sum _{e^{\prime }}P(e^{\prime }|e)MEU(e,e^{\prime })\\ VPI(E^{\prime }|e)& =MEU(e,E^{\prime })-MEU(e)\end{array}$$

Properties of VPI

• Nonnegativity.
• Nonadditivity. It’s true because generally observing some new evidence E j might change how much we care about Ek
• Order-independence.