However, I don’t see where “value of information” shows up in this framework anywhere. Do I need to distinguish some choice nodes as “information gathering”, and apply a default strategy to them as “don’t gather any information”, and then compute value of information as the difference between best I can do with my eyes closed and the best I can do flat out?
Think of VoI as going in the reverse direction. That is, beforehand you would have modeled your test outcome as a nature node because you didn’t consider the option of not running the test. Now you stick in a choice node of “run the test” that leads to the nature node of the test output on the one branch, and the tree where you don’t know the test output on the other branch. Like you suggest, you then use the work-backwards algorithm to figure out the optimal decision at the “run the test” node, and the difference between the branch node values is the absolute value of the VoI minus the test cost.
What if there is no natural partition of some actions as information gathering and not-information-gathering?
Then VoI won’t help you very much. VoI is a concept that helps in specifying decision problems- building the tree- not computing a tree to find an optimal policy. It suggests modeling information-gathering activities as choice nodes leading to nature nodes, rather than just nature nodes. If you’ve got a complete decision problem already, then you don’t need VoI.
I should point out that most tests aren’t modeled as just information-gathering. If a test is costless, then why not run it, even if you throw the results away? Typically the test will have some cost, in either utility or prospects, and so in some sense there’s rarely actions that are purely information gathering.
Think of VoI as going in the reverse direction. That is, beforehand you would have modeled your test outcome as a nature node because you didn’t consider the option of not running the test. Now you stick in a choice node of “run the test” that leads to the nature node of the test output on the one branch, and the tree where you don’t know the test output on the other branch. Like you suggest, you then use the work-backwards algorithm to figure out the optimal decision at the “run the test” node, and the difference between the branch node values is the absolute value of the VoI minus the test cost.
The problem with this model is that it doesn’t necessarily give you the value of INFORMATION. Making the ‘get info’ node a choice point on the tree essentially allows arbitrary changes between the with info and without info branches of the tree. In other words it’s not clear we are finding the value of information and not some other result of this choice.
That is why I choose to phrase my model in terms of getting to look at otherwise hidden results of nature nodes.
Think of VoI as going in the reverse direction. That is, beforehand you would have modeled your test outcome as a nature node because you didn’t consider the option of not running the test. Now you stick in a choice node of “run the test” that leads to the nature node of the test output on the one branch, and the tree where you don’t know the test output on the other branch. Like you suggest, you then use the work-backwards algorithm to figure out the optimal decision at the “run the test” node, and the difference between the branch node values is the absolute value of the VoI minus the test cost.
Then VoI won’t help you very much. VoI is a concept that helps in specifying decision problems- building the tree- not computing a tree to find an optimal policy. It suggests modeling information-gathering activities as choice nodes leading to nature nodes, rather than just nature nodes. If you’ve got a complete decision problem already, then you don’t need VoI.
I should point out that most tests aren’t modeled as just information-gathering. If a test is costless, then why not run it, even if you throw the results away? Typically the test will have some cost, in either utility or prospects, and so in some sense there’s rarely actions that are purely information gathering.
The problem with this model is that it doesn’t necessarily give you the value of INFORMATION. Making the ‘get info’ node a choice point on the tree essentially allows arbitrary changes between the with info and without info branches of the tree. In other words it’s not clear we are finding the value of information and not some other result of this choice.
That is why I choose to phrase my model in terms of getting to look at otherwise hidden results of nature nodes.