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.
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.