Value of Perfect Information

${\mathrm{VPI}}_E(F):=\left({\sum }_{f\in D}P(F=f\mid E)\cdot \mathrm{EU}\left({\alpha }_f\mid E,F=f\right)\right)-\mathrm{EU}(\alpha \mid E)$

Given this, the best possible Action out of all actions $A$ is ${\alpha }_f=\underset{a\in A}{\mathrm{argmax}}\mathrm{EU}(a\mid E,F=f)$

When more than one piece of evidence can be gathered this becomes a sequential decision problem.

Properties

Non-negative

In Expectation, not Post-Hoc

$VP{I}_E(F)\ge 0$

Non-additive

For example when obtaining the same information twice, or the new information contains parts of the old information that we already have.

$VP{I}_E(F,G)\ne VP{I}_E(F)+VP{I}_E(G)$

Order-independent

It does not matter in what order we gather information. At the point we make the decision, we would still have the same value of information.

$VP{I}_E(F,G)=VP{I}_E(F)+VP{I}_{E,F}(G)=VP{I}_E(G)+VP{I}_{E,G}(F)$