So let's look at some of. Properties of the value of perfect

information. So, the first important property of the

value of perfect information, is assuming that there's no cost of the information,

so not counting in how much it might cost say to conduct the survey.

One can show that the value of perfect information is always greater than or

equal to zero. So let's first go ahead and convince

ourselves that this is true. So lets look at this expression over here

which compares the maximum expected utility between two different influence

diagrams. And remember that each of these is

obtained by optimizing. Over a decision rule, this one is

optimized as the MU of the original decision V, is optimizing a decision rule

delta which is a CPD of A given it's current set of parents Z.

In this one the new influence diagram is optimizing a decision rule delta where A

has Z, all the original parents Z plus an additional parent X.

And the point that one, that becomes obvious when you think of it this way, is

that this is a strictly larger class of CPDs then this.

That is, any CPD. Of the form delta of A given Z is also a

CPD. Of the form delta of A given dx, which

means any decision rule that I have could implemented in my original influence

diagram I can also implement in the context of my current influence diagram

and if it had a particular value there it will still have that same expected

utility value in the original diagram. So to go back to our example for exam-

for instance. If the agent.

Has a, decision role that, found, the sides say to found the company,

regardless of the value of the survey, that is a still a legitamate decision

role, even when they get to observe the survey and it would have the same,

expected utility. And so, that means that the set of

decisions that I get to consider is, just larger in the context of the richer of

the richer influence diagram, and therefore, one cannot possibly lose, by

exploring a larger set of, a larger space over which to optimize.

Okay. So, now let's think about the second

property. Which is, when this value of perfect

information. Is equal to zero.

And this, follows from very similar reason to the one.

That, we just talked about. So if, the optimal decision rule for, D.

And for my original influence diagram B is still optimal for the exntended

influence diagram, then I've gained nothing, from the information, that is,

I, any, any decision that I could, any decision rule that I could have applied

before, I can still apply and therefore there, I have gained nothing from this

additional observation. And so this gives us a very clear notion

of when information is useful. Information is useful precisely when it

changes my decision. In at least one case.