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We showed how influence diagrams can allow an agent to make decisions

regarding what course of action the agent should take given a set of observations.

But often we want to answer a different type of question, which is what

observations should I even make before making a decision?

For example, a doctor encountering a particular patient might have to decide

which set of tests to perform on that patient.

Tests are not free, they cause pain to the patient, they come

with a risk, and they cost money. So which ones are worthwhile and which

ones are not? The same kind of question comes up in

many other scenarios. So for example, if you're running a

sensor network, which sensors should I measure?

The sensor might require energy in order to transmit the information and that may

be something that we want to consider carefully.

And there's many other examples of that. It turns out that the same framework of

influence diagrams can also be used to answer that question using rigorous

formal foundations. But how do we provide a formal semantics

for the notion of the value of getting information or the value of making an

observation? So the, the formal definition that one

can provide for this is the value of perfect information.

So this, this stands for value of perfect information about a variable X, is the

value that we have by observing X, before choosing an action in A.

And, perfect means that we observe X with perfectly without any, without any noise.

How do we make that a formal how do we give that a formal value?

Well, if D was our original influence diagram before I had the opportunity to

observe X. We can compare the value of D to the

value of a different influence diagram, which is the one where I introduce an

edge from X to A. Because that tells me what the value of

the situation would be if if I had that, the ability to make that

observation. So we can now define the value of perfect

information to be simply a difference between the maximum expected utility that

I have in the situation where I have this observation.

Minus the value, the expected utility to the agent in a scenario where I don't.

So in this example that we've presented before, we saw that we'd compared two

decision situations. One where the agent has found the company

without any kind of additional information about the value of the

market. And the other is where the agent gets to

make an observation regarding the survey variable prior to making the decision

whether to found the company. So we can compare the value of the

decision making situation with a variable from F to F.

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Minus. The value of the decision making

situation. Assuming the agent makes optimal

decisions, of the original decision making situation d.

And we can compare that and see how much the agent's gained by this.

And if you recall, we computed this to be 3.25.

And this was two. So the value of perfect information was

1.25. Which means that the agent should be

willing to pay anything up to 1.25 utility points bef- in order to conduct

the survey because doing that will increase his expected utility.

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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.

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Let's see how this intuition manifests in an actual decision making scenario.

So, let's imagine that our entrepreneur has decided against founding a widget

company, and is now starting to trying to pick between two companies that he can

choose to join. For each company, there is the state that

the company is in. So, S1 is that the company is not,

doesn't have that great of a management. Things are not necessarily going so well.

So that's S1. S two is medium and s three is the

company is doing great. And the same thing holds for both

companies. We are assuming that the company funders

have access to some of the info, to this information about the company's state

because they can do some very in depth due diligence.

And so the chances of a company to get funding.

Depends on the state of the company, so you can see if the company state is poor,

S1, then the chances of getting funding are zero point one.

Where as if the company is doing great the chances of getting funding are zero

point nine. And we're seeing that the agent's utility

is one if the company that he chose, that he joins.

It's funded, and zero otherwise. So now let's think about the two

strategies that the agent can take without any information, and so if the

agent chooses to join company one one can see that company one is

that the expected utility now is 0.72, and the expected utility of company two

which is not doing as great is only 0.33. That's, you know, if you look at the

state of the company that makes perfect sense.

Now what happens if the agent now gets to make an observation?

And specifically, we're going to let the agent make the observation.

Of s2, regarding s2. Which is, in this case the weaker of the

two companies. The agent has a little mole inside the

company, and can get access to that information before making decision.

What happens then? Well

the if you look at the utility values you can see that if company one is in state,

sorry if company two is in state one. Then, which is a not unlikely scenario,

it happens with probability 40%. But chances of getting funding are 0.1.

And so the agents expected utility in this case, so the expected utility if.

The agent chooses c2, and s and the state of the second company is s1, is 0.1.

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The accepted utility if C equals C, if the company, if the agent chooses the

second company and it's doing. Moderatly well in 0.4, both of these are

lower then 0.72 that the agent can guarentee on expectation if he choses

company one, even without any additional information on company one.

And so in both of these cases the agent is going to prefer.

Stick with his original. Choice of going with company one.

It is only in the one scenario that that we have where.

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The changes in opinion and go with c2. But that happens with very little

probability, it only happens with probability with 0.1 and so that means

that the value of information here is going to be very low, because although

there is a situation in which the agent changes his mind, it is an unlikely

scenario. And, sure enough if you look at the

expected utility in the influence diagram with that edge that I just added.

It only goes up from 0.7 to 0.743 which means that the agent shouldn't be willing

to pay his [INAUDIBLE] company too much money in order to get information about

the detail. 'Kay, now let's look at a slightly

different situation. Where now, neither company's doing so

great. So, you can see that now company one is

also kind of this sort of rocky start-up without a very good management structure

and a, and an unclear business model. In this case, what happens?

So once again we can compute the expected utility of the two actions and now we can

see that the expected utility of choosing company one is 0.35 as compared to the

expected utility of company two which is 0.33.

So now decisions are much more finely balanced, relative to each other and so

you would think that there would be a much higher value of information to be

gained because the chances that the agent would change his mind are considerably

larger so let's work our way through that.

And see that once again if we consider adding this edge from the mole in company

two, we can now see that the agent is going to want to change his mind either

when he observes s2 or when he observes s3, because both of these, both 0.4 and

0.9 are larger than, than the expected utility he expects from sticking with

company one. And now indeed the expected utility goes

up, in the case where we have this influent diagram.

And it goes up to 0.43, which is a much more significant increase in their

expected utility relative to what we had before.

Because now there is more value to the information.

We change the agent changes their opinion in two out of three scenarios.

And that's zero. What happens with probability 0.6.

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Now let's look at. Yet a third scenario where now we've

changed the probability that the company gets funded.

Now we're back in the bubble days of the Internet boom, and basically, pretty much

every company gets funded with a, pretty high probability, even if their business

model is totally dubious. And in this case, what happens.

So now we can once again compute the expected utility of C1 which is 0.788 the

expected utility of C2 which is 0.779 and we can see that again these expected

utilities are really close to each other. And intuitively what that's going to mean

is that even if the agent changes their mind.

It doesn't make much of a difference in terms of their expected utility.

So, here we see that because of. In this case, we can see that 0.8, which

is their expected utility in the case of the observed s two.

This value S2 is zero is bigger than 0.788, and so they're going to pick.

They're going to decide to change their mind.

And go from C1 to C2 and similarly for S3 but the actual utility gains in this case

are fairly small and so now the utility, the expected utility that we have in this

scenario where, where the edge didn't get observe this variable without before

making a decision is zero point. 8412, which is only a fairly small

increase over the 0.788 that they could have guaranteed themselves without making

that observation, so once again this is a case where the poor mole in company two

doesn't get that much money. So, to summarize, influence diagrams

provide a very clear and elegant interpretation for what it means to make

an observation. As simply the val-, the difference in the

expected utility values, or the NEU values, rather.

Between two influence diagrams. And this allows us to provide a concrete

intuition about when information is valuable.

And that is only and exactly when it induces a change in the action in at

least one context. And now quantitatively it means that, the

extent in which information is valuable, depends on both how much my utility

improves based on that change, and on how likely the context are in which I changed

the decision.