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

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

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

Â observations should I even make before making a decision?

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

Â which set of tests to perform on that patient.

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

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

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

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

Â sensor network, which sensors should I measure?

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

Â be something that we want to consider carefully.

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

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

Â formal foundations. But how do we provide a formal semantics

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

Â observation? So the, the formal definition that one

Â can provide for this is the value of perfect information.

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

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

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

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

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

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

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

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

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

Â observation. So we can now define the value of perfect

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

Â I have in the situation where I have this observation.

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

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

Â decision situations. One where the agent has found the company

Â without any kind of additional information about the value of the

Â market. And the other is where the agent gets to

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

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

Â decision making situation with a variable from F to F.

Â 2:55

Minus. The value of the decision making

Â situation. Assuming the agent makes optimal

Â decisions, of the original decision making situation d.

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

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

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

Â 1.25. Which means that the agent should be

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

Â the survey because doing that will increase his expected utility.

Â 3:34

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

Â information. So, the first important property of the

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

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

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

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

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

Â which compares the maximum expected utility between two different influence

Â diagrams. And remember that each of these is

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

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

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

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

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

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

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

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

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

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

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

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

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

Â for instance. If the agent.

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

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

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

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

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

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

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

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

Â property. Which is, when this value of perfect

Â information. Is equal to zero.

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

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

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

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

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

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

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

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

Â changes my decision. In at least one case.

Â 7:42

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

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

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

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

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

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

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

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

Â companies. We are assuming that the company funders

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

Â because they can do some very in depth due diligence.

Â And so the chances of a company to get funding.

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

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

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

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

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

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

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

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

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

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

Â state of the company that makes perfect sense.

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

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

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

Â two companies. The agent has a little mole inside the

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

Â What happens then? Well

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

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

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

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

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

Â 10:55

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

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

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

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

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

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

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

Â 12:01

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

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

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

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

Â scenario. And, sure enough if you look at the

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

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

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

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

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

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

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

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

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

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

Â expected utility of company two which is 0.33.

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

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

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

Â larger so let's work our way through that.

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

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

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

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

Â company one. And now indeed the expected utility goes

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

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

Â expected utility relative to what we had before.

Â Because now there is more value to the information.

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

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

Â 14:30

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

Â changed the probability that the company gets funded.

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

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

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

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

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

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

Â is that even if the agent changes their mind.

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

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

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

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

Â They're going to decide to change their mind.

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

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

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

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

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

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

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

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

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

Â expected utility values, or the NEU values, rather.

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

Â intuition about when information is valuable.

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

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

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

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

Â the decision.

Â