Expected value of perfect information
Alright, imagine you are playing a game where you have to decide between two doors. Behind one door, there is a big bag of candy that you really want, and behind the other door, there is a big pile of dirty socks that you definitely don't want. You have to make a decision and choose one of the doors.
Now, let's say that I come along and tell you that I know which door has the candy and which door has the socks. But here's the catch: I'm kind of mean and I won't just tell you outright which door is which. Instead, I'm going to make you pay me some of your candy to give you some information. You might ask, "Why would I give you my candy?" Well, because the information I'm giving you is really valuable and it could help you make the best decision.
The expected value of perfect information is a fancy term that basically means how much your decision would improve if you had all the information you needed. In this case, it means how much better your decision would be if you knew for certain which door had the candy and which door had the socks.
To calculate the expected value of perfect information, we need to do a little bit of math. We can start by figuring out the expected value without any extra information. Let's say there's a 50% chance that the candy is behind Door 1 and a 50% chance that it's behind Door 2. So, if you choose Door 1, there's a 50% chance you get the candy and a 50% chance you get the socks. The same applies if you choose Door 2.
So, the expected value before any extra information is just the average of the candy and sock values. Let's say the candy is worth 10 pieces and the socks are worth -5 pieces (because you really don't want them). The expected value before any information will be:
(0.5 * 10) + (0.5 * -5) = 5 - 2.5 = 2.5
Okay, now let's calculate the expected value of perfect information. This means we know for sure which door has the candy and which door has the socks. If we had this information, we would always choose the door with the candy, right? So, the expected value of perfect information is just the value of the candy, which is 10 pieces in this case.
If we compare the expected value before any information (2.5 pieces) with the expected value of perfect information (10 pieces), we can see that having all the information would improve our decision by 7.5 pieces. That's the expected value of perfect information!
In simpler terms, the expected value of perfect information tells us how much better our decision would be if we knew everything there is to know. In our candy and socks example, it tells us how much more candy we would get if we knew which door had the candy beforehand.
Now, let's say that I come along and tell you that I know which door has the candy and which door has the socks. But here's the catch: I'm kind of mean and I won't just tell you outright which door is which. Instead, I'm going to make you pay me some of your candy to give you some information. You might ask, "Why would I give you my candy?" Well, because the information I'm giving you is really valuable and it could help you make the best decision.
The expected value of perfect information is a fancy term that basically means how much your decision would improve if you had all the information you needed. In this case, it means how much better your decision would be if you knew for certain which door had the candy and which door had the socks.
To calculate the expected value of perfect information, we need to do a little bit of math. We can start by figuring out the expected value without any extra information. Let's say there's a 50% chance that the candy is behind Door 1 and a 50% chance that it's behind Door 2. So, if you choose Door 1, there's a 50% chance you get the candy and a 50% chance you get the socks. The same applies if you choose Door 2.
So, the expected value before any extra information is just the average of the candy and sock values. Let's say the candy is worth 10 pieces and the socks are worth -5 pieces (because you really don't want them). The expected value before any information will be:
(0.5 * 10) + (0.5 * -5) = 5 - 2.5 = 2.5
Okay, now let's calculate the expected value of perfect information. This means we know for sure which door has the candy and which door has the socks. If we had this information, we would always choose the door with the candy, right? So, the expected value of perfect information is just the value of the candy, which is 10 pieces in this case.
If we compare the expected value before any information (2.5 pieces) with the expected value of perfect information (10 pieces), we can see that having all the information would improve our decision by 7.5 pieces. That's the expected value of perfect information!
In simpler terms, the expected value of perfect information tells us how much better our decision would be if we knew everything there is to know. In our candy and socks example, it tells us how much more candy we would get if we knew which door had the candy beforehand.
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