oblique fact: when the monty hall host reveals the door that contains the goat, they are technically revealing 1/3 bits of information about which door the car is behind.
that information allows you to improve your odds of winning the car, and it just so happens that switching is how you act on this information
what is a fractional bit? it's a bit with uncertainty.
For example, if you ask me a yes/no question and you know that half of the time I just answer randomly, and the other half I tell the truth, then my answer will contain 1/2 bits of information.
I'm thinking now about how you can compare the entropy of a prior and posterior distribution, and the difference will be the number of bits of information gained between them. I'm not sure if this is valid 🤔
if you believe that you have a 90% likelihood of winning a lottery, and then 100 of your friends lose and you use bayes rule to update your likelihood to 50%, did you just lose information
(I didn't actually do the calculation to see what bayes theorem tells us what the posterior likelihood would be I just took 50% for examples sake lmao)
@SuricrasiaOnline One way I've tried to explain this... suppose there are 1000 doors. You pick one. The host opens 998 wrong doors. Do you still think of the two doors he didn't open, both are equally likely to contain the prize? You know he has to avoi the door you picked and the door with the goat.
People often still don't get it when you put it this way, though.
@SuricrasiaOnline this was my first attempt at an explanation to my partner and they were like uhh,,,
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