This post is one that has been restored after the hacking. All original comments were lost.
This post is really an excuse to highlight a cute video that’s being passed around (shown above, linked here; part one is here). Describes a simple bet in which 100 people take a dollar bill and write their number on it (1-100). The bills are hidden in 100 numbered boxes in a closed room. Participants enter the room singly and can open up to 50 boxes (and must re-close them); participants may not communicate after exiting the room.
If everybody discovers their own bill, all receive $101, else all forfeit their buck. What’s the best strategy, i.e. the one that maximizes the possibility of all winning?
Before watching, can you guess the best solution? Here’s one possible way. Each person goes into the room and opens 50 boxes. Now, conditional on the premises of the rules of the game and on this simple opening algorithm, what are the chances you find your number (bill)? Obviously 50%.
That is to say, the probability of you finding your bill is deduced as 50%. So the chance that all win is 0.5 to the 100th power, a small number. This really is the probability. And the probability, like all probabilities, is a measure of information. This claim is proved easily because premised on another guessing strategy, the probability everybody guesses correctly is different, around 30%, a not-so-small number.
I’ll let you watch the video to discover the better guessing strategy. It hinges on the idea of permutations. Which is to say, there is more information lurking in the problem that you might not have suspected.
Point is this: the probability changes based on the premises. Nothing physical has changed. The dollars do not move boxes. The people remain the same. No boxes are even opened! The probability changes because the information changes. And the only way that can happen is if probability is a measure of truth, of information.
Probability can’t therefore be subjective or some relative frequency or anything else. If probability were subjective you could claim, based on the way you feel, baby, that the probability all win is 82.141%. And why not? Emotions are infinitely variable, thus so too is probability if probability is a matter of feeling. If probability were physical or some relative frequency, then the probability must remain fixed because the situation is fixed.
All that changes is the information. And when it changes, so does the probability. Thus probability must be a measure of information.
The Monty Hall problem is easier to understand than the permutation video, but it’s the same idea. Different premises, i.e. changing information, means different probabilities. (Read the linked article for a fuller description of the setup.)
Three doors, a prize behind one. You pick a door, and Monty opens one you didn’t pick. There are now two choices, the door you originally picked and another. Should you switch or stay? Premised on “there are two doors, one of which conceals the prize”, the chance really is 50% you have picked the correct door. Thus switching or staying is the same. That means people who insist on this probability are correct, as long as they also insist on just these premises.
But like the permutation problem, there is more information available. Frequently forgotten is that Monty knows where the prize is. He will not open the door which conceals the prize. And it is that information which, if conditioned on, changes the probability of winning to 2/3 if you switch. Different information, different probability. (See the article for why.)
Once again, probability is a measure of information.
Remember the interocitor example?
Given the evidence, or premises, “In this box are six green interocitors and four red ones. One interocitor will be pulled from the box” the probability of “A green interocitor will be pulled” is 6/10. Even though there are no such things as interocitors. Hence no real relative frequencies.
Subjectivity is dangerous in probability. A subjective Bayesian could, relying on the theory, say, “I ate a bad burrito. The probability of pulling a green interocitor is 97.121151%”. How could you prove him wrong?
There are no frequencies nor anything else physical. Probability can’t be relative frequencies or some real tangible, i.e. physical thing. Nor do feelings come into it. Probability isn’t subjective. The probability is strictly 6/10, deduced based on the given information.