Let’s say that we have a box of weighted coins. Some are more likely to fall heads; others tails. We pull one out and flip it many times. The flips are identical, so we can switch the order. They are independent conditional on knowing which coin was chosen, but ahead of time they are dependent, the one telling us about the choice of coin and thus about the other. De Finetti’s theorem says that all exchangeable sequences take this form.
Added: Actually, de Finetti’s theorem only applies to infinite sequences. Here’s an example of a finite exchangeable sequence that doesn’t fit the theorem: draw balls from a box without replacement. This can only go on until the box is empty. And of course you can combine the two: randomly choose a box with at least n balls and then pull out n balls without replacement.
Added: A crazy model that is exchangeable is Pólya’s urn. It is not obvious that it is exchangeable, let alone that the conclusion of de Finetti’s theorem applies. Pólya’s urn contains balls of two colors, the initial numbers of which are known. Every time you draw one out, you put k of the same color back. If k=1, this is drawing with replacement; if k=0, this is drawing without replacement, both of which are exchangeable. And if k is a larger integer, it is also exchangeable. Here is an idea of how to the exchangeability. What if are somehow confused about the size of the balls, and think that they are r times bigger than they really are? Then each time we remove an actual ball, we’re removing 1/r part of a confused ball. That’s like removing 1 confused ball and putting back 1-1/r balls. Thus k=1-1/r is like drawing without replacement, but with this confusion. This is exchangeable. Thus the model is exchangeable for infinitely many values of k, which verifies some identities for infinitely many values of k, which is probably enough to verify it as an algebraic identity.
Let’s say that we have a box of weighted coins. Some are more likely to fall heads; others tails. We pull one out and flip it many times. The flips are identical, so we can switch the order. They are independent conditional on knowing which coin was chosen, but ahead of time they are dependent, the one telling us about the choice of coin and thus about the other. De Finetti’s theorem says that all exchangeable sequences take this form.
Added: Actually, de Finetti’s theorem only applies to infinite sequences. Here’s an example of a finite exchangeable sequence that doesn’t fit the theorem: draw balls from a box without replacement. This can only go on until the box is empty. And of course you can combine the two: randomly choose a box with at least n balls and then pull out n balls without replacement.
Added: A crazy model that is exchangeable is Pólya’s urn. It is not obvious that it is exchangeable, let alone that the conclusion of de Finetti’s theorem applies. Pólya’s urn contains balls of two colors, the initial numbers of which are known. Every time you draw one out, you put k of the same color back. If k=1, this is drawing with replacement; if k=0, this is drawing without replacement, both of which are exchangeable. And if k is a larger integer, it is also exchangeable.
Here is an idea of how to the exchangeability. What if are somehow confused about the size of the balls, and think that they are r times bigger than they really are? Then each time we remove an actual ball, we’re removing 1/r part of a confused ball. That’s like removing 1 confused ball and putting back 1-1/r balls. Thus k=1-1/r is like drawing without replacement, but with this confusion. This is exchangeable. Thus the model is exchangeable for infinitely many values of k, which verifies some identities for infinitely many values of k, which is probably enough to verify it as an algebraic identity.