3 ms·

Regarding the "money in envelopes" version in the second half of the post: When I first encountered this, I came up with the author's answer (about how the "alw

by fenomas 11y ago

Regarding the "money in envelopes" version in the second half of the post: When I first encountered this, I came up with the author's answer (about how the "always switch" strategy implicitly assumes they come from an infinite distribution). But over the years I've come to believe that it confuses things to arbitrarily claim that the opponent is choosing their numbers from a distribution, and then talking about what kind of a distribution it was. I think it's much clearer to examine things in Bayesian terms, by noting that before opening an envelope the player expects it's 50% likely that they chose low, and after opening it they can update their expectations. (The math works out the same either way, it just seems weird to me to impose rules on the opponent that weren't part of the puzzle.) (Also: the fact that people always seem to examine this puzzle by imagining what kind of distribution the opponent chooses from seems, to me, an artifact of just how accustomed we are to standard statistics, and how uncomfortable we are with probability puzzles that don't involve values being chosen from predefined distributions.)

dllthomas 11y ago
It's not that the opponent is "choosing their numbers from a distribution", so much as that we need to model them that way, and that running probability calculations when violating probability axioms (of course) fails. I think that's roughly as true whether we're talking frequentist or Bayesian.
- fenomas 11y ago
  > It's not that the opponent is "choosing their numbers from a distribution", so much as that we need to model them that way, That's what I mean - the statistics we're mostly taught can only deal with distributions, so we wind up inventing them even when none is specified. Which would be fine if it made the math easier, but it seems to me that it makes it worse. Consider a concrete question, like "if I open the envelope and find $10, how confident would I need to be that it's the low envelope before I switch?". In Bayesian terms that's easy to answer, but with the frequentist version one is left asking "how likely is it that my opponent used a distribution which when sampled once yielded $10, as opposed to a distribution which.." or some such thing. I mean, the envelopes puzzle basically says, "you start out expecting two things to be equally likely, then you learn a piece of information, now how does that change your expectations?" It's as Bayesian a question as ever there was.
  - dllthomas 11y ago
    The question of how you change your expectations, though, boils down to asking "how do I think the person chose?" Interestingly, I think that given any particular choice of maximum entropy distribution, the recommendations boil down to "there is a value above which you should stick and below which you should switch." Since I don't think - from the question alone - there is any particular reason to favor a specific distribution, I think it amounts to a free choice of that number. If you have some idea of, say, the bankroll of the game runners (or, at worst, the total amount of money in circulation) you should be able to do better, of course.
    - fenomas 11y ago
      > boils down to asking "how do I think the person chose?" I agree, but the operative word is "think". We can't talk about how the opponent chose, only about the player's expectations of how they chose. And talking about how evidence affects expectations is the problem Bayes exists to solve, is my reasoning. For your second point, that sounds right but assuming a well-formed high-entropy distribution seems like a very arbitrary thing to do. The player might very naturally, say, consider round numbers more or less likely, or at any rate believe that $50 is a more likely value than $50.01, etc, right? And actually, this point highlights why I think this should be considered as a Bayesian problem. In frequentist terms, one talks about the opponent choosing numbers from some distribution, then the player choosing a strategy based on their expectations about the distribution, then they open their envelope and apply the strategy. If nothing else this approach greatly obscures the most important feature of the puzzle, which is that the player gains new information when they open their envelope. More, it seems somewhat bonkers when considered - if the player chooses a switch value of $10, then opens their envelope and finds $50,000, should they really just apply the strategy they chose based on expectations they now know to have been wildly incorrect? Effectively the player winds up deciding whether to switch based on the fact that they thought $10 was more likely than $20, when one would expect them to reconsider their expectations that they now know to have been wildly off-base. In contrast, the Bayesian version is straightforward, and we can completely describe the player's strategy without making any assumptions on how it's formed. Specifically, suppose the player starts off with some expectation P(T) that that the total value of both envelopes would be T. Then they open their envelope, find a value of x, and and update their expectations. Then, their strategy depends solely on the ratio between P(1.5x) and P(3x). And we can say all this without even considering whether P is continuous/discrete/etc, or what factors it's based on. All we assume is that it's defined at the two points that are relevant to a given instance of the puzzle - i.e. that the player can form some expectation of how likely one value is compared to twice as much. I fear I can't describe this very well, but does that make any sense?

by fenomas 11y ago

dllthomas 11y ago

It's not that the opponent is "choosing their numbers from a distribution", so much as that we need to model them that way, and that running probability calculations when violating probability axioms (of course) fails. I think that's roughly as true whether we're talking frequentist or Bayesian.

fenomas 11y ago
> It's not that the opponent is "choosing their numbers from a distribution", so much as that we need to model them that way, That's what I mean - the statistics we're mostly taught can only deal with distributions, so we wind up inventing them even when none is specified. Which would be fine if it made the math easier, but it seems to me that it makes it worse. Consider a concrete question, like "if I open the envelope and find $10, how confident would I need to be that it's the low envelope before I switch?". In Bayesian terms that's easy to answer, but with the frequentist version one is left asking "how likely is it that my opponent used a distribution which when sampled once yielded $10, as opposed to a distribution which.." or some such thing. I mean, the envelopes puzzle basically says, "you start out expecting two things to be equally likely, then you learn a piece of information, now how does that change your expectations?" It's as Bayesian a question as ever there was.
- dllthomas 11y ago
  The question of how you change your expectations, though, boils down to asking "how do I think the person chose?" Interestingly, I think that given any particular choice of maximum entropy distribution, the recommendations boil down to "there is a value above which you should stick and below which you should switch." Since I don't think - from the question alone - there is any particular reason to favor a specific distribution, I think it amounts to a free choice of that number. If you have some idea of, say, the bankroll of the game runners (or, at worst, the total amount of money in circulation) you should be able to do better, of course.
  - fenomas 11y ago
    > boils down to asking "how do I think the person chose?" I agree, but the operative word is "think". We can't talk about how the opponent chose, only about the player's expectations of how they chose. And talking about how evidence affects expectations is the problem Bayes exists to solve, is my reasoning. For your second point, that sounds right but assuming a well-formed high-entropy distribution seems like a very arbitrary thing to do. The player might very naturally, say, consider round numbers more or less likely, or at any rate believe that $50 is a more likely value than $50.01, etc, right? And actually, this point highlights why I think this should be considered as a Bayesian problem. In frequentist terms, one talks about the opponent choosing numbers from some distribution, then the player choosing a strategy based on their expectations about the distribution, then they open their envelope and apply the strategy. If nothing else this approach greatly obscures the most important feature of the puzzle, which is that the player gains new information when they open their envelope. More, it seems somewhat bonkers when considered - if the player chooses a switch value of $10, then opens their envelope and finds $50,000, should they really just apply the strategy they chose based on expectations they now know to have been wildly incorrect? Effectively the player winds up deciding whether to switch based on the fact that they thought $10 was more likely than $20, when one would expect them to reconsider their expectations that they now know to have been wildly off-base. In contrast, the Bayesian version is straightforward, and we can completely describe the player's strategy without making any assumptions on how it's formed. Specifically, suppose the player starts off with some expectation P(T) that that the total value of both envelopes would be T. Then they open their envelope, find a value of x, and and update their expectations. Then, their strategy depends solely on the ratio between P(1.5x) and P(3x). And we can say all this without even considering whether P is continuous/discrete/etc, or what factors it's based on. All we assume is that it's defined at the two points that are relevant to a given instance of the puzzle - i.e. that the player can form some expectation of how likely one value is compared to twice as much. I fear I can't describe this very well, but does that make any sense?