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> Bayesian inference requires all unknowns to be represented by probability distributions, which awkwardly implies that the probability of an event for which we

> Bayesian inference requires all unknowns to be represented by probability distributions, which awkwardly implies that the probability of an event for which we are completely ignorant must be assigned a particular numerical value such as 1/2, as if it were known as precisely as the probability of a truly random event (e.g., a coin flip). This reads as a complete non-sequitur to me. I thought we'd moved past these misunderstandings. The coin flip is just as much a problem of ignorance as the boxing match. It's only easier if you give up on attempting to use any of your initial information about eg. the initial orientation of the coin, the method of launch, etc, and just assume you don't know anything. The same is true with boxing vs wrestling. If I don't know anything about boxing vs wrestling or their players, it's an easy 50-50, because that best reflects my state of ignorance. It's only difficult if I attempt to improve my prediction by applying what I know about the sport. Then I need to make a real effort, which is hard.

nurettin 4y ago
They are just saying that you can't start adjusting without any priors and personally, I don't think .5 means "I don't know". However from a practical standpoint if you are saying that the method converges to the same distribution at the same rate in any case no matter where the prior is, that would make sense.
- inimino 4y ago
  If the two outcomes are both possible and that's all you know, that's exactly what .5 means. I think the confusion comes from that, if the coin flip is repeated, we know (if the coin is fair) it will approach N/2, and in the other case, we don't know any such thing.
inimino 4y ago
Indeed. Obviously Gelman is accomplished in the field, so it's a bit surprising. But it's easy to get confused on these issues, for anyone apparently. There is a part towards the end of the paper where he suggests that the coin flip is really a "point distribution" at .5, while the outcome of the fight is something else. I think the intuition that he was going for was that if the fight was repeated N times, the outcome likely would not be N/2 wins for each. This suggests that there is "something" about the problem that could probably be known, and that is different from 1/2. But obviously that "something" is not our prior in the scenario given before the fight occurs. Determining the prior should be as simple as: Are both outcomes possible? Yes. Do you have any reason to expect one over the other? No. Well, then you have your prior. I think this "something" is where the idea of taking a uniform distribution on [0,1] comes from -- the idea that there is some "true" probability of the outcome of the fight, but we just don't know what it is, rather than realizing that no such thing exists. Really it seems like re-reading Jaynes should just clear the whole thing up.
- jbay808 4y ago
  I totally agree! There is definitely a difference in that repeated boxing matches are not independent probabilities. But honestly, if I put on my precision engineering hat, the fact that repeated coin flips are independently 1/2 just tells me that they're being repeated badly. With enough improvements to the test setup, we can get the variability out and they'll all be heads.
Strilanc 4y ago
This reminds me of a simpler situation [1]. I give you a coin, and promise it is biased, but don't tell you the bias. I flip the coin and ask you what is the probability that it landed heads. - A bayesian says the probability of the coin landing heads is 50/50. - A frequentist says the probability of the coin landing heads is anything except 50/50, because the coin was promised to be biased; if it was 50/50 then the coin would be fair not biased. They are of course using the word "probability" slightly differently. The bayesian is talking about an aggregation of the available hypotheses that is useful for betting and predictions, while the frequentist is talking about the individual possible hypotheses going into that aggregation. [1]: https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probability-is-in-the-mind https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probabilit...
- kgwgk 4y ago
  > I flip the coin and ask you what is the probability that it landed heads. > - A frequentist says the probability of the coin landing heads […] A frequentist says the concept of “the probability that it landed heads” makes no sense whatsoever.
kgwgk 4y ago
> The coin flip is just as much a problem of ignorance as the boxing match. Would you say that “the next coin flip” is _just_ as much a problem of ignorance as “the previous coin flip”? You may assign the same probability 1/2 to “getting heads is the next flip” and to “having got heads in the last flip” but these cases are not completely equivalent and there are practical differences.
- inimino 4y ago
  I might be missing the distinction you're making here. If you mean if the flip has already occurred, but the result was concealed (for example, the coin is under my hand but I have not looked at it), since my state of knowledge is the same as before the flip, I'd say it's equivalent.
  - kgwgk 4y ago
    "the next flip" We're going to flip a coin now. Before the flip, I give you the choice between a) $1 for sure and b) $10 if it comes heads "the previous flip" We have just flipped a coin. After the flip, I give you the choice between a) $1 for sure and b) $10 if it came heads Do you think that these scenarios are equivalent - i.e. is your choice a) or b) the same?
    - inimino 4y ago
      If you know the result of the flip, then no. Otherwise, yes (b for both).
      - kgwgk 4y ago
        The coin flip is different from the boxing match in that sense. Your ignorance regarding the former is the same as everyone else’s - while for the latter others may be better informed.
        jbay808 4y ago
        For sure! I also wouldn't bet on a coin flip offered by a guy standing next to a 500-kg servo-driven coin-tossing machine. But their confidence in offering me that bet is also useful information to update my probabilities. (This is also why poker is not purely a game of chance.) Probabilities are subjective, so we can always account for other people having more information and different probabilities than we do ourselves. Unfortunately the language of probability does not communicate this very explicitly, so it's easy to forget. With, say, radioactive decay, yes there's no chance anyone has less ignorance than we do, so we can at least be certain the result will be fair. (I don't think we are in any disagreement about this, though).
UncleMeat 4y ago
The paper mentions this. If you don't like the coin flip example you can use radioactive decay, which does not appear to depend on initial conditions and cannot be predictively modeled by physical measurement.

nurettin 4y ago

They are just saying that you can't start adjusting without any priors and personally, I don't think .5 means "I don't know". However from a practical standpoint if you are saying that the method converges to the same distribution at the same rate in any case no matter where the prior is, that would make sense.

inimino 4y ago
If the two outcomes are both possible and that's all you know, that's exactly what .5 means. I think the confusion comes from that, if the coin flip is repeated, we know (if the coin is fair) it will approach N/2, and in the other case, we don't know any such thing.

inimino 4y ago

Indeed. Obviously Gelman is accomplished in the field, so it's a bit surprising. But it's easy to get confused on these issues, for anyone apparently. There is a part towards the end of the paper where he suggests that the coin flip is really a "point distribution" at .5, while the outcome of the fight is something else. I think the intuition that he was going for was that if the fight was repeated N times, the outcome likely would not be N/2 wins for each. This suggests that there is "something" about the problem that could probably be known, and that is different from 1/2. But obviously that "something" is not our prior in the scenario given before the fight occurs. Determining the prior should be as simple as: Are both outcomes possible? Yes. Do you have any reason to expect one over the other? No. Well, then you have your prior. I think this "something" is where the idea of taking a uniform distribution on [0,1] comes from -- the idea that there is some "true" probability of the outcome of the fight, but we just don't know what it is, rather than realizing that no such thing exists. Really it seems like re-reading Jaynes should just clear the whole thing up.

jbay808 4y ago
I totally agree! There is definitely a difference in that repeated boxing matches are not independent probabilities. But honestly, if I put on my precision engineering hat, the fact that repeated coin flips are independently 1/2 just tells me that they're being repeated badly. With enough improvements to the test setup, we can get the variability out and they'll all be heads.

Strilanc 4y ago

This reminds me of a simpler situation [1]. I give you a coin, and promise it is biased, but don't tell you the bias. I flip the coin and ask you what is the probability that it landed heads. - A bayesian says the probability of the coin landing heads is 50/50. - A frequentist says the probability of the coin landing heads is anything except 50/50, because the coin was promised to be biased; if it was 50/50 then the coin would be fair not biased. They are of course using the word "probability" slightly differently. The bayesian is talking about an aggregation of the available hypotheses that is useful for betting and predictions, while the frequentist is talking about the individual possible hypotheses going into that aggregation. [1]: https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probability-is-in-the-mind https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probabilit...

kgwgk 4y ago
> I flip the coin and ask you what is the probability that it landed heads. > - A frequentist says the probability of the coin landing heads […] A frequentist says the concept of “the probability that it landed heads” makes no sense whatsoever.

kgwgk 4y ago

> The coin flip is just as much a problem of ignorance as the boxing match. Would you say that “the next coin flip” is _just_ as much a problem of ignorance as “the previous coin flip”? You may assign the same probability 1/2 to “getting heads is the next flip” and to “having got heads in the last flip” but these cases are not completely equivalent and there are practical differences.

inimino 4y ago
I might be missing the distinction you're making here. If you mean if the flip has already occurred, but the result was concealed (for example, the coin is under my hand but I have not looked at it), since my state of knowledge is the same as before the flip, I'd say it's equivalent.
- kgwgk 4y ago
  "the next flip" We're going to flip a coin now. Before the flip, I give you the choice between a) $1 for sure and b) $10 if it comes heads "the previous flip" We have just flipped a coin. After the flip, I give you the choice between a) $1 for sure and b) $10 if it came heads Do you think that these scenarios are equivalent - i.e. is your choice a) or b) the same?
  - inimino 4y ago
    If you know the result of the flip, then no. Otherwise, yes (b for both).
    - kgwgk 4y ago
      The coin flip is different from the boxing match in that sense. Your ignorance regarding the former is the same as everyone else’s - while for the latter others may be better informed.
      - jbay808 4y ago
        For sure! I also wouldn't bet on a coin flip offered by a guy standing next to a 500-kg servo-driven coin-tossing machine. But their confidence in offering me that bet is also useful information to update my probabilities. (This is also why poker is not purely a game of chance.) Probabilities are subjective, so we can always account for other people having more information and different probabilities than we do ourselves. Unfortunately the language of probability does not communicate this very explicitly, so it's easy to forget. With, say, radioactive decay, yes there's no chance anyone has less ignorance than we do, so we can at least be certain the result will be fair. (I don't think we are in any disagreement about this, though).

UncleMeat 4y ago

The paper mentions this. If you don't like the coin flip example you can use radioactive decay, which does not appear to depend on initial conditions and cannot be predictively modeled by physical measurement.