A Paradox of Robust Bayesian Inference and Belief Functions [pdf] (2005)
stat.columbia.edu
stat.columbia.edu
The distinction between the two is that in the context of ignorance your belief state is very unstable and susceptible to change with the injection of new information. On the other hand, in the context of a "random" event like a coin flip, your belief state is very stable and unsusceptible to change with the injection of new information.
Whereas you may initially apply the same probability to both a coin flip and the outcome of a wrestling match, those probabilities will evolve in very different ways as you acquire new information such as inspecting the coin/combatants. The probability of the outcome of the coin flip is unlikely to change at all, whereas the probability of the outcome of the wrestling match could change substantially.
This can be modeled by assigning the probability of an outcome to a continuous rate parameter, and assigning a probability distribution to that rate parameter. The entropy of that probability distribution can describe how ignorant your belief state is, and hence how susceptible it is to change.
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.
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.
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.
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.
- 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/probabilit...
> - 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.
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.
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?
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).