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If you were holding a physical coin in your hand, you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. This is sort o

by bjterry 13y ago

If you were holding a physical coin in your hand, you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. This is sort of a really minor pet peeve of mine when people use coin-flipping as an example for these things. If it's even vaguely coin-like, even the most ridiculous distortion (maybe it's made of uranium on one side and aluminum on the other) probably couldn't bring the true probability past 70% or something.

spicyj 13y ago
So what would you use as a reasonable prior here?
- dthunt 13y ago
  For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair' definition), you would concentrate most of your probability mass there, and find a mutually agreeable shape. The 'it's not shaped like a sphere' thing suggests that some people would take into account visual information about the object to inform their priors, and be prone to rejecting the fitness of the uniform. That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends? Edit: And for formal mathy purposes, ask someone else :)
- bermanoid 13y ago
  If it's a normal looking coin being flipped in the air and caught before presentation, the only reasonable prior is that almost all of the probability mass should be that it's completely fair. Maybe I'd leave 1% to spread amongst the rest just in case the adversary has some seriously devious tricks up their sleeve. Because physics - it's not possible to bias a rigid body so that it rotates with non-constant angular speed when flipped, as long as air resistance can be neglected, and that means that a fair flip gives 50/50 odds as to what side you catch it on. (Edit: clarity) If other stuff is going on, like you're letting it bounce or something, then it depends on the particulars. It's rather easy to load a die, for instance.
- darkxanthos 13y ago
  One of the beautiful aspects of Bayesian methods is that they are subjective. Due to that I think you should really be asking "What sort of priors are reasonable here?" From that perspective the uniform prior I use is reasonable (though non-committal and fairly uninformative), a triangular distribution with its peak at 50% (http://en.wikipedia.org/wiki/Triangular_distribution http://en.wikipedia.org/wiki/Triangular_distribution), the prior I have after updating tails and then heads, a normal distribution, or using the beta distribution with an equal number of both parameters. Honestly you may decide there's a fair chance the coin is heads biased so you choose a prior that has most of the mass of the probability above 50%. As long as you have a reason, it's reasonable. You don't have to worry too much that you've chosen the optimal prior since more data makes equals of priors in the long run. The only terrible thing you can do, is pick a prior that absolutely excludes certain hypotheses with 0% likelihood. No amount of data can overcome that.
khafra 13y ago
> you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. The distribution for a coin is uniform over the event space [heads,tails]; not over the angle of rotation it lands on. You can have continuous or discrete event spaces.
- bjterry 13y ago
  Sure. My point was that you can't physically construct a flat cylinder a couple centimeters across that you can flip fairly and achieve a 98% bias. And if you don't meet those physical requirements, it's not a coin. If you had a sphere shaped object and the head condition was a particular point being above the equator, maybe you could include some distribution of internal weighting that could achieve that level of bias. But really the sphere thing was just tongue-in-cheek. If you have something shaped like a gömböc[1] you can have a "coin" that lands on heads ~100% of the time, except for the whole not-actually-being-a-coin part. 1: https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c

by bjterry 13y ago

spicyj 13y ago

So what would you use as a reasonable prior here?

dthunt 13y ago
For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair' definition), you would concentrate most of your probability mass there, and find a mutually agreeable shape. The 'it's not shaped like a sphere' thing suggests that some people would take into account visual information about the object to inform their priors, and be prone to rejecting the fitness of the uniform. That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends? Edit: And for formal mathy purposes, ask someone else :)
bermanoid 13y ago
If it's a normal looking coin being flipped in the air and caught before presentation, the only reasonable prior is that almost all of the probability mass should be that it's completely fair. Maybe I'd leave 1% to spread amongst the rest just in case the adversary has some seriously devious tricks up their sleeve. Because physics - it's not possible to bias a rigid body so that it rotates with non-constant angular speed when flipped, as long as air resistance can be neglected, and that means that a fair flip gives 50/50 odds as to what side you catch it on. (Edit: clarity) If other stuff is going on, like you're letting it bounce or something, then it depends on the particulars. It's rather easy to load a die, for instance.
darkxanthos 13y ago
One of the beautiful aspects of Bayesian methods is that they are subjective. Due to that I think you should really be asking "What sort of priors are reasonable here?" From that perspective the uniform prior I use is reasonable (though non-committal and fairly uninformative), a triangular distribution with its peak at 50% (http://en.wikipedia.org/wiki/Triangular_distribution http://en.wikipedia.org/wiki/Triangular_distribution), the prior I have after updating tails and then heads, a normal distribution, or using the beta distribution with an equal number of both parameters. Honestly you may decide there's a fair chance the coin is heads biased so you choose a prior that has most of the mass of the probability above 50%. As long as you have a reason, it's reasonable. You don't have to worry too much that you've chosen the optimal prior since more data makes equals of priors in the long run. The only terrible thing you can do, is pick a prior that absolutely excludes certain hypotheses with 0% likelihood. No amount of data can overcome that.

khafra 13y ago

> you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. The distribution for a coin is uniform over the event space [heads,tails]; not over the angle of rotation it lands on. You can have continuous or discrete event spaces.

bjterry 13y ago
Sure. My point was that you can't physically construct a flat cylinder a couple centimeters across that you can flip fairly and achieve a 98% bias. And if you don't meet those physical requirements, it's not a coin. If you had a sphere shaped object and the head condition was a particular point being above the equator, maybe you could include some distribution of internal weighting that could achieve that level of bias. But really the sphere thing was just tongue-in-cheek. If you have something shaped like a gömböc[1] you can have a "coin" that lands on heads ~100% of the time, except for the whole not-actually-being-a-coin part. 1: https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c