3 ms·

So what would you use as a reasonable prior here?

by spicyj 13y ago

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.

So what would you use as a reasonable prior here?

by spicyj 13y ago

So what would you use as a reasonable prior here?

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.