I don’t understand you. Neither “a 51% percent coin” nor “a fair coin” are probability distributions, and the choice of metric in question is “metric on spaces of probability distributions”. Could you clarify?
Although, I could take your statement at face value, too. Want to make a few million $1 bets with me? We’ll either be using “rand < .5” or “rand < .51″ to decide when I win; since trying to distinguish between the two is useless you don’t need to bother.
You could call them Bernoulli distributions representing aleatory uncertainty on a single coin flip, I suppose. Bayesian updates of purely aleatory uncertainty aren’t very interesting, though, are they? Your evidence is “I looked at it, it’s heads”, and your posterior is “It was heads that time”.
I suppose you could add some uncertainty to the evidence; maybe we’re looking at a coin flip through a blurry telescope? But in any context, Bernoulli distributions from a finite-dimensional probability distribution space mean that Bayesian updates on them are still well-posed. The concern here is that infinite-dimensional spaces of probability distributions don’t always lead to well-posed Bayesian updates, depending on what metric you use to define well-posed. If there’s also a concern that this can happen on Bernoulli distributions then I’d like to see an example; if not then that’s a red herring.
Also, once you are not limited to a single flip and can flip the coins multiple times, you graduate to binomial distributions which are highly useful and for which Bayesian updates are sufficiently interesting :-)
Any metric whereby a 51% percent coin isn’t close to a fair coin is useless in practice.
I don’t understand you. Neither “a 51% percent coin” nor “a fair coin” are probability distributions, and the choice of metric in question is “metric on spaces of probability distributions”. Could you clarify?
Although, I could take your statement at face value, too. Want to make a few million $1 bets with me? We’ll either be using “rand < .5” or “rand < .51″ to decide when I win; since trying to distinguish between the two is useless you don’t need to bother.
Of course they are, they represent Bernoulli distributions.
You could call them Bernoulli distributions representing aleatory uncertainty on a single coin flip, I suppose. Bayesian updates of purely aleatory uncertainty aren’t very interesting, though, are they? Your evidence is “I looked at it, it’s heads”, and your posterior is “It was heads that time”.
I suppose you could add some uncertainty to the evidence; maybe we’re looking at a coin flip through a blurry telescope? But in any context, Bernoulli distributions from a finite-dimensional probability distribution space mean that Bayesian updates on them are still well-posed. The concern here is that infinite-dimensional spaces of probability distributions don’t always lead to well-posed Bayesian updates, depending on what metric you use to define well-posed. If there’s also a concern that this can happen on Bernoulli distributions then I’d like to see an example; if not then that’s a red herring.
I also don’t understand the downvote. Is there a single sentence in the above post that’s mistaken? If so then a correction would be appreciated.
Also, once you are not limited to a single flip and can flip the coins multiple times, you graduate to binomial distributions which are highly useful and for which Bayesian updates are sufficiently interesting :-)