Even if two models eventually come to the exact same conclusion about the outcome distribution from rolls of a particular die, they can still make different predictions about the physical properties of the die itself.
That’s surprising. I thought the “biased coins model” would come to agree with the “unbiased coins model” by converging (close) to the same (uniform) distribution.
They do converge to the same distribution. But they make different predictions about a physical die: the unbiased model predicts uniform outcomes because the die is physically symmetric, whereas the general biased model doesn’t say anything about the geometry of a physical die. So if I see uniform outcomes from a physical die, then that’s Bayesian evidence that the die is physically symmetric.
That’s surprising. I thought the “biased coins model” would come to agree with the “unbiased coins model” by converging (close) to the same (uniform) distribution.
They do converge to the same distribution. But they make different predictions about a physical die: the unbiased model predicts uniform outcomes because the die is physically symmetric, whereas the general biased model doesn’t say anything about the geometry of a physical die. So if I see uniform outcomes from a physical die, then that’s Bayesian evidence that the die is physically symmetric.
See Wolf’s Dice II for more examples along these lines.