Sure. Any practical implementation will have to figure out all the practical details including the ones that you mention. But that’s implementation issues of something that is still straightforward Bayes, at least for a single individual. If you have a history of predictions and know the actual outcomes, you can even just plot the empirical calibration curve without any estimation involved.
Now, if you have multiple people involved, things become more interesting and probably call for something like Gelman’s favourite multilevel/hierarchical models. But that’s beyond what OP asked for—he wanted a “rigorously mathematically defined system” and that’s plain-vanilla Bayes.
Sure. Any practical implementation will have to figure out all the practical details including the ones that you mention. But that’s implementation issues of something that is still straightforward Bayes, at least for a single individual. If you have a history of predictions and know the actual outcomes, you can even just plot the empirical calibration curve without any estimation involved.
Now, if you have multiple people involved, things become more interesting and probably call for something like Gelman’s favourite multilevel/hierarchical models. But that’s beyond what OP asked for—he wanted a “rigorously mathematically defined system” and that’s plain-vanilla Bayes.