I’m not convinced that we can have real probability distributions over impossible possible worlds. At the very least, a real probability distribution must sum its exhaustive and exclusive possibilities to 1, but in fact it seems to me that the same type of effort that is needed to show that a set of impossible possibilities sums to 1 also changes the degree to which they have been examined, changing their subjective probabilities. It specifically seems to me that pseudo-probability distribution over impossible possible worlds will generally contain non-correctable biases from framing such as overconfidently narrow probability distributions, or conversely conjunction fallacies and subadditivity.
Well, for a simpler example (with only finitely many cases), it can look a bit better. For example, take the Riemann Hypothesis— please. Mathematicians have informally been saying for years that based on the exact precision of the first trillion cases and on the rich consequences derived from it, that they think it’s almost certainly true, pending a proof.
Clearly, two of the three universes “The RH can be proved”, “The RH can be disproved”, “The RH is undecidable from ZFC” are logically impossible; we just don’t know which. Still, it doesn’t seem illegitimate to portion probability between these three cases.
This isn’t strictly Bayes, it seems, because the theoretical Bayesian reasoner contains all logical consequences; but I don’t think it’s thereby an invalid process to arrive at subjective probabilities.
Michael:
I’m not convinced that we can have real probability distributions over impossible possible worlds. At the very least, a real probability distribution must sum its exhaustive and exclusive possibilities to 1, but in fact it seems to me that the same type of effort that is needed to show that a set of impossible possibilities sums to 1 also changes the degree to which they have been examined, changing their subjective probabilities. It specifically seems to me that pseudo-probability distribution over impossible possible worlds will generally contain non-correctable biases from framing such as overconfidently narrow probability distributions, or conversely conjunction fallacies and subadditivity.
Well, for a simpler example (with only finitely many cases), it can look a bit better. For example, take the Riemann Hypothesis— please. Mathematicians have informally been saying for years that based on the exact precision of the first trillion cases and on the rich consequences derived from it, that they think it’s almost certainly true, pending a proof.
Clearly, two of the three universes “The RH can be proved”, “The RH can be disproved”, “The RH is undecidable from ZFC” are logically impossible; we just don’t know which. Still, it doesn’t seem illegitimate to portion probability between these three cases.
This isn’t strictly Bayes, it seems, because the theoretical Bayesian reasoner contains all logical consequences; but I don’t think it’s thereby an invalid process to arrive at subjective probabilities.