Right. There’s also a somewhat stronger desideratum that we want to expect sequences to be simple rather than complex.

But I think there is something lots of logical uncertainty schemes are missing, which is estimation of numerical parameters. We should be able to care whether the target region is of size 0.001 or 0.000000000000000000000000000001, even if we have no positive examples, but sequence-prediction approaches don’t do that.

If we’re willing to “cheat” a bit and use as an input to our logical uncertainty method the class of objects that we’re drawing from and comparing to some numerical parameter, then we can just treat prior examples as being drawn from the distribution we’re trying to learn. And this captures our intuition very well, but it has some trouble fitting into schemes for logical uncertainty because of the requirement for cheating.

Right. There’s also a somewhat stronger desideratum that we want to expect sequences to be simple rather than complex.

But I think there is something lots of logical uncertainty schemes are missing, which is estimation of numerical parameters. We should be able to care whether the target region is of size 0.001 or 0.000000000000000000000000000001, even if we have no positive examples, but sequence-prediction approaches don’t do that.

If we’re willing to “cheat” a bit and use as an input to our logical uncertainty method the class of objects that we’re drawing from and comparing to some numerical parameter, then we can just treat prior examples as being drawn from the distribution we’re trying to learn. And this captures our intuition very well, but it has some trouble fitting into schemes for logical uncertainty because of the requirement for cheating.