I’m not aware of any problems for “0 and 1 are not probabilities” if
Only applied to statements with short descriptions (otherwise they can’t be “picked out in advance” and it seems fine for e.g. an ML random infinite sequence of coin flips to have 0 probability), and of course the empty set gets 0 probability etc.
Your decision theory requires bounded utility functions, which tends to be an analytically useful assumption.
Here are examples of short statements with probabilities respectively 0 and 1: “0 = 1” and “1 = 1″. Or a little less trivially, the uniform distribution on the unit interval assigns 0 probability to each value in that interval and 1 to the whole interval. There is a class of theorems that the answers to certain probability questions about stochastic processes can only be 0 or 1.
I don’t think there is a good justification for bounded utility functions, any more than for supposing the real numbers to be bounded. It’s as artificial as supposing 0 and 1 to actually be not probabilities, and it does not solve any problems anyway.
yes, I know that the uniform distribution assigns probability zero to every point, that doesn’t mean I ever have to, which is why I specified ML random reals in the unit interval (Equivalently binary sequences). im well aware that many events have probability zero under some particular stochastic process (including the 0-1 laws), but in order to do that calculation you have to arrive at that stochastic process as a model and how will you assign it probability 1 in the first place?
The statement “0=1” is equivalent to the empty set, so also already covered in what I said.
In fact, assuming that the utility function is bounded solves many problems, such as pascal’s muggings and the saint Petersburg paradox. For history based RL, assuming continuity can also be useful, implying both boundedness and the existence of optimal policies.
I’m not aware of any problems for “0 and 1 are not probabilities” if
Only applied to statements with short descriptions (otherwise they can’t be “picked out in advance” and it seems fine for e.g. an ML random infinite sequence of coin flips to have 0 probability), and of course the empty set gets 0 probability etc.
Your decision theory requires bounded utility functions, which tends to be an analytically useful assumption.
Here are examples of short statements with probabilities respectively 0 and 1: “0 = 1” and “1 = 1″. Or a little less trivially, the uniform distribution on the unit interval assigns 0 probability to each value in that interval and 1 to the whole interval. There is a class of theorems that the answers to certain probability questions about stochastic processes can only be 0 or 1.
I don’t think there is a good justification for bounded utility functions, any more than for supposing the real numbers to be bounded. It’s as artificial as supposing 0 and 1 to actually be not probabilities, and it does not solve any problems anyway.
yes, I know that the uniform distribution assigns probability zero to every point, that doesn’t mean I ever have to, which is why I specified ML random reals in the unit interval (Equivalently binary sequences). im well aware that many events have probability zero under some particular stochastic process (including the 0-1 laws), but in order to do that calculation you have to arrive at that stochastic process as a model and how will you assign it probability 1 in the first place?
The statement “0=1” is equivalent to the empty set, so also already covered in what I said.
In fact, assuming that the utility function is bounded solves many problems, such as pascal’s muggings and the saint Petersburg paradox. For history based RL, assuming continuity can also be useful, implying both boundedness and the existence of optimal policies.