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Comments (44)
All recursive probability spaces converge to the same probabilities, as the information increases.
Not that those people making up probabilities knows anything about that.
If you want an universal probability space, just take some universal computer, run all programs on it, and keep those that output event A. Then you can see how many of those that output event B, and thus you can get p(B|A) whatever A and B are.
This is algorithmic information theory, and should be known by any black belt bayesian.
Kim Øyhus
Google gives 0 hits on "recursive probability space". Blanket assertions like this need to be technically precise.
I refer interested readers to the Algorithmic probability article on Scholarpedia.
The technically precise reference was this part:
"This is algorithmic information theory,.."
But if you claim my first line was too obfuscated, I can agree.
Kim Øyhus
Please specify in what sense the first line was correct, or declare it an error. Pronouncing assertions known to be incorrect and then just shrugging that off shouldn't be acceptable on this forum.
O.K.
One wants an universal probability space where one can find the probability of any event. This is possible:
One way of making such a space is to take all recursive functions of some universal computer, run them, and storing the output, resulting in an universal probability space because every possible set of events will be there, as the results of infinitely many recursive functions, or programs as they are called. The probabilities corresponds to the density of these outputs, these events.
A counterargument is that it is too dependent on the actual universal computer chosen. However, theorems in algorithmic information theory shows that this dependence converges asymptotically as information increases, because the difference of densities of different outputs from different universal computers can at most be 2 to the power of the shortest program simulating the universal computer in another universal computer.
Kim Øyhus
OK....
what!? You haven't yet described a probability space. The aforementioned set is infinite, so the uniform distribution is unavailable. What probability distribution will you have on this set of recursive-function-runs. And in what way is the resulting probability space universal?
As far as I can tell, you are talking absolute gibberish.
If I'm wrong, please explain.
edit: if someone who downvoted me could please explain what the heck a "recursive probability space" is supposed to be, I'd appreciate it.