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Eliezer_Yudkowsky comments on 0 And 1 Are Not Probabilities - Less Wrong

34 Post author: Eliezer_Yudkowsky 10 January 2008 06:58AM

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Comment author: Eliezer_Yudkowsky 10 January 2008 07:16:49PM 8 points [-]

If there's a map you're trying to use all over the place (and you do seem to), then I claim it makes no sense to put a little region on the map labelled "maybe this map doesn't make any sense at all". If the map seems to make sense and you're still following it for everything, you'll have to ignore that region anyway.

Janos, are you saying that it is in fact impossible that your map in fact doesn't make any sense? Because I do, indeed, have a little section of my map labelled "maybe this map doesn't make any sense at all", and every now and then, I think about it a little, because there are so many fundamental premises of which I am unsure even in their definitions. (E.g: "the universe exists", and "but why?") Just because this area of my map drops out of my everyday decision theory due to failure to generate coherent advice on preferences, does not mean it is absent from my map. "You must ignore" or rather "You should usually ignore" is decision theory, and probability theory should usually be firewalled off from preferences.

Computable numbers are the largest countable class I know of.

Either all countable sets are the same size anyway, or you can generate a larger set by saying "all computable reals plus the halting probability". How about computable with various oracles?

What's false is the tempting statement that probability 0 events are impossible. It's only the converse that's true: impossible events have probability 0.

If you cannot repose probability 1 in the statement "all events to which I assign probability 0 are impossible" you should apply a correction and stop reposing probability 0 to those events. Do you mean to say that all impossible events have probability 0, plus some more possible events also have probability 0? This makes no sense, especially as a justification for using "probability 0" in a meaningfully calibrated sense.

To use "probability 0" without a finite expectation of being infinitely surprised, you must repose probability 1 in the belief that you use "probability 0" only for actually impossible events; but not necessarily believe that you assign probability 0 to every impossible event (satisfying both conditions implies logical omniscience).

I should mention that I'm also an infinite set atheist.