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Wes_W 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: Wes_W 20 August 2015 05:02:33PM 0 points [-]

If we're asking what the author "really meant" rather than just what would be correct, it's on record.

The argument for why zero and one are not probabilities is not, "All objects which are special cases should be cast out of mathematics, so get rid of the real zero because it requires a special case in the field axioms", it is, "ceteris paribus, can we do this without the special case?" and a bit of further intuition about how 0 and 1 are the equivalents of infinite probabilities, where doing our calculations without infinities when possible is ceteris paribus regarded as a good idea by certain sorts of mathematicians. E.T. Jaynes in "Probability Theory: The Logic of Science" shows how many probability-theoretic errors are committed by people who assume limits directly into their calculations, without first showing the finite calculation and then finally taking its limit. It is not unreasonable to wonder when we might get into trouble by using infinite odds ratios. Furthermore, real human beings do seem to often do very badly on account of claiming to be infinitely certain of things so it may be pragmatically important to be wary of them.

I... can't really recommend reading the entire thread at the link, it's kind of flame-war-y and not very illuminating.

Comment author: EHeller 20 August 2015 05:14:30PM *  3 points [-]

I think the issue at hand is that 0 and 1 aren't special cases at all, but very important for the math of probability theory to work (try and construct a probability measure where some subset doesn't have probability 1 or 0).

This is incredibly necessary for the mathematical idea of probability ,and EY seems to be confusing "are 0 and 1 probabilities relevant to Bayesian agents?" with "are 0 and 1 probabilities?" (yes, they are, unavoidably, not as a special case!).

Comment author: Lumifer 20 August 2015 05:18:06PM *  0 points [-]

It seems that EY position boils down to

Pragmatically speaking, the real question for people who are not AI programmers is whether it makes sense for human beings to go around declaring that they are infinitely certain of things. I think the answer is that it is far mentally healthier to go around thinking of things as having 'tiny probabilities much larger than one over googolplex' than to think of them being 'impossible'.

And that's a weak claim. EY's ideas of what is "mentally healthier" are, basically, his personal preferences. I, for example, don't find any mental health benefits in thinking about one over googolplex probabilities.

Comment author: Wes_W 20 August 2015 05:27:16PM *  0 points [-]

Cromwell's Rule is not EY's invention, and relatively uncontroversial for empirical propositions (as opposed to tautologies or the like).

If you don't accept treating probabilities as beliefs and vice versa, then this whole conversation is just a really long and unnecessarily circuitous way to say "remember that you can be wrong about stuff".

Comment author: EHeller 20 August 2015 05:44:34PM 2 points [-]

The part that is new compared to Cromwell's rule is that Yudkowsky doesn't want to give probability 1 to logical statements (53 is a prime number).

Because he doesn't want to treat 1 as a probability, you can't expect complete sets of events to have total probability 1, despite them being tautologies. Because he doesn't want probability 0, how do you handle the empty set? How do you assign probabilities to statements like "A and B" where A and B are logical exclusive? (the coin lands heads AND the coin lands tails).

Removing 0 and 1 from the math of probability breaks most of the standard manipulations. Again, it's best to just say "be careful with 0 and 1 when working with odds ratios."

Comment author: Lumifer 20 August 2015 05:48:30PM 1 point [-]

Nobody is saying EY invented Cromwell's Rule, that's not the issue.

The issue is that "0 and 1 are not useful subjective certainties for a Bayesian agent" is a very different statement than "0 and 1 are not probabilities at all".

Comment author: Wes_W 20 August 2015 06:05:37PM *  0 points [-]

You're right, I misread your sentence about "his personal preferences" as referring to the whole claim, rather than specifically the part about what's "mentally healthy". I don't think we disagree on the object level here.