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TobyBartels 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: TobyBartels 28 July 2010 04:13:51AM 4 points [-]

My intution as a mathematician declares that nobody will never develop an elegant mathematical formulation of probability theory that does not allow for statements that are logically impossible or certain, such as statements of the form p AND NOT p. And it is necessary, if the theory is to be isomorphic to the usual one, that these statements have probability 0 (if impossible) or 1 (if certain). However, I believe that it is quite reasonable to declare, as a condition demanded of any prior deemed rational, that only truly impossible or certain statements have those probabilities. I think that this gives you what you want.

It's obvious that you can make this very demand when working with discrete probability distributions. It may not be obvious that you can make this demand when working with continuous probability distributions. Certainly the usual theory of these, based on so-called ‘measure spaces’ and ‘σ-algebras’ (I mention those in case they jog the reader's memory), cannot tolerate this requirement, at least not if anything at all similar to the usual examples of continuous distributions are allowed.

One answer is that only discrete probability distributions apply to the real world, in which one can never make measurements with infinite precision or observe an infinite sequence of events. Even if the world has infinite size or is continuous to infinitesimal scales, you will never observe that, so you don't need to predict anything about that.

However, even if you don't buy this argument, never fear! There is a mathematical theory of probability based on ‘pointless measure spaces’ and ‘abstract σ-algebras’. In this theory, it again makes perfect sense to demand that any prior must assign probability 0 or 1 only to impossible or certain events. The idea is that if something can never be observed, even in principle, then it is effectively impossible, and the abstract pointless theory allows one to treat it as such.

Then I agree that one should require, as a condition on considering a prior to be rational, that it should assign probability 0 only to these impossible events and assign probability 1 only to their certain complements.

Comment author: TobyBartels 28 July 2010 04:47:40AM *  0 points [-]

PS: cumulant-nimbus above gives a brief summary of the usual approach to measure theory. The pointless approach that I advocate can be suggested from that as follows: taboo \Omega. Neel Krishnamurti's comment is implicitly using the pointless approach; his event space is cumulant-nimbus's \mathcal{F}, and he works entirely in terms of events.