[SEQ RERUN] 0 And 1 Are Not Probabilities

MinibearRex

Today's post, 0 And 1 Are Not Probabilities was originally published on 10 January 2008. A summary (taken from the LW wiki):

In the ordinary way of writing probabilities, 0 and 1 both seem like entirely reachable quantities. But when you transform probabilities into odds ratios, or log-odds, you realize that in order to get a proposition to probability 1 would require an infinite amount of evidence.

Discuss the post here (rather than in the comments to the original post).

This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Infinite Certainty, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.

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Today's post, 0 And 1 Are Not Probabilities was originally published on 10 January 2008. A summary (taken from the LW wiki):

In the ordinary way of writing probabilities, 0 and 1 both seem like entirely reachable quantities. But when you transform probabilities into odds ratios, or log-odds, you realize that in order to get a proposition to probability 1 would require an infinite amount of evidence.

I think this is kind of funny considering that the second axiom of probability states that an elementary event has probability one. It's just a simple way to define the system, like how the axioms of euclidean geometry are simpler if you have a point at infinity. It doesn't necessarily mean anything. I just find it kind of funny.

the probability that some elementary event in the entire sample space will occur is 1

I believe that a part of the post's point is that the entire sample space is hard to find in most real-life cases. From the post:

However, in the real world, when you roll a die, it doesn't literally have infinite certainty of coming up some number between 1 and 6. The die might land on its edge; or get struck by a meteor; or the Dark Lords of the Matrix might reach in and write "37" on one side.

EDIT: Another example, this time from the Martin Gardner's ex... (read more)

-3endoself14y

Jaynes didn't like Kolmogorov's axioms, and I expect Eliezer would agree. I remember he mentioned somewhere in the sequences that he thought probability could be axiomatized without reference to probabilities of 0 or 1, but it wouldn't have much practical use to do so.

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