# Perplexed comments on If a tree falls on Sleeping Beauty... - Less Wrong

64 12 November 2010 01:14AM

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Comment author: 12 November 2010 01:43:19AM *  4 points [-]

I tend to agree with you, but apparently E. T. Jaynes does not:

It has always seemed objectionable to some, including this writer, to base probability theory on such vulgar things as betting, expectation of profit, etc. We think that the principles of logic ought to be on a higher plane.

He then goes on to point out (rather illogically, I think) that if probability estimates actually do depend on how the wager is structured, then

a well posed question would have more than one "right" answer, with nothing to choose between them. This, in our view, is another aspect of the superiority of [the approach of PT:TLOS based on Cox's theorem] which stresses logical consistency instead and, just for that reason, is more likely to have a lasting place in probability theory.

Quotes from the final paragraph of Chapter 13 of PT:TLOS

Comment author: 12 November 2010 04:29:40PM *  6 points [-]

We have an apparently ambiguous word: "probability". That we accept it's ambiguous (the thesis of this post), means that there are situations where either of its multiple possibly contradictory meanings is intended. This doesn't negate a possibility of there being a concept that can be applied to most situations, doesn't itself suffer from ambiguity, and is usually referred to using the same word.

There is no apparent contradiction here: the disagreement you point out results from equivocating between the ambiguous word "probability" and (presumably) the more specific concept of probability that Jaynes refers to.

Comment author: 12 November 2010 03:06:52AM 1 point [-]

The sleeping beauty problem is not really a paradox of probability theory. The question is what Beauty's beliefs "should" be. As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed the probability is still 1/2. I think it is reasonable to say that probability theory should be based on a strict logical formalism independent of feedback, but that the solution to this paradox involves recognizing the role of feedback in determining what Beauty "should" believe.

Comment author: 12 November 2010 04:48:52AM *  9 points [-]

As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed the probability is still 1/2.

Here we see an illustration of OP's argument that stating things in terms of probability can be ambiguous. But this ambiguity could be cleared up by instead saying either,

"As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed at least once during the week, the probability is still 1/2."

or,

"As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads; but conditioned on the observation that today is either Monday or Tuesday, and Beauty is being interviewed, the probability is 2/3."

Or, a commentator could be equally ambiguous using the language of wagers by saying,

"Beauty will be asked whether the coin came up heads or tails, and given \$1 if she is correct."

Using the language of probability does not make one suddenly unable to speak clearly; and using the language of wagers doesn't guarantee precision.

Maybe the language of probability makes it easier to shoot yourself in the foot, but that doesn't mean there's something nonsensical about it.