On Jun 13, 9:25 am, Russell Standish <[EMAIL PROTECTED]> wrote:
> I'm not sure his application of Bayes is correct. Given the facts of
> his hypothetical scenario, and writing e=10^{-4050}
>
> p(1|A) = e
> p(2|A) = 1-e
> p(1|B) = 1-e
> p(2|B) = e
>
> This is my translation of:
>
> "Now suppose that (somehow) we're able to extract the following (somewhat
> fanciful) predictions: theory A implies that in the entire history of
> the universe, there will be 10^50 observers* of class 1 and 10^5000 observers
> of class 2, while theory B implies that in the entire history of
>the universe, there will be 10^5000 observers of class 1 and 10^50 observers
>of class 2."
Hi Russell
The p(2|A) you give above is the probability for selecting one
observer at random from the totality of all observers throughout the
history of the universe, and finding that he/she/it belongs to class 2
(given theory A). But no such selection process has taken place.
Given that humans are class 2 observers, all we have is the fact H:
H := "The number of class 2 observers in the history of the
universe is at least of the order 10^10."
(We could argue that this ought to be somewhat higher than 10^10,
depending on how we classify our ancestors, but the point is that any
reasonable number we pick will be less than 10^50. And of course this
whole scenario is just a toy model for the sake of having a concrete
example to discuss.)
We then have:
P(H|A) = P(H|B) = 1
P(A) = P(B) = 1/2
P(H) = P(A) P(H|A) + P(B) P(H|B) = 1
P(A|H) = P(H|A) P(A) / P(H) = 1/2
P(B|H) = P(H|B) P(B) / P(H) = 1/2
In other words, the data we have, expressed in the observation H, does
nothing to discriminate between theory A and theory B, and leaves the
initial prior probabilities unchanged.
We, in the here and now, have no access to any process that randomly
samples the set of all observers in the history of the universe. Of
course it's possible to construct various sums over the set of *all*
observers and seek to maximise some kind of global average, and to ask
questions such as "What strategy, if adopted uniformly by every single
observer in the history of the universe, would maximise the
expectation value for the number of observers in the history of the
universe who correctly guessed whether A or B was the true description
of the universe." But whether or not there are any plausible
scenarios in which maximising that number could be a desirable
goal ... the fact remains that if we're discussing the *information*
available to *us* -- the human population of Earth at the present
moment -- we do not have access to the probabilities p(1|A), p(2|A),
p(1|B), p(2|B) that you describe.
The context in which I was discussing this at the N-Category Café is
the claim by some cosmologists that we ought to favour A-type
cosmological theories in which class 2 observers like us, with a clear
Darwinian history, will not be outnumbered (over the whole history of
the universe) by class 1 observers (Boltzmann brains). My contention
is that we have no empirical data at the present time that tells us
anything at all about the relative frequencies (over the whole history
of the universe) of class 1 and class 2 observers, and that our own
existence should not be mistaken for the outcome of a random sampling
of that whole-of-spacetime population. These issues are discussed in
more detail in:
"Are We Typical?" by James Hartle and Mark Srednicki,
http://arxiv.org/abs/0704.2630
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