Hi Greg, and welcome to the list. Your ears must be burning - you have
often been talked about here, always in a good light!

On Fri, Jun 13, 2008 at 09:28:07PM -0700, Greg Egan wrote:
> 
> 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.

There may be no physical process doing the sampling like pulling balls
from an urn, but it is nevertheless a sampling. My attributes (eg
height, weight and so on) are all drawn from distributions of such
attributes. Why not some hypothetical property like "observer class"
as set up in this toy problem?

Of course, in reality, there may be no well defined meaning to terms
like p(2|A), particularly if, as I suspect, observer moments satisfy a
complex valued measure. However, in this toy problem you presented,
the terms are well defined.

> 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 also have the fact that I am of class 2.

> (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.
> 

H does not discriminate, but 2 (I am of class 2) does. And all the
result does is give a preference to theory A rather than B, assuming
no prior preference (eg Occams razor). 

> 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.
> 

That is largely what we do with applications of Occams razor. We
choose the simpler theory on the basis that it is more likely to
continue being right when tested with future observations.

> 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

I would agree that this subject is marred by too many
unknowns. However on the issue of the likelihood of having found
ourselves being a Bolztmann brain, it is no different IMHO to that of
finding oneself in a "White Rabbit" universe (or "Harry Potter"
universe - as discussed in some other forums). Whilst there are an
overwhelming number of white rabbits, most of these would be
indistinguishable from ordinary reality by the observer
concerned. Only a very tiny number of experiences are "magical". To
quantify how tiny is tiny, consider a magical process that introduces
a million bits of information into the world (this is about the
complexity of a simple bacterium). The chance of experiencing this is
2^{-1e6}, a truly minuscule number, and equivalent to the proverbial
tornado assembling a jumbo jet from a scrapyard.

Cheers
-- 

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A/Prof Russell Standish                  Phone 0425 253119 (mobile)
Mathematics                              
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au
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