Back in June, Nick Bostrom <N.Bostrom.domain.name.hidden> wrote
> Hi, I've just submitted my doctoral dissertation titled
> "Observational selection effects and probability", which presents
> the first mathematically explicit "observation theory". I use it to
> resolve a range of puzzles related to the "anthropic principles",
> multiverse models, the Doomsday argument, etc. The text is
> available at
> http://www.anthropic-principle.com/phd
> as a MS-Word-file (77,000 words; 1.2 KB).
> Feel free to download it if you think it might be interesting to you.
> Any constructive comments, criticism or suggestions would be
> welcome.
It took a while, but I finally managed to get round to reading it and
I'm very glad I did. It is a clear and thoughtful exploration of an
interesting topic.
At the heart of the thesis is the Self-Sampling Assumption (SSA)
> Every observer should reason as if they were a random sample
> drawn from the set of all observers
and its various generalizations. Having motivated this assumption
by its usefulness in justifying multiverse theories in cosmology,
Bostrom is mainly concerned with working out its consequences and
in particular with whether or not some of those consequences are
paradoxical.
Bostrom leaves two crucial questions open:
1) The reference class problem -- what counts as an observer?
2) Why should the self-sampling assumption be true?
For many years now, I have been developing a detailed version of
the many-minds interpretation of quantum theory. In the papers on
my web site
http://www.poco.phy.cam.ac.uk/~mjd1014
I attempt to give a mathematically-precise definition of
an observer. This provides a putative solution to question 1.
More interestingly, my definition also suggests a solution to
question 2:
The self-sampling assumption is true because all observers
are different possible developments of a single null observer.
At first sight this looks a bit crazy. It is based however on the
simple principle that any observer (any mind) is defined by his
history (by his mental life).
Quantum theory in a many-minds formulation suggests that all
possible observer histories will exist within the universal quantum
state. My explicit definition of an observer sees such a history as
a finite path of a Markov process on a discrete state space. The set
of these paths is a tree with a unique root (the observer at the
beginning of his history).
When we consider an application of the self-sampling assumption
we should look at the likelihood of different observed paths.
Fundamentally, this is exactly what we do whenever we consider
any empirical proposition.
The correct (objective) a priori probability of our observations is
defined by whatever is the true physical theory. I assume that
some version of quantum field theory is true, and much of my work
has involved exploring the mathematical definition of probability in
the context of localized observers in a theory compatible with
special relativity. This allows different observers to have
different probabilities dependent on their observations.
As a first approximation to the correct probabilities, however, it is
not reasonable to guess that we are about as likely as most
observers like us. This is the self-sampling assumption.
One of the interesting problems that Bostrom raises for the
self-sampling assumption is what an observer with an obviously
expectional birth rank should believe. In particular, to what extent
is the doomsday argument justified, and what should Adam and
Eve have believed?
My response is that most of our empirical testing of the objective
probabilities which define our lives does not directly relate to our
birth rank. If Eve has not studied quantum theory then it is hardly
suprising if her ignorance leads her astray. However, if she has
learnt quantum theory, then she knows that, like all humans, her a
priori probability within the universal quantum state is very very
small. Indeed, it is so tiny that factoring in the information that
Eve gained when she first learned the improbability of her birth
rank has negligible effect on the logarithm of her total a priori
probability. Moreover, her net a priori probability changes rapidly
with time. When she first sees the stars in the sky, for example,
she fixes them in her mind; collapsing one particular possible
stellar pattern out of all the possibilities that might have come to
her from the big bang.
Nevertheless, Eve still must expect to be typical as an observer.
Otherwise, she has no reason to believe her physical theories. Her
birth rank puts her in the position of someone who has tossed a fair
coin 1,000 times and it has always come up heads. But she is
tossing a million other coins at the same time so it isn't really that
surprising. And if she tosses any of her coins again, she would be
better advised to predict the results using the physics of the coins
rather than her past experience with them.
Matthew Donald (matthew.donald.domain.name.hidden)
***************************************************
Received on Tue Sep 19 2000 - 08:18:55 PDT