Regarding the interpretation of a passage involving statistical
inference in cosmology, reproduced at the end of this post

Steve Willner wrote:

The article has apparently been accepted by Phys. Rev., so at least
one referee thinks it's OK. :-)

Argument by authority. ;-)

As I understand it, they are suggesting that physical constants might
in principle vary throughout the Universe. However, observers will
preferentially measure values that obtain _within galaxies_, and
theoretical values at those locations -- not at "a random location in
space" -- are the ones to be compared with observations. This
appears to be a very weak version of the Anthropic Principle.

Hmm... I hadn't thought of that possibility. I took it that the
constants were varying over non-communicating universes. You may be
right, or it may come to the same thing: if the regions of given values
of physical constants are large enough, they may as well be separately
evolving universes.


Tito wrote:

Thanks for calling attention to this article. It is in my field of interest.
Yes it is very dense but I think I get most of it. The article wants to
narrow the field of what is probable in the universe by eliminating what is
impossible and all the interrelated things that are impossible, leaving a
smaller field of the probable.

The passage you quote was focusing on the two main components that can
reduce the # of probable outcomes of the 31 parameter probability
distribution f(p). Our 31 mysterious constants that defy explanation!

There are two main components. f (prior) of p and f(selection) of p.
f (prior) refers to the reduced number of possible setups for our universe,
prior to its observation. The content of this part of the paper focuses on
astrophysical selection effects which include the dark matter density
parameter, dark energy density, and seed fluctuation density, to ensure the
formation of dark matter halos, galaxies and stable solar systems.
f(selection) refers to the observer or selection effect that corresponds to
f (prior). Go to page 10 for this discussion.

I am not much enlightened by page 10.

The paper is very dense, and no doubt I am also, but perhaps somewhat
less dense about questions of statistical inference, which somehow we
should be able to disentangle from the astrophysics -- though I may
overestimate my ability and even "basic" probabilistic inference
contains elements maddening enough to spawn deathless story problems,
like the Monty Hall problem (well, maybe that's the only one, I'm not
sure).

The use of the "prior" might suggest they are performing some kind of
Bayesian inference, but I don't think this is the case. This would
require us to modify the prior into the "posterior" distribution by
conditioning on some event. Yet, if the "event" is to observe
particular values of the jointly distributed (31) variables, the
postiori is simply a constant (vector): we know we have observed just
those values. It may be that if I read carefully I would see we
_haven't_ observed these values, but must leave some significant range
of error, and these error bars define our event. This would make the
posterior distribution more interesting, but I don't think this is the
argument: the product from given is not the Bayesian update form.

My working hypothesis is that we are working some kind of hypothesis
test: Given the hypothesis that our universe was drawn from the
distribution f(p), how unlikely in the observed outcome? This
question involves some subtlety. Following Feynman, and his comment
about observing license plate numbers (We just saw Nevada 245-UVX! How
incredibly unlikely was that among all the millions of US plates!), we
might argue that each particular value of the parameters drawn from the
continuous distribution is in fact only infinitesimally likely, and
that we can't make an further inference from seeing just this one, and
not that one -- something must in fact be drawn.

However, if we group the outcome space into events, we feel we can
begin to plausibly say that such an outcome was either plausible or
implausible given the assumed -- ok, "prior", distribution. "We only
would be this far into the tail 0.01% of the time, so it seems unlikely
our prior assumption (hypothesis) was correct. Applying this idea to
our "observation" of our one universe, suppose that according to our
assumed prior distribution life would be possible only 0.01% of the
time. Given that we are alive, we might argue, it then seems very
improbable that our prior was correct.

Ok... I think this may be the argument must be what they are
addressing. The escape clauses they list support this assumption. For
example, it might be argued that even if life as we know it was
extremely improbable, _some_ form of introspective life was much more
probable, and this is the event partition of the space we should be
inferring from -- the detailed form of life is just a license plate
number, whereas the existence of something able to ask the question is
seeing a Pennsylvania plate in New York:

If on the other hand the existence of an intelligent observer is deemed
to be very improbable, and we have only one throw of the dice, then the
observation of quintuple box cars is indeed unlikely and a plausible
basis for inferring our assumption of fair dice is wrong. A different
escape clause -- and this is I think their model -- is to claim that
there is not just one role of the dice, but a large, even infinite
ensemble of dice rolls! In that case, even if the probability of an
observer arising in a given throw is 10^(-100), the probability of an
observer arising _somewhere_ is 1. In fact, the probability of an
infinite numbers of observers arising is 1; and in this case, sitting
back from our meta-god position, we see each of these infinite
observers questioning the miraculous improbability of their existence,
whereas we argue that only the universes with observers in them can ask
the question, and their existence and questioning is not therefore
improbable, but insured.

Ok... there is something Monty-Hall like about this all after all!
However, supposing I have in fact inferred their universe of inference,
I still don't see what they are trying to accomplish. If there is in
fact an ensemble of universes, and we accept the argument that the
existence of observers, however vanishingly improbable in individual
instance, is bound to happen so long as the probability of existence is
not zero, and that each individual observer is therefore not correctly
inferring the improbability (in some yet prior distribution of models)
of the model which produces him only improbably, then we are finished
inferring. How can we make any further progress? The only inference
we can make from our thinking and therefore am-ing is that, assuming a
prior meta-meta-verse of possible meta-verses, we are not in a
meta-verse which makes the probability of our existence in each (plain
old) universe zero! I therefore fail to find a role for "f_selection".

?

____

"Let us group the 31 parameters of Table 1 into a 31-
dimensional vector p. In a fundamental theory where
inflation populates a landscape of possibilities, some or
all of these parameters will vary from place to place as
described by a 31-dimensional probability distribution
f(p). Testing this theory observationally corresponds
to confronting that theoretically predicted distribution
with the values we observe. Selection effects make this
challenging [9, 12]: if any of the parameters that can
vary affect the formation of (say) protons, galaxies or ob-
servers, then the parameter probability distribution dif-
fers depending on whether it is computed at a random
point, a random proton, a random galaxy or a random
observer [12, 14]. A standard application of conditional
probabilities predicts the observed distribution


f(p) ~ f_prior(p) f_selec(p) (1)


where f_prior(p) is the theoretically predicted distribution
at a random point at the end of inflation and f_selec(p) is
the probability of our observation being made at that
point. This second factor f_selec(p), incorporating the
selection effect, is simply proportional to the expected
number density of reference objects formed (say, protons,
galaxies or observers)."

http://arxiv.org/PS_cache/astro-ph/pdf/0511/0511774.pdf