Wei writes:

> On Tue, Apr 20, 2004 at 12:42:16PM -0700, Hal Finney wrote:
> > This is the part which really confuses me.  Assigning a prior of 1 would
> > mean that you are certain that the actual world is the class of all sets.
> > Assigning a prior of one in a billion would mean that you thought it
> > very unlikely.  Yet these two possibilites won't make any difference
> > in behavior?  Why not?
>
> Because after changing your prior for the actual world being the class of
> all set to 1, you can always find a new measure to adopt (expressing how
> much you care about each part of the world) that would ensure that your
> behavior is exactly the same as before.

Okay, so going back to the original dialog:

: Do you assign a non-zero prior to the class of all sets being the actual
: world?
:
: (yes)
:
: Pragmatically, how does that differ from assigning a prior of 1 for the
: class of all sets?
:
: (What do you mean by "pragmatically"?)
:
: I mean are there any circumstances in which you'd act differently if you
: assigned a prior of 1 instead?
:
: (no)

Let me see if I follow this.  According to decision theory, we take
those actions which maximize expected utility.  Expected utility for
the outcome of an action is a weighted average of the utility of each
possible outcome, weighted by the probability that each outcome occurs.
The probability for an outcome is ultimately based on one's belief in
the nature of the world.

Knowing the nature of the world is hard, and we can have only uncertain
information about it.  One way to express this uncertainty is to
say that we might live in any one of many different possible worlds,
and to estimate a probability distribution over the possibilities.
Bayesian reasoning then gives us rules for how we should update these
probabilities, based on observations.

Multiverse theory introduces a new complication.  If a Tegmarkian or
even Schmiduberian multiverse model is true, or might be true, then we
need a rule to estimate how likely we are to live in one element of that
multiverse or another.  This rule produces a numerical probability value
for each element of the ensemble of universes, which we call a measure.
The measure is necessary for us to go from the abstract multiverse model
to a set of rules that allows us to make predictions about what we will
see, and therefore to take effective actions.

(Actually, the measure is usually considered as being a property of
an element of the multiverse ensemble, i.e. a property of an entire
universe, rather than the probability of a specific observer being in
that world.  Then we need yet another rule to go from the measure of a
multiverse branch to the probability that we are actually in that branch.
But for the present purpose we can combine these two rules and just let
the measure be the probability of a branch being "real" for a particular
observer.)

In the context of Bayesian reasoning, the measure is problematic,
because it introduces an extra degree of freedom in our universe model.
Rather than having simply a probability distribution over possible
universes, we have to consider that among the possible universes are
multiverses, and then there are an infinite number of possible measures
for each possible kind of multiverse.

These measures are a hard problem and it's not clear what the right
way is to think of them.  There are various philosphical arguments that
one measure or another is the most likely to be true.  However it would
seem that no such argument can be conclusive, hence we must always admit
some uncertainty in the possible measure.  It's also not clear whether
observation allows us to constrain measure, because in a multiverse model
we observe only a small part of the multiverse, and so we can't judge
the conditional probabilities necessary to apply Bayesian reasoning.
Can we say, for example, that our observations of the universe imply
(if a multiverse model were true) that the measure must be such that
our universe has a relatively large measure?  Maybe not, because even
if our universe had a small measure, we would still be here observing it.

Anyway, I think Wei's point is that if we admit even the possibility that
a multiverse model is true, then the shape of our mental possible-universe
space becomes inconsistent or redundant.  We have all these various
possible universes as in standard Bayesian reasoning, and among them we
add the multiverse.  But the multiverse includes all possibilities, and
in particular it includes all of the non-multiverse possible universes
we were considering originally.  So it's redundant to consider these
possible universes outside the context of the multiverse, since we
can get the same set of possible universes by looking solely within
the multiverse.  And given our freedom to choose the measure for the
universes within the multiverse, we can reproduce all of our probability
estimates for non-multiverse possible universes, even if we just consider
the multiverse.

Therefore, as Wei says, we can give the multiverse a probability of 1,
and choose a measure over the worlds within the multiverse to match any
desired probability distribution over possible worlds.

I see two possible problems with this.  The first is that it's not clear
that we have this much freedom in our choice of measure.  Some Bayesians
may accept the philosophical arguments which say that certain measures
are preferred.  The second is that if we did have complete freedom in
choosing a measure, how can Bayesianism work in this model?  Specifically,
how should we update our beliefs in response to observations?  In the
standard Bayesian model we have specified rules to follow.  Would that
imply that if we accept the multiverse as certain, we must still apply
those specific rules to update our estimates of measure?  That seems
inconsistent with the assumption that measure is freely chosen.

Hal Finney

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