Re: Probabilities with infinite sets ( Was many (identical) universes)

From: <GSLevy.domain.name.hidden>
Date: Sat, 19 Feb 2000 16:47:42 EST

In a message dated 02/17/2000 2:12:51 PM Pacific Standard Time,
fritzgriffith.domain.name.hidden writes:

> >From: GSLevy.domain.name.hidden
> >To: everything-list.domain.name.hidden
> >Subject: Re: Probabilities with infinite sets ( Was many (identical)
> >universes)
> >Date: Thu, 17 Feb 2000 01:05:21 EST
> >
> >In a message dated 02/14/2000 11:42:56 PM Pacific Standard Time,
> >flipsu5.domain.name.hidden writes:
> >
> > >
> > > Selwyn St Leger wrote:
> > >
> > > > > Date: Sun, 13 Feb 2000 12:58:59 -0800
> > > > > From: Fred Chen <flipsu5.domain.name.hidden>
> > > > > To: Saj Malhi <sajm.domain.name.hidden>
> > > > > Cc: everything-list.domain.name.hidden
> > > > > Subject: Re: many (identical) universes
> > > >
> > > > > Saj Malhi wrote:
> > > >
> > > > > > 2. The infinity of even numbers is as large as the infinity of
> >all
> > > real
> > > > > > numbers. Only different classes / types of infinity have been
> >shown
> >to
> > > be of
> > > > > > different size.
> > > > > >
> > > > >
> > > > > Hmmm...I would have thought the set of even numbers is much
smaller
> >than
> > > the set
> > > > > of reals, because between any two even numbers, you have
infinitely
> >more
> > > reals.
> > > > >
> > > > > Fred
> > > >
> > > > Surely the even mumbers are a sub-set of the integers and thus of a
> > > > lower order of infinity than the reals?
> > > >
> > > > Selwyn St Leger
> > >
> > > My previous intuition would have led me to think that there are fewer
> >evens
> > > than
> > > integers, but more rigorously it seems possible to map evens to
> >integers
> >1:1
> > > by using
> > > n->2n, and likewise for odds. However, it is difficult to imagine a
way
>
> >of
> > > doing this
> > > for rationals (or reals).
> > >
> > > I guess the issue for AUH would be: can we have meaningful
probability
> >with
> > > infinities? Is the total number of evens 'half' the total number of
> >integers,
> > > so that
> > > any randomly picked integer has a 50% chance of being even? Perhaps
> >instead
> > > of
> > > discussing total number, we need number density.
> > >
> > > Fred
> > >
> > >
> >
> >Fred has put his finger on one of the most important issues involving MWI
> >and
> >quantum immortality. The meaning of probability in the context of the MW
> >branching.
> >
> >Either the Shroedinger wave function splits into a finite number of
> >branches
> >or into an infinite number of branches.
> >
> >If the split is finite, calculating the probability of a given outcome x,
> >given a starting situation S is trivial. Let's assume that the total
number
> >of branchesfrom S is N, of which n<N contain the given outcome. Then,
> >assuming a **random** selection of any branch from the set of N branches,
> >the probability of outcome x is given by:
> >
> >P{x} = n/N
> >
> >When the split is infinite, the probability calculation is not so obvious.
> >Consider an infinite split of cardinality aleph null. Let's index each
> >branch
> >so that branch index = natural number. So we have branches "1", "2",
> >"3",..... Now let us define the outcome x as (branch index = even). Let's
> >calculate the probability of outcome x.
> >Common sense indicates that total number N of branches (all natural
> >numbers)
> >= twice number n of branches with outcome x (i.e., even branches). Common
> >sense however is misleading.
> >
> >Here is the key issue. Depending how the branches are sorted and the
> >**random** selection is performed, the probability calculation will
produce
> >different results. If we sort the branches in the order "1", "2", "3",...
> >then it seems that the even outcome x will occur 50% of the time. However,
> >this particular sorting is entirely arbitrary. I could have sorted the
> >branches as "1", "3", "2", "5", "7", "4", "9", "11", "6", "13", "15",
> >"8",...
> >that is having the density of the odd numbers = twice the density of the
> >even
> >numbers. With this arrangement all natural numbers are guaranteed to
occur
> >in
> >the series. Yet a **random** sampling will generate a probability of
> >picking
> >an even number = 1/3. It seems then that the definition of probability in
> >the
> >context of infinite sets must include the specification of how these sets
> >are
> >ordered!
>
> But you could say this about any set, infinite or finite. You could have
a
> set containing 100 random numbers valued from 1 to 100. Then you could
put
> all the low numbers (less than 10) first and only look at the first 10
> numbers. With the collected data you could then conclude that the numbers
> are only random numbers from 1 to 10. With your argument, you are saying
> that first you assume some non-random ordering of the branches (twice as
> many odds as evens) and then you only look at a small sample. Of course,
> with an infinite set it is impossible to look at all values, but in most
> cases it is safe to assume a random arrangement. This is the same concept
> as when polls are taken - only a small portion of the population is
> interviewed, but it is hoped that that sample will represent the whole
> population accurately.
> In addition, it is not clear how a random sampling can be performed
> >over the natural number, since we seems always to be biased toward the
> >beginning of the scale (near zero) since for any finite number that is
> >selected, we are guaranteed to have many more numbers above it than under
> >it!
> >
> >In an attempt to find a way out of this dilemma, we could assume that the
> >branching is really a continuum. So let us consider a line interval from
0
> >to
> >1 as a model. Consider an outcome x specified for every real real number.
> >That is any real number either has property x or does not have it. This
> >property could be for example the 2nd digit = 3. The sorting problem
crops
> >up
> >again! I could order my reals any way I please with all the reals
> >containing
> >property x crowded in a tiny subinterval, say between 0 and 0.001. Any
> >**random** sampling over interval (0,1) will be very unlikely to pick any
> >real with the property 2nd digit = 3. So assuming a continuum does not
> >solve
> >the dilemma!.
> >
> >We know from experience that probability calculations in the MW do make
> >sense!!! There must be something wrong with the math. Does any one of you
> >know what? In this context the concept of limit seems to be suspect. I
> >don't
> >know if probability theory has been extended to cover infinite sets. Does
> >any
> >one of you know what the state of the art is in this subject?
> >


With finite set of numbers say 1-100, if my sampling procedure generates a
selection index with a uniform distribution over the interval 1-100, then the
number being sampled will also have a uniform distribution INDEPENDENTLY of
the way the numbers are sorted in the interval.

However, with an infinite set of natural numbers, how can I have a selection
procedure with a uniform distribution? As an exercise, please come up with an
algorithm that could produce a uniformly distributed pseudo-random selection
index where the index ranges over all the natural number. In addition come up
with a procedure that generates a uniformly distributed pseudo-random sorting
of all the natural number. What would the first three numbers of a
"randomly selected" sequence look like?
:-)

And even if you could come up with an algorithm for generating a random
number from among all the natural numbers, any finite number that it would
produce would necessarily be biased toward zero since there are more numbers
above it then under it. Since the sampling is biased toward zero, the
ordering of the number is critical in defining the distribution.

Taking polls of finite population is different than taking polls of infinite
populations! For example in the set of natural numbers we can show that the
number of even numbers is equal to the number of multiples of three.

George Levy
Received on Sat Feb 19 2000 - 13:51:41 PST