On 3/8/2020 3:56 AM, Bruce Kellett wrote:
On Sun, Mar 8, 2020 at 7:46 PM Russell Standish <[email protected]
<mailto:[email protected]>> wrote:
On Sun, Mar 08, 2020 at 06:50:52PM +1100, Bruce Kellett wrote:
> On Sun, Mar 8, 2020 at 5:32 PM Russell Standish
<[email protected] <mailto:[email protected]>> wrote:
>
> On Fri, Mar 06, 2020 at 10:44:37AM +1100, Bruce Kellett wrote:
>
> > That is, in fact, false. It does not generate the same
strings as
> flipping a
> > coin in single world. Sure, each of the strings in Everett
could have
> been
> > obtained from coin flips -- but then the probability of a
sequence of
> 10,000
> > heads is very low, whereas in many-worlds you are
guaranteed that one
> observer
> > will obtain this sequence. There is a profound difference
between the two
> > cases.
>
> You have made this statement multiple times, and it appears
to be at
> the heart of our disagreement. I don't see what the profound
> difference is.
>
> If I select a subset from the set of all strings of length
N, for example
> all strings with exactly N/3 1s, then I get a quite specific
value for the
> proportion of the whole that match it:
>
> / N \
> | | 2^{-N} = p.
> \N/3/
>
> Now this number p will also equal the probability of seeing
exactly
> N/3 coins land head up when N coins are tossed.
>
> What is the profound difference?
>
>
>
> Take a more extreme case. The probability of getting 1000 heads
on 1000 coin
> tosses is 1/2^1000.
> If you measure the spin components of an ensemble of identical
spin-half
> particles, there will certainly be one observer who sees 1000
spin-up results.
> That is the difference -- the difference between probability of
1/2^1000 and a
> probability of one.
>
> In fact in a recent podcast by Sean Carroll (that has been
discussed on the
> list previously), he makes the statement that this rare event
(with probability
> p = 1/2^1000) certainly occurs. In other words, he is claiming
that the
> probability is both 1/2^1000 and one. That this is a flat
contradiction appears
> to escape him. The difference in probabilities between coin
tosses and
> Everettian measurements couldn't be more stark.
That is because you're talking about different things. The rare event
that 1 in 2^1000 observers see certainly occurs. In this case
certainty does not refer to probability 1, as no probabilities are
applicable in that 3p picture. Probabilities in the MWI sense refers
to what an observer will see next, it is a 1p concept.
And that 1p context, I do not see any difference in how probabilities
are interpreted, nor in their numerical values.
Perhaps Caroll is being sloppy. If so, I would think that could be
forgiven.
Yes, I think the Carroll's comment was just sloppy. The trouble is
that this sort of sloppiness permeates all of these discussions. As
you say, probability really has meaning only in the 1p picture. So the
guy who sees 1000 spin-ups in the 1000 trials will conclude that the
probability of spin-up is very close to one. That is why it makes
sense to say that the probability is one. The fact that this one guy
sees this is certain in Many-worlds (This may be another meaning of
probability, but an event that is certain to happen is usually
referred to as having probability one.).
The trouble comes when you use the same term 'probability' to refer to
the fact that this guy is just one of the 2^N guys who are generated
in this experiment. The fact that he may be in the minority does not
alter the fact that he exists, and infers a probability close to one
for spin-up. The 3p picture here is to consider that this guy is just
chosen at random from a uniform distribution over all 2^N copies at
the end of the experiment. And I find it difficult to give any
sensible meaning to that idea. No one is selecting anything at random
from the the 2^N copies because that is to how the copies come about
-- it is all completely deterministic.
The guy who gets the 1000 spin-ups infers a probability close to one,
so he is entitled to think that the probability of getting an
approximately even number of ups and downs is very small:
eps^1000*(1-eps)^1000 for eps very close to zero. Similarly, guys who
see approximately equal numbers of up and down infers a probability
close to 0.5. So they are entitled to conclude that the probability of
seeing all spin-up is vanishingly small, namely, 1/2^1000.
The main point I have been trying to make is that this is true
whatever the ratio of ups to downs is in the data that any individual
observes. Everyone concludes that their observed relative frequency is
a good indicator of the actual probability, and that other ratios of
up:down are extremely unlikely. This is a simple consequence of the
fact that probability is, as you say, a 1p notion, and can only be
estimated from the actual data that an individual obtains. Since
people get different data, they get different estimates of the
probability, covering the entire range [0,1]; no 3p notion of
probability is available -- probabilities do not make sense in the
Everettian case when all outcomes occur.
I think this is wrong. The is both a 3p and 1p notion of probability.
The 1p notion is that I'm ignorant of the probability of getting a 1 or
0 but those are some fixed values so I can estimate the probability from
Bernoulli trials. The 3p notion is that there is a fixed ensemble of
sequences produced by a certain fixed branching ratio of 1s and 0s. If
I pick one sequence at random from the ensemble I can estimate that
branching ratio. My claim is that these are equivalent. The 1p is the
ergodic process model of the 3p.
This is the basic argument that Kent makes in arxiv:0905.0624.
The difference from the deterministic coin tossing situation is that
in that case, only one outcome occurs in any trial, so the sequence of
N trials generates a single bit sting of length N, indicating a
particular value of the probability for success on any toss. The
situation could not be more different from the case in which all
outcomes always occur.
Yes it could be more different. The N->oo 1p and 3p statistics could
disagree, but they don't. The expected values are the same, and the
std-deviation is the same.
Brent
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