Jesse Mazer wrote:
> > Saibal Mitra wrote:
> > >
> > >This means that the relative measure is completely fixed by the
>absolute
> > >measure. Also the relative measure is no longer defined when
>probabilities
> > >are not conserved (e.g. when the observer may not survive an experiment
>as
> > >in quantum suicide). I don't see why you need a theory of
>consciousness.
> >
> > The theory of consciousness is needed because I think the conditional
> > probability of observer-moment A experiencing observer-moment B next
>should
> > be based on something like the "similarity" of the two, along with the
> > absolute probability of B. This would provide reason to expect that my
>next
> > moment will probably have most of the same memories, personality, etc.
>as
>my
> > current one, instead of having my subjective experience flit about
>between
> > radically different observer-moments.
>
>Such questions can also be addressed using only an absolute measure. So,
>why
>doesn't my subjective experience ''flit about between radically different
>observer-moments''? Could I tell if it did? No! All I can know about are
>memories stored in my brain about my ''previous'' experiences. Those
>memories of ''previous'' experiences are part of the current experience. An
>observer-moment thus contains other ''previous'' observer moments that are
>consistent with it.
But I would expect this consistency to be a matter of degree, because
sharing "memories" with other observer-moments also seems to be a matter of
degree. Normally we use the word "memories" to refer to discrete episodic
memories, but this is actually a fairly restricted use of the term, episodic
memories are based on particular specialized brain structures (like the
hippocampus, which if damaged can produce an inability to form new episodic
memories like the main character in the movie 'Memento') and it is possible
to imagine conscious beings which don't have them. The more general kind of
memory is the kind we see in a basic neural network, basically just
conditioned associations. So if a theory of consciousness determined
"similarity" of observer-moments in terms of a very general notion of memory
like this, there'd be a small degree to which my memories match those of any
other person on earth, so I'd expect a nonzero (but hopefully tiny)
probability of my next experience being that of a totally different person.
>Therefore all one needs to show is that the absolute
>measure assigns a low probability to observer-moments that contain
>inconsistent observer-moments.
But if observer-moments don't "contain" past ones in discrete way, but just
have some sort of fuzzy "degree of similarity" with possible past
observer-moments, then you could only talk about some sort of probability
distribution on possible pasts, one which might be concentrated on
observer-moments a lot like my current one but assign some tiny but nonzero
probability to very different ones.
In any case, surely my current observer-moment is not complex enough to
contain every bit of information about all observer-moments I've experienced
in the past, right? If you agree, then what do you mean when you say my
current one "contains" past ones?
> >
> > As for probabilities not being conserved, what do you mean by that? I am
> > assuming that the sum of all the conditional probabilities between A and
>all
> > possible "next" observer-moments is 1, which is based on the quantum
> > immortality idea that my experience will never completely end, that I
>will
> > always have some kind of next experience (although there is some small
> > probability it will be very different from my current one).
>
>I don't believe in the quantum immortality idea. In fact, this idea arises
>if one assumes a fundamental conditional probability.
Yes, it depends on whether one believes there is some theory that would give
an objective truth about first-person conditional probabilities. But even if
one does assume such an objective truth about conditional probabilities,
quantum immortality need not *necessarily* be true--perhaps for a given
observer-moment, this theory would assign probabilities to various possible
future observer-moments, but would also include a nonzero probability that
this observer-moment would be a "terminal" one, with no successors. However,
I do have some arguments for why an objective conditional probability
distribution would at least strongly suggest the quantum immortality idea,
which I outlined in a post at
http://www.escribe.com/science/theory/m4805.html
>I believe that
>everything should follow from an absolute measure. From this quantity one
>should derive an effective conditional probability. This probability will
>no
>longer be well defined in some extreme cases, like in case of quantum
>suicide experiments. By probabilities being conserved, I mean your
>condition
>that ''the sum of all the conditional probabilities between A and all
> possible "next" observer-moments is 1'' should hold for the effective
>conditional probability. In case of quantum suicide or amnesia (see below)
>this does not hold.
I'm not sure what you mean by "effective conditional probability"...is it
just the P(S'|S) = P(S')/P(S) idea you suggested earlier? This equation
would seem to suggest that the degree of similarity between two
observer-moments is irrelevant when deciding the conditional probability of
experiencing the second after the first, that if an observer-moment of
Charlie Chaplin's brain in 1925 has the same absolute probability as an
observer-moment of my brain 1 second from now, I should expect the same
probability of either one as my next experience. But from your other
comments I guess you're also adding that if an observer-moment S' doesn't
"contain" my current one S then there's 0 probability I will experience it
next.
>
> >
> > Finally, as for your statement that "the relative measure is completely
> > fixed by the absolute measure" I think you're wrong on that, or maybe
>you
> > were misunderstanding the condition I was describing in that post.
>
>I agree with you. I was wrong to say that it is completely fixed. There is
>some freedom left to define it. However, in a theory in which everything
>follows from the absolute measure, I would say that it can't be anything
>else than P(S'|S)=P(S')/P(S)
Only if you also impose the condition that each observer-moment has a unique
past, that there can be no "merging". If merging is possible, the
conditional measure could still follow from the absolute measure (my
suggestion is that the two measures mutually determine each other), but the
probabilities would be different.
>
> Imagine
> > the multiverse contained only three distinct possible observer-moments,
>A,
> > B, and C. Let's represent the absolute probability of A as P(A), and the
> > conditional probability of A's next experience being B as P(B|A). In
>that
> > case, the condition I was describing would amount to the following:
> >
> > P(A|A)*P(A) + P(A|B)*P(B) + P(A|C)*P(C) = P(A)
> > P(B|A)*P(A) + P(B|B)*P(B) + P(B|C)*P(C) = P(B)
> > P(C|A)*P(A) + P(C|B)*P(B) + P(C|C)*P(C) = P(C)
> >
> > And of course, since these are supposed to be probabilities we should
>also
> > have the condition P(A) + P(B) + P(C) = 1, P(A|A) + P(B|A) + P(C|A) = 1
>(A
> > must have *some* next experience with probability 1), P(A|B) + P(B|B) +
> > P(C|B) = 1 (same goes for B), P(A|C) + P(B|C) + P(C|C) = 1 (same goes
>for
>
> > C). These last 3 conditions allow you to reduce the number of unknown
> > conditional probabilities (for example, P(A|A) can be replaced by (1 -
> > P(B|A) - P(C|A)), but you're still left with only three equations and
>six
> > distinct conditional probabilities which are unknown, so knowing the
>values
> > of the absolute probabilities should not uniquely determine the
>conditional
> > probabilities.
>
>Agreed. The reverse is true. From the above equations, interpreting the
>conditional probabilities P(i|j) as a matrix, the absolute probability is
>the right eigenvector corresponding to eigenvalue 1.
Yes, that occurred to me after I had posted this. But I don't remember
enough linear algebra to say what conditions have to be met for a
nonnegative matrix to have a unique eigenvector for a given eigenvalue, and
the situation is complicated by the fact that I'm really imagining an
infinite number of distinct possible observer-moments and thus the matrix
would have an infinite number of components (but the sum of each row and
each column would be finite).
I also thought of a possible simplification: I said earlier that I thought
the function for the conditional probability between A and B would involve
both the absolute probability of B, P(B), and 'formal properties' of A and B
that, like the vague notion of 'similarity' I have been talking about, which
would have to be quantified by a theory of consciousness. But it seems like
there's a good argument for saying something a bit more specific, namely
that the function would involve the *product* of P(B) with the 'similarity'
(or whatever you call it) between A and B. Think of a duplication experiment
where the initial difference between the two duplicates is very small, like
they initially have identical brainstates but then diverge as one sees he's
in a room with green walls and the other sees he's in a room with red walls.
Presumably any quantity based on a comparison of formal properties between
two observer-moments, such as 'similarity', would be basically the same
whether you compared my current observer-moment with the one in the green
room or the one in the red room, so if the observer-moment in the green room
had twice the absolute probability as the one in the red room (say, because
the one in the green room was scheduled to be duplicated again later while
the one in the red room was not), it makes intuitive sense that my
conditional probability of becoming the one in the green room would also be
twice as large.
Obviously this isn't a watertight argument, but if it's true then we could
say P(B|A) = P(B)*Sab, where Sab is the 'similarity' between A and B (Don't
take the term 'similarity' too literally since this function might be quite
different from the ordinary sense of the term...for example, ordinarily we
think of the word similarity as something symmetrical, so the similarity of
A to B is the same as that of B to A, but the subjective directionality of
time and memory suggests this probably shouldn't be true for whatever
function is used here, because I'd expect to have a much higher conditional
probability of my next experience being that of my brain 1 second from now
than he should have of his next experience being my current one.) So the
equations would look like this:
P(A)*Saa*P(A) + P(A)*Sab*P(B) + P(A)*Sac*P(C) = P(A)
P(B)*Sba*P(A) + P(B)*Sbb*P(B) + P(B)*Sbc*P(C) = P(B)
P(C)*Sca*P(A) + P(C)*Scb*P(B) + P(C)*Scc*P(C) = P(C)
Which simplifies to:
Saa*P(A) + Sab*P(B) + Sac*P(C) = 1
Sba*P(A) + Sbb*P(B) + Sbc*P(C) = 1
Sca*P(A) + Scb*P(B) + Scc*P(C) = 1
Which would mean the "similarity matrix" operating on the
absolute-probability vector equals the unit vector, so as long as the
similarity matrix has an inverse, this inverse operating on the unit vector
would give the vector of absolute probabilities. Again though, I don't know
much about how linear algebra works for infinite matrices, or whether they'd
have inverses.
>
> >
> > >Let P(S) denote the probability that an observer finds itself in state
>S.
> > >Now S has to contain everything that the observer knows, including who
>he
> > >is
> > >and all previous observations he remembers making. The ''conditional''
> > >probability that ''this'' observer will finds himself in state S' given
> > >that
> > >he was in state S an hour ago is simply P(S')/P(S).
> >
> > This won't work--plugging into the first equation above, you'd get
> > (P(A)/P(A)) * P(A) + (P(B)/P(A)) * P(B) + P(P(C)/P(A)) * P(C), which is
>not
> > equal to P(A).
>You meant to say:
>
>''P(A)/P(A)) * P(A) + (P(A)/P(B)) * P(B) + P(A)/P(C) * P(C), which is not
> equal to P(A).''
Actually I just got confused about whether S' or S was the current state,
but yeah, that's what I should have written. Anyway, as I said, for
something like this to be true in general you'd need a 1/N factor, where N
is the total number of possible observer-moments. But from your comments
about amnesia below I take it you're saying that S' has a unique previous
state S, so if B's unique past state was A, then P(B|A) = P(B)/P(A) while
P(B|C) = 0 and P(B|B) = 0, so the condition P(B|A)*P(A) + P(B|B)*P(B) +
P(B|C)*P(C) = P(B) would be satisfied. But does this also mean that each
observer-moment has a unique future? Consider the matrix of conditional
probabilities:
P(A|A) P(A|B) P(A|C)
P(B|A) P(B|B) P(B|C)
P(C|A) P(C|B) P(C|C)
You're saying that only one entry in each row can be nonzero. But this means
either that each column has exactly one entry that's nonzero (every
observer-moment has a unique future), or that some columns have multiple
nonzero entries while others have all zero entries--maybe these might
correspond to "terminal" observer-moments where death is certain? Anyway, I
guess this conclusion wouldn't hold for a matrix whose rows and columns
contained an infinite number of components, where you could have something
like this:
.5 0 0 0 0 0 . . .
.5 0 0 0 0 0
0 .5 0 0 0 0
0 .5 0 0 0 0
0 0 .5 0 0 0
0 0 .5 0 0 0
. .
. .
. .
>
>This shows that in general, the conditional probability cannot be defined
>in
>this way. In P(S')/P(S), S' should be consistent with only one S. Otherwise
>you are considering the effects of amnesia.
By "amnesia", you're talking about the idea that streams of consciousness
can merge as well as split, correct? That a given observer-moment can be
compatible with multiple pasts? If so, then yes, I would assume something
like that is possible, if splitting is possible.
Jesse
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Received on Mon Feb 09 2004 - 05:10:17 PST