Eric's comments made me think about these two articles:
http://arxiv.org/abs/math-ph/0008018
Change, time and information geometry
Authors: Ariel Caticha
''Dynamics, the study of change, is normally the subject of mechanics.
Whether
the chosen mechanics is ``fundamental'' and deterministic or
``phenomenological'' and stochastic, all changes are described relative to
an external time. Here we show that once we define what we are talking
about, namely, the system, its states and a criterion to distinguish among
them, there is a single, unique, and natural dynamical law for irreversible
processes that is compatible with the principle of maximum entropy. In this
alternative dynamics changes are described relative to an internal,
``intrinsic'' time which is a derived, statistical concept defined and
measured by change itself. Time is quantified change.''
And:
http://arxiv.org/abs/gr-qc/0109068
Entropic Dynamics
Authors: Ariel Caticha
''I explore the possibility that the laws of physics might be laws of
inference rather than laws of nature. What sort of dynamics can one derive
from well-established rules of inference? Specifically, I ask: Given
relevant information codified in the initial and the final states, what
trajectory is the system expected to follow? The answer follows from a
principle of inference, the principle of maximum entropy, and not from a
principle of physics. The entropic dynamics derived this way exhibits some
remarkable formal similarities with other generally covariant theories such
as general relativity.''
Instead of identifying an observer moment with the exact information stored
in the ''brain'' of an observer, one could identify it with a probability
distribution over such precisely defined states. This seems more realistic
to me. No observer can be aware of all the information stored in his brain.
When I think about who I am, I am actually performing a measurement of some
average of the state my brain is in. After this measurement the probability
distribution will be updated. To apply Caticha's ideas, one has to identify
the measurements with taking averages over an ensemble of observers
described by the same probability distribution. In general this cannot be
true, but like in statistical mechanics, under certain conditions one is
allowed to replace actual averages involving only one system with averages
over a (hypothetical) ensemble.
Saibal
----- Original Message -----
From: Eric Hawthorne
To: everything-list.domain.name.hidden
Sent: Saturday, February 07, 2004 5:26 AM
Subject: Re: measure and observer moments
Given temporal proximity of two states (e.g. observer-moments),
increasing difference between the states will lead to dramatically lower
measure/probability
for the co-occurrence as observer-moments of the same observer (or
co-occurrence in the
same universe, is that maybe equivalent?) .
When I say two states S1, S4 are more different from each other whereas
states S1,S2 are less different
from each other, I mean that a complete (and yet fully abstracted i.e. fully
informationally compressed) informational
representation of the state (e.g. RS1) shares more identical (equivalent)
information with RS2 than it does with RS4.
This tells us something about what time IS. It's a dimension in which more
(non-time) difference between
co-universe-inhabiting states can occur with a particular probability
(absolute measure) as the states
get further from each other in the time of their occurrence. Things (states)
which were (nearly) the same can only
become more different from each other (or their follow-on most-similar
states can anyway) with the passage
of time (OR with lower probability in a shorter time.)
Maybe?
Eric
Saibal Mitra wrote:
----- Original Message -----
From: Jesse Mazer <lasermazer.domain.name.hidden>
To: <everything-list.domain.name.hidden>
Sent: Thursday, February 05, 2004 12:19 AM
Subject: Re: Request for a glossary of acronyms
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. Therefore all one needs to show is that the absolute
measure assigns a low probability to observer-moments that contain
inconsistent observer-moments.
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. 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.
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)
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.
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).''
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. In such cases, you would expect
the probability to increase.
Saibal
Received on Sat Feb 07 2004 - 08:48:26 PST