On 13 Sep., 00:48, Russell Standish wrote:
> It would be possible to construct an ensemble of purely finite strings
> (all strings of length googol bits, say). This wouldn't satisfy the
> zero information principle, or your no-justification, as you still
> have the finite string size to justify (why googol and not googol+1,
> for instance). I suspect the observable results would be
> indistinguishable from the infinite string ensembles for large enough
> string string size, however.
We've a little misunderstanding in this point. I did never suggest
strings of an overall fixed length, but only of a finite length that
may vary from string to string without being limited. The idea behind
this was that imaginable things should be describable completely (e.g.
by a person telling me about them) and not only asymptotically
(which---I thought---could be the case if the descriptions were
infinite).
On the other hand, I do see two arguments in favor of the infinite
strings:
1. It may be that something can be described by a finite description
in one "language", but must be described by an infinite description in
another "language". A simple example is the number pi which can be
defined by finite expressions (e.g. by writing down formally the
Gregory-Leibniz series). But if we restrict ourselves to describe
numbers by writing down their digits in the decimal numeral system,
then the description of pi is infinite. This can be seen as a
motivation to allow infinite strings.
2. The difference between finite and infinite strings is somehow
similar to the difference between natural and real numbers (at least
as far as their cardinalities are concerned) in mathematics. If, in a
far future, we want to establish analytical methods to study the
Everything ensemble (this of course is a very, very problematic task
and cannot be our concern here) it may turn out useful to allow
infinite strings as it turned out useful for ordinary mathematics to
allow real numbers instead of natural or rational numbers.
> Where differences lie is in the measure attached to these strings. I
> take each string to be of equal weight to any other, so that there are
> twice the measure of strings satisfying 01* as 011*. This leads
> naturally to a universal prior.
I'm still hesitant to accept the idea that the Everything ensemble by
itself comes up with a measure. Although undoubtedly the measure is a
fundamental ingredient of our theories, I think that it should only be
introduced for practical reasons, i.e. whenever we are interested in
probabilities. Then the measure is adapted to our state of ignorance.
The standard case will be that one has no information whether to
prefer a given description which leads to your measure of equal weight
and the universal prior. This is very analogous to statistical physics
where we usually assign equal measure to every microstate.
I am not yet familiar with Schmidhuber's ideas but I am going to read
up on this topic soon, in particular in the context of the White
Rabbit paradox.
Youness Ayaita
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Received on Wed Sep 12 2007 - 20:24:03 PDT