Re: Predictions & duplications

From: Russell Standish <>
Date: Fri, 26 Oct 2001 08:56:20 +1000 (EST) wrote:
> > From: Russell Standish <>
> > To:
> >
> > I think we got into this mess debating whether an infinite set could
> > support a uniform measure. I believe I have demonstrated this.
> > I've yet to see anything that disabuses me of the notion that a
> > probability distribtuion is simply a measure that has been normalised
> > to 1. Not all measures are even normalisable.
> Russell, at the risk of beating a dead horse: a uniform measure is _not_ a
> uniform probability distribution. Why were measures invented in the first
> place? To deal with infinite sets. You cannot have a uniform probability
> distribution on infinitely many things. That's why measures isolate just
> finitely many things, say, every bitstring of size n, and for each x of
> size n look at the infinite set of strings starting with x. A uniform
> measure assigns equal probability to each such set. Of course, then you
> have a uniform probability distribution on those finitely many things
> which are sets. But that's not a uniform probability distribution on
> infinitely many things, e.g., on the bitstrings themselves! The measure
> above is _not_ a probability distribution; it is an infinite _set_ of
> _finite_ probability distributions, one for string size 0, one for string
> size 1, one for string size 2,...

Good grief, you've missed the point entirely! Measures do measure
infinite sets - consider the set [0,1] - it is an infinite set, and
the uniform probaility distribution and uniform measure are the same
thing. We can also consider the set [0,\infty) which has a uniform
measure on it (the same constant probability distibution as before,
but extended to the full set). Clearly it cannot have a uniform
probability distribution, as the set has infinite measure. You seem
fixated on the peculiar definition of measure on strings that you gave
before, rather than the more general notion.

> > I realise that the Halting theorem gives problems for believers of
> > computationalism.
> It does not. Why should it?

Restating what I said before simply:

1) Halting theorem implies the set of halting programs is of measure
zero within the set of all programs

2) Computationalism is the assumption that the universe's observer is
a universal Turing machine.

3) Uniform measure over the Plenitude of descriptions implies that
observer will never see simple strings.

> > I never subscribed to computationalism at any time,
> > but at this stage do not reject it. I could conceive of us living in
> > a stupendous virtual reality system, which is in effect what your GP
> > religion Mark II is. However, as pointed out by others, it does suffer
> > from "turtle-itis", and should not be considered the null
> > hypothesis. It requires evidence for belief.
> By turtle-itis you mean: in which universe do the GP and his computer
> reside? Or the higher-level GP2 which programmed GP? And so on? But
> we cannot get rid of this type of circularity - computability and
> mathematical logic are simply taken as given things, without any
> further explanation, like a religion. The computable multiverse, or
> the set of logically possible or mathematically possible or computable
> universes, represents the simplest explanation we can write down formally.
> But what exactly does it mean to accept something as a formal statement?
> What does it mean to identify the messy retina patterns caused by this
> text with abstract mathematical symbols such as x and y? All formal
> explanations in our formal papers assume that we agree on how to read
> them. But reading and understanding papers is a complex physical and
> cognitive process. So all our formal explanations are relative to this
> given process which we usually do not even question. Essentially, the
> GP program is the simplest thing we can write down, relative to the
> unspoken assumption that it is clear what it means to write something
> down, and how to process it. It's the simplest thing, given this use
> of mathematical language we have agreed upon. But here the power of the
> formal approach ends - unspeakable things remain unspoken.

The GP program (aka Universal Dovetailer) is not the simplest
thing. The simplest describable thing is the set of all descriptions,
that simply exist without being generated. That has precisely zero
information or complexity, whereas the UD has some complexity of the
order of a few 100 bits. The simplest prior is the uniform one, ie
there is no bias whatsoever in the selection.

The only reason for not accepting the simplest thing is if it can be
shown to be logically inconsistent. This far, you have shown no such
thing, but rather demonstrated an enormous confusion between measure
and probability distribution.

The set of all descriptions and the uniform measure do not suffer from
turtle-itis. However, they are susceptible to criticism of the
Church-Turing thesis - namely why should our idea of information in
terms of strings of symbols from a discrete alphabet have universal
applicability. Yet, I don't see anyone here seriously arguing against the
CT thesis, so there I will let it lie.

> Juergen Schmidhuber

Dr. Russell Standish Director
High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile)
UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (")
Room 2075, Red Centre
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Received on Thu Oct 25 2001 - 16:15:49 PDT

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