# Re: Predictions & duplications

From: Russell Standish <R.Standish.domain.name.hidden>
Date: Thu, 11 Oct 2001 18:20:25 +1000 (EST)

juergen.domain.name.hidden wrote:
>
>
>
> > From R.Standish.domain.name.hidden :
> > juergen.domain.name.hidden wrote:
> > >
> > > So you NEED something additional to explain the ongoing regularity.
> > > You need something like the Speed Prior, which greatly favors regular
> > > futures over others.
> >
> > I take issue with this statement. In Occam's Razor I show how any
> > observer will expect to see regularities even with the uniform prior
> > (comes about because all observers have resource problems,
> > incidently). The speed prior is not necessary for Occam's Razor. It is
> > obviously consistent with it though.
>
> First of all: there is _no_ uniform prior on infinitely many things.
> Try to build a uniform prior on the integers. (Tegmark also wrote that
> "... all mathematical structures are a priori given equal statistical
> weight," but of course this does not make much sense because there is
> _no_ way of assigning equal nonvanishing probability to all - infinitely
> many - mathematical structures.)

I don't know why you insist on the prior being a PDF. It is not
necessary. With the uniform prior, all finite sets have vanishing
probability. However, all finite descriptions correspond to infinite
sets, and these infinite sets have non-zero probability.

>
> There is at best a uniform measure on _beginnings_ of strings. Then
> strings of equal size have equal measure.
>
> But then regular futures (represented as strings) are just as likely
> as irregular ones. Therefore I cannot understand the comment: "(comes
> about because all observers have resource problems, incidently)."
>

Since you've obviously barked up the wrong tree here, it's a little
hard to know where to start. Once you understand that each observer
must equivalence an infinite number of descriptions due to the
boundedness of its resources, it becomes fairly obvious that the
smaller, simpler descriptions correspond to larger equivalence classes
(hence higher probability).

> Of course, alternative priors lead to alternative variants of Occam's
> razor. That has been known for a long time - formal versions of Occam's
> razor go at least back to Solomonoff, 1964. The big question really
> is: which prior is plausible? The most general priors we can discuss are
> those computable in the limit, in the algorithmic TOE paper. They do not
> allow for computable optimal prediction though. But the more restrictive
> Speed Prior does, and seems plausible from any programmer's point of view.
>
> > The interesting thing is of course whether it is possible to
> > experimentally distinguish between the speed prior and the uniform
> > prior, and it is not at all clear to me that it is possible to
> > distinguish between these cases.
>
> I suggest to look at experimental data that seems to have Gaussian
> randomness in it, such as interference patterns in split experiments.
> The Speed Prior suggests the data cannot be really random, but that a
> fast pseudorandom generator PRG is responsible, e.g., by dividing some
> seed by 7 and taking some of the resulting digits as the new seed, or
> whatever. So it's verifiable - we just have to discover the PRG method.
>

I can't remember which incompleteness result it is, but it is
impossible to prove the randomness of any sequence. In order to
falsify your theory one would need to prove a sequence to be
random. However, of course if all known sequences are provably
pseudo-random (ie compressible), then this would constitute pretty
good evidence. However, this is a tall order, as there is no algorithm
for generating the compression behind an arbitrary sequence.

Unless someone else has some brilliant ideas, its all looking a bit grim.

> Juergen Schmidhuber
>
> http://www.idsia.ch/~juergen/
> http://www.idsia.ch/~juergen/everything/html.html
> http://www.idsia.ch/~juergen/toesv2/
>
>
>

----------------------------------------------------------------------------
Dr. Russell Standish Director
High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile)
UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (")
Australia R.Standish.domain.name.hidden
Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks
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Received on Thu Oct 11 2001 - 01:35:09 PDT

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