From: Ben Goertzel (ben@goertzel.org)
Date: Sun Aug 25 2002 - 00:05:46 MDT
> > Bayes' Theorem is a powerful tool (and one of the central parts of the
> > Novamente reasoning module).
>
> Explicit, abstract use of Bayes' Theorem, seen deliberatively as Bayes'
> Theorem, is a tiny subset of the universal domain governed by the BPT.
This is true.
In fact, however, the Novamente design explicitly uses bayesian inference
not only for some deliberative thought, but also for some nondeliberative
thought...
> > However, like the rest of elementary probability theory, it pertains to
> > a set of probabilities defined over a fixed "universal set."
> >
> > Defining the universal set is a serious issue in any practical
> > situation. You can say "define it as the set of all entities ever
> > observed by the mind doing the reasoning". But this doesn't really
> > work, because we *hear* about entities via linguistic communication,
> > including many entities we haven't seen. I may want to include Beijing
> > in my universal set for my internal probabilistic inference, even
> > though I have never been there or seen it.
>
> You must learn to see the BPT flowing underneath the surface of all
> cognition, like blood beneath skin.
I made a very practical mathematical point, and you responded with poetry,
which doesn't seem to resolve the issue I raised at all.
> In this case, you have encountered a set of sensory experiences
> which lead
> you to believe that Beijing exists - other people producing reports about
> Beijing. Since you expect that people will report that Beijing exists in
> cases where Beijing exists, and that they will not report that Beijing
> exists if it does not exist, this is, under the BPT, evidence; it is a
> case of a thing being evidence about another thing.
>
> In terms of the BPT, this is no different from seeing photons reflecting
> from Beijing in person, or from accessing the deposited memories
> afterward. These forms of evidence differ in strength and
> detail, but are
> all equally governed by the BPT.
I would like to ask you to carry out this particular discussion in more
concrete, mathematical terms. Bayes Theorem is a mathematical theorem, not
a philosophical or scientific theory. If you mean something different by
BPT, than the branch of math I am familiar with, please articulate what you
mean.
Bayes' Theorem is a result in elementary probability theory.
We are concerned here with discrete probability theory.
All theorems from discrete elementary prob. theory, take place within an
assumed finite universal set U.
Now, suppose we have an intelligent system that is *explicitly or
implicitly* making use of elementary probability theory to reason about the
world.
I am making the following argument:
*For each step of probabilistic inference the system does*, it must assume
some specific universal set U.
Therefore:
*The system must use some process besides prob. inference to decide on the
universal set U to use, in at least some cases.*
This statement is not refuted by arguing that the system can, in some cases,
(explicitly or implicitly) use prob. inference to derive a universal set U
to use in some contexts. Because one then asks: what is the universal set U
for the prob. inference used in that derivation?
If the universal set U cannot always be provided by prob. inference, how
else can it be provided?
a) By biological wiring
b) By some nonprobabilistic-inference-based cognitive process
I suggest b). Perhaps you believe a).
> Not at all. You must learn to see the BPT flowing underneath the surface
> of all reality.
If we take an abstract perspective, and consider a universal set consisting
of the entire physical universe, then we can do probabilistic inference over
the whole cosmos in an "objective" way.
But, this is not a perspective that any mind can take in practice, because
no finite mind is complete; its subjective universe is continually changing
and expanding.
And in fact, modern physics theories do not support the notion of the
universe as a fixed "universal set" either.
So, I think that YOU must learn to see probability theory as a tool for
doing inference *in a fixed, defined context*.
It can be used recursively, so that one prob. inference defines the context
for another prob. inference, etc.
But it is not objective -- it is subjective, dependent upon the context.
This is not a philosophical point, it's a mathematical point -- the actual
probabilistic inferential conclusions you get depend upon what universal set
you assume.
The universal fixed universe you seem to assume is a mathematical
abstraction, convenient for proving theorems in probability theory, but not
really universal and fixed in any real situation.
-- Ben G
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