On Aug 27, 7:35 pm, Bruno Marchal <marc....domain.name.hidden> wrote:
>
> Zermelo Fraenkel theory has full transfinite induction power, but is
> still limited by Gödel's incompleteness. What Gentzen showed is that
> you can prove the consistency of ARITHMETIC by a transfinite induction
> up to epsilon_0. This shows only that transfinite induction up to
> epsilon_0 cannot be done in arithmetic.
Yes. That's all I need for the purposes of my criticism of Bayes.
SInce ZF theory has full transfinite induction power, it is more
powerful than arithmetic.
The analogy I was suggesting was:
Arithmetic = Bayesian Inference
Set Theory =Analogical Reasoning
If the above match-up is valid, from the above (Set/Category more
powerful than Arithmetic), it follows that analogical reasoning is
more powerful than Bayesian Inference, and Bayes cannot be the
foundation of rationality as many logicians claim.
The above match-up is justified by (Brown, Porter), who shows that
there's a close match-up between analogical reasoning and Category
Theory. See:
‘"Category Theory: an abstract setting for analogy and
comparison" (Brown, Porter)
http://www.maths.bangor.ac.uk/research/ftp/cathom/05_10.pdf
‘Comparison’ and ‘Analogy’ are fundamental aspects of knowledge
acquisition.
We argue that one of the reasons for the usefulness and importance
of Category Theory is that it gives an abstract mathematical setting
for analogy and comparison, allowing an analysis of the process of
abstracting
and relating new concepts.’
This shows that analogical reasoning is the deepest possible form of
reasoning, and goes beyond Bayes.
> I agree with your critics on Bayesianism, because it is a good tool
> but not a panacea, and it does not work for the sort of credibility
> measure we need in artificial intelligence.
The problem of priors in Bayesian inference is devastating. Simple
priors only work for simple problems, and complexity priors are
uncomputable. The deeper problem of different models cannot be
solved by Bayesian inference at all:
See:
http://74.125.155.132/search?q=cache:_XQwv9eklmkJ:eprints.pascal-network.org/archive/00003012/01/statisti.pdf+%22bayesian+inference%22+%22problem+of+priors%22&cd=9&hl=en&ct=clnk&gl=nz
"One of the most criticized issues in the Bayesian approach is related
to
priors. Even if there is a consensus on the use of probability
calculus to
update beliefs, wildly different conclusions can be arrived at from
different
states of prior beliefs. While such differences tend to diminish with
increas-
ing amount of observed data, they are a problem in real situations
where
the amount of data is always finite. Further, it is only true that
posterior
beliefs eventually coincide if everyone uses the same set of models
and all
prior distributions are mutually continuous, i.e., assign non-zero
probabili-
ties to the same subsets of the parameter space (‘Cromwell’s rule’,
see [67];
these conditions are very similar to those guaranteeing consistency
[8]).
As an interesting sidenote, a Bayesian will always be sure that her
own
predictions are ‘well-calibrated’, i.e., that empirical frequencies
eventually
converge to predicted probabilities, no matter how poorly they may
have
performed so far [22].
It is actually somewhat misleading to speak of the aforementioned
crit-
icism as the ‘problem of priors’, as it were, since what is meant is
often at
least as much a ‘problem of models’: if a different set of models is
assumed,
differences in beliefs never vanish even with the amount of data going
to
infinity."
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Received on Fri Aug 28 2009 - 01:47:13 PDT