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From: Brent Meeker <meekerdb.domain.name.hidden>

Date: Fri, 28 Aug 2009 10:21:06 -0700

marc.geddes wrote:

*>
*

*>
*

*> 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,
*

From analogies are only suggestive - not proofs.

*>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.
*

But did Brown and Porter justify Arithmetic=Bayesian inference? ISTM

that Bayesian math is just rules of inference for reasoning with

probabilities replacing modal operators "necessary" and "possible".

*> 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.
*

Look at Winbugs or R. They compute with some pretty complex priors -

that's what Markov chain Monte Carlo methods were invented for.

Complex =/= uncomputable.

*> The deeper problem of different models cannot be
*

*> solved by Bayesian inference at all:
*

Actually Bayesian inference gives a precise and quatitative meaning to

Occam's razor in selecting between models.

http://quasar.as.utexas.edu/papers/ockham.pdf

*>
*

*> 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.
*

A feature, not a bug.

*>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.
*

And beliefs do not converge, even in probability - compare Islam and

Judaism. Why would any correct theory of degrees of belief suppose

that finite data should remove all doubt?

*>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."
*

But some models are more probable than others.

Brent

*>
*

*>
*

*> >
*

*>
*

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Received on Fri Aug 28 2009 - 10:21:06 PDT

Date: Fri, 28 Aug 2009 10:21:06 -0700

marc.geddes wrote:

From analogies are only suggestive - not proofs.

But did Brown and Porter justify Arithmetic=Bayesian inference? ISTM

that Bayesian math is just rules of inference for reasoning with

probabilities replacing modal operators "necessary" and "possible".

Look at Winbugs or R. They compute with some pretty complex priors -

that's what Markov chain Monte Carlo methods were invented for.

Complex =/= uncomputable.

Actually Bayesian inference gives a precise and quatitative meaning to

Occam's razor in selecting between models.

http://quasar.as.utexas.edu/papers/ockham.pdf

A feature, not a bug.

And beliefs do not converge, even in probability - compare Islam and

Judaism. Why would any correct theory of degrees of belief suppose

that finite data should remove all doubt?

But some models are more probable than others.

Brent

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To post to this group, send email to everything-list.domain.name.hidden

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Received on Fri Aug 28 2009 - 10:21:06 PDT

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