marc.geddes wrote:
> On Aug 27, 7:35 pm, Bruno Marchal <> 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)
> ‘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.

> See:
> "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.


> > 

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