> On Aug 27, 7:35 pm, Bruno Marchal <marc...@ulb.ac.be> 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
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".
> ‘"Category Theory: an abstract setting for analogy and
> comparison" (Brown, Porter)
> ‘Comparison’ and ‘Analogy’ are fundamental aspects of knowledge
> 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
> 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
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.
> "One of the most criticized issues in the Bayesian approach is related
> priors. Even if there is a consensus on the use of probability
> calculus to
> update beliefs, wildly different conclusions can be arrived at from
> states of prior beliefs.
A feature, not a bug.
>While such differences tend to diminish with
> ing amount of observed data, they are a problem in real situations
> 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
> beliefs eventually coincide if everyone uses the same set of models
> and all
> prior distributions are mutually continuous, i.e., assign non-zero
> ties to the same subsets of the parameter space (‘Cromwell’s rule’,
> see ;
> these conditions are very similar to those guaranteeing consistency
> As an interesting sidenote, a Bayesian will always be sure that her
> predictions are ‘well-calibrated’, i.e., that empirical frequencies
> converge to predicted probabilities, no matter how poorly they may
> performed so far .
> It is actually somewhat misleading to speak of the aforementioned
> 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
> differences in beliefs never vanish even with the amount of data going
But some models are more probable than others.
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