marc.geddes wrote: > > > On Aug 27, 7:35 pm, Bruno Marchal <[email protected]> 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 > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
