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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, 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." --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---