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Scientific Reasoning: The Bayesian Approach by Colin Howson (2005-04-10)

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  • By Michael Emmett Brady on December 25, 2008

    Howson and Urbach are totally committed to the subjectivist,Bayesian approach to probability founded by Ramsey,De Finetti,and Savage.This approach is based on the misbelief that all probability estimates are precise,exact ,unique single numbers.The decision maker can initially hold any subjective belief he wants to entertain as long as he is willing to incorporate additional evidence over time using Bayesian updating.This means that the decision maker uses the mathematical laws of the probability calculus (the addition and multiplication rules)to update his subjective probabilities so that they are consistent over time with the mathematical laws of the probablity calculus.His beliefs will be coherent and not subject to having a Dutch book made against him.Whatever his initial a priori ,subjective estimates of the probabilities were,the updated versions will start to converge to the correct a posteriori probabilities.The allegiance to the use of the mathematical laws of the probability calculus guarantee a decision maker that over time his assessments will be consistent. The problems with this approach are based on the unspecified assumptions that there is always sufficient evidence to specify a sample space or unique probability distribution.This requires that there be NO uncertainty,vagueness,unclearness,ambiguity,or conflict in the evidence used to specify the probability distribution.Keynes specifies this by saying that the weight of the evidence is complete or equal to 1 on the unit interval [0,1].Ellsberg would specify this by requiring that rho =1,where rho is specified on the unit interval[0,1].Urbach and Howson simply assume that there will always be sufficient information that is clear in the present or future so that they can specify their distribution.Nowhere in this book do they show or discuss the difficulty in showing that w=1 or rho=1.Urbach and Howson just assume it. The second problem is the assumption that all probability estimates are precise exact ,single numbers. Keynes founded his logical theory of probability on the interval estimate approach of George Boole.Theodore Hailperin has shown since 1965 that all of the Boole-Keynes problems using lower-upper probabilities (Howson and Urbach are completely ignorant of the fact that Keynes and Boole,and not Good,Smith,Dempster,etc.,are the founders of the interval,upper-lower probabilities approach to probability )can be translated into linear programming problems.This completely destroys the claims that (a) all a priori logical probabilities must be specified using the principle of indifference and (b) that the logical probability approach is not operational.Urbach and Howson need to completely rewrite their discussions of logical probabiity and upper-lower probability to take into account what it was that Keynes actually did and not what Frank Ramsey and Bruno De Finetti think that he did.The discussions in the book about Logical probability contain so many errors of commission and omission that I recommend that a reader skip them entirely.It turns out that the Bayesian ,Subjectivist approach of Howson and Urbach is a special case that arises when (a) rho and /or w =1 on the interval between 0 and 1 and (b) the lower probability estimates are equal to the upper probability estimates.Recently, Kadane ,Schervish,and Siedenfeld have called into question the entire logical framework of the Bayesian,Subjectivist approach to probability in their " Rethinking the Foundations of Statistics "(Cambridge University Press,1999).They have completely thrown in the towel and admitted implicitly that Keynes was right and Ramsey,De Finetti,and Savage were incorrect.The book is composed of 16 essays,all of which have been previously published in academic journals and/or other books.Throughout the book,the authors concede that there are many holes and deficiencies in the logical foundations of the subjectivist approach.The biggest hole is vastly understated by KSS: " In fact,it seems reasonable to deny that there are consequences in practical decisions.Thus,our position is that,lacking consequences,expected utility theory must treat probability distributions as extraneous(italicized)..."(KSS,1999,p.195).Of course,"...lacking consequences...", means that the outcomes are state independent.On pp.157-160,KSS had already demonstrated the near impossibility in the real world of being able to specify outcomes that would make their utilities state independent.Given that the subjective probabilities are completely extraneous,there is no longer any way in which the a priori beliefs of a decision maker can be represented by a unique probability distribution on purely decision theoretic grounds alone.This means that unique,definite,precise,numerical ,single,exact,hard,sharp point estimates of subjective probabilities do not exist.This result goes to the heart of the entire edifice erected by Ramsey,De Finetti,and Savage,in particular.Savage argued that,based on a careful elicitation of subjective preferences based on betting quotients,a unique probability measure(distribution)can be defined to represent the agent's preference relation.Nowhere is it stated by KSS that this position( of Savage) goes to the heart of the dispute between Keynes and Ramsey about the inherent indeterminateness of many probability estimates.The comparative -interval estimate-approach to estimating probabilities,presented by Keynes for the first time in the A Treatise on Probability(TP;1921,pp.160-163,pp.186-194),is fully operational since all of the problems Keynes presented as examples in the TP in chapters 15 and 17, that used the difficult Boolean approach, can instead use the substantially easier integer-mixed integer linear programming approach of Theodore Hailperin in order to obtain solutions.It appears that Ramsey's approach is a very special case of Keynes's approach that is applicable only when a single,unique probability distribution can be specified a priori. KSS's technical results,first published in 1990 in the Journal of the American Statistical Association(JASA), totally undermines the logical ,decision theoretic foundations of the subjectivist approach to estimating probabilities.Howson and Urbach have overlooked the numerous logical deficiencies in the current Bayesian approach.Keynes has been right all along.Howson and Urbach's book needs to be completely rewritten .Nevertheless, I recommend that the book be bought.The reader ,however,needs to realize that the theory presented in this book by Urbach and Howson is a special theory,not a general one.

  • By Viktor Blasjo on May 24, 2009

    Science is probability. Science is about degrees of belief: should I or should I not believe such-and-such a hypothesis? Such degrees of belief are probabilities (i.e., satisfy the probability axioms). This follows from the assumption that our degrees of belief determine a coherent assignment of fair odds, and the theorem that odds are coherent (i.e., cannot be Dutch-booked) if and only if they satisfy the probability axioms.Beliefs are updated in accordance with Bayes' theorem. This does actually not follow from the above: one could in principle go from one coherent set of beliefs to another in an erratic manner without being susceptible to Dutch books (despite what some people say, "diachronic" Dutch books do obviously not indicate inconsistency or irrationality). But it follows with a reasonable additional assumption: namely, that a scientists should not change his P(h/e)'s as the evidence e comes in, i.e., the new P'(h/e)'s should be equal to the old P(h/e)'s. This is reasonable because when forming P(h/e) we have already imagined that e were true; that its truth is not factual rather than imagined should not make a difference. With this assumption we see that P'(h)=P'(h/e)=P(h/e), which is precisely the conditionalisation rule.Bayes' theorem P(h/e)=P(e/h)P(h)/P(e) reflects scientific practice. To what extent e supports h obviously depends on the extent to which h implies e, which is reflected in the term P(e/h), on a scale from entailment P(e/h)=1 to refutation P(e/h)=0. The term P(h) reflects the fact that some hypotheses are a priori more reasonable (simple, non-ad hoc, etc.) than others. The term P(e) assures that surprising evidence carries more weight.The alleged further explanatory success of Bayesianism is unimpressive. No one should be impressed, for example, that the theory can be made to conform with various aspects of science (e.g., "solve" the Duhem problem) by just-so assignments of probabilities. Our authors also make a big deal out of the rather plain fact their approach is successful in purely probabilistic settings (e.g., statistical tests); but these are not impressive arguments for Bayesianism, as they are merely instances of probability theory conforming with itself.Objections:Problem of old evidence. Known e has P(e)=1, so by Bayes' theorem it can never support any hypothesis. The proposed solution is that conditionalisation should take place with respect to background knowledge "minus e." But whatever this is supposed to mean our authors cannot say, except in the trivial case where e is independent of everything else in the background knowledge.Bayesian conditionalisation does not allow prediction any privilege over accommodation. Our authors' reply is that such a difference can be accounted for by appeal to priors or background knowledge. Sure enough, since Bayesianism does not regulate these factors in any way, they can of course be used to account for anything. The irony in this being an obvious case of ad hoc accommodation should be clear.Paradox of evidence reinforcing priors. Before I flip a coin for the first time I do not know if it is biased; I assign priors 1/2. Now suppose I flip it 1000 times and it comes out precisely that way. Bayesianism makes the posterior probability the same as the prior without reflecting the fact that the probability is now more securely established. This problem can be solved by allowing probabilities to be distributions rather than numbers.Subjectivism. Subjective Bayesianism allows arbitrary priors, so it is a "confirmation theory" that relies on opinion as much as fact. The reply is confused. On the one hand there is the analogy with logic suggesting that it is not the premises but the rules of inference that are the proper subject matter, which is fine but obviously dodges the question (i.e., what are we justified in believing?). On the other hand we read that: "The prescription of the same 'objective' prior probability for everybody in the same knowledge state is a prescription for stagnation and eventual catastrophe, as is the suppression of dissent quite generally." This makes no sense. Why celebrate dissent in the priors but suppress it in conditionalisation? Indeed, it is claimed on the same page that "experience is allowed to dominate prior beliefs ...; disagreement is not eradicated at once, but its effect is usually falls off quickly." Bayesianism suppresses dissent, in other words. Or so our authors claim. But this "usual" washing out of the priors is a convenient fiction: the word "usually" here is in effect code fore "when one plugs in the kind of numbers that gives the desired results."


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