Inference to the Best Explanation, Dutch Books, and Inaccuracy Minimisation

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    Bayesians have traditionally taken a dim view of the Inference to the Best Explanation (IBE),arguing that, if IBE is at variance with Bayes rule, then it runs afoul of the dynamic Dutchbook argument. More recently, Bayes rule has been claimed to be superior on grounds of condu-civeness to our epistemic goal. The present paper aims to show that neither of these argumentssucceeds in undermining IBE.

    The inference rule called Inference to the Best Explanation (IBE) assignsconfirmation-theoretic import to explanatory considerations. According tosome, IBE is the cornerstone of scientific methodology.1 But critics haveargued that, if IBE is at variance with Bayes rule, then, like any othersuch rule, it is to be rejected as leading to irrational belief updates.2 Formany years, the standard argument for this claim has been the so-calleddynamic Dutch book argument, which purports to show that updating byany rule other than Bayes makes one liable to sure financial losses. How-ever, even Bayesians themselves have increasingly come to regard this ar-gument as addressing the wrong issue, to wit, that of whether it isrational from a practical, rather than an epistemic, viewpoint to deviatefrom Bayes rule. This has led some theorists to pursue a different strategy

    1 See, for example, R. Boyd, The Current Status of Scientific Realism in J. Leplin(ed.), Scientific Realism (University of California Press, 1984), pp. 4182, and E. McMullin,The Inference that Makes Science (Marquette University Press, 1992).

    2 We will be throughout concerned with update rules applicable to learning events inwhich an agent becomes certain of a proposition of which he or she was previously uncer-tain. Bayesians acknowledge that other types of learning event call for different updaterules. See, for example, R. Jeffrey, The Logic of Decision (University of Chicago Press, 2nded., 1983), Ch. 11, and B. C. van Fraassen, Laws and Symmetry (Oxford UP, 1989), Ch. 13.

    The Philosophical Quarterly Vol. 63, No. 252 July 2013ISSN 0031-8094 doi: 10.1111/1467-9213.12032

    2013 The Author The Philosophical Quarterly 2013 The Editors of The Philosophical QuarterlyPublished by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford ox4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA

  • in defence of Bayes rule, a strategy that is purported to offer a distinc-tively epistemic argument in favour of that rule, one spelled out in termsof inaccuracy minimisation. Roughly, the argument is that by updatingvia any non-Bayesian rule, ones degrees of belief are not as accurate asthey would have been had one updated via Bayes rule.

    This paper aims to show that neither of the aforesaid arguments succeedsin undermining IBE. The first part of the paper argues that, while currentdevelopments in mainstream epistemology may help to deflect some of thecriticism the dynamic Dutch book argument has met with, the argumentfails nonetheless, since it rests on an unfounded (and unstated) premise. Thesecond part focuses on the inaccuracy-minimisation defence of Bayes rule,arguing that there appear to be several equally legitimate ways to interpretthe notion of inaccuracy minimisation, and using computer simulations toshow that under some of them it may be IBE rather than Bayes rule thatdoes best with regard to inaccuracy minimisation.


    For many decades, the Ramseyde Finetti Dutch book argument hasbeen viewed as key to the Bayesian account of rationality. According tothis argument, we are susceptible to Dutch bookscollections of betsensuring a negative net pay-off come what mayprecisely if ourdegrees of belief violate the axioms of probability. From this, Ramseyand de Finetti concluded that rational degrees of belief are formallyprobabilities.

    Ian Hacking may have been the first to observe that the RamseydeFinetti argument in fact does nothing to justify Bayes rule.3 However, afew years after the publication of Hackings paper, Paul Teller reported aDutch book argumentwhich he attributed to David Lewisaimed atjustifying Bayes rule as the only rational update rule.4 This dynamicDutch book argument (as it is now called) purports to show that if a per-son updates by some rule other than Bayes, she can be offered a series ofbets at different points in time such that each bet will seem fair at thetime it is offered, yet jointly the bets guarantee a financial loss. What isworsethe argument continuesthe person could have seen this loss

    3 I. Hacking, Slightly More Realistic Personal Probabilities, Philosophy of Science, 34,1967, pp. 31125.

    4 P. Teller, Conditionalization and Observation, Synthese, 26, 1973, pp. 21858. Thesource of the argument reported by Teller was later published as D. Lewis, Why Condi-tionalize? in his Papers on Metaphysics and Epistemology (Cambridge UP, 1999), pp. 4037.


    2013 The Author The Philosophical Quarterly 2013 The Editors of The Philosophical Quarterly

  • coming. This vulnerability to dynamic Dutch books has convinced manythat non-Bayesian updating is a mark of irrationality.5

    Recently, however, the Dutch book approach to defending Bayesianismhas come under a cloud. Critics have argued that when we are concerned withthe rationality of degrees of belief as well as the change thereof over time,we are concerned with questions of epistemic rather than practical rationality.Given that being vulnerable to cunning bookies seems primarily a practicalliability, the Dutch book arguments have been said to be beside the point.6

    In response to this, some (e.g., Brian Skyrms7) have claimed that Dutchbook vulnerability does flag an underlying epistemic defect: it is a mani-festation of the fact that a person deems one and the same bet or seriesof bets as both fair and not fair and thus is in an inconsistent state ofmind. Even if this claim is true for the (static) Ramseyde Finetti argu-ment,8 the point does not carry over to the dynamic Dutch book argu-ment. An agent may be susceptible to engage in the kind of betting overtime that figures in that argument without at any one time holding incon-sistent views on the fairness of any bets. Naturally, after a learning eventshe may regard a bet as unfair that previously she regarded as fair, butthe same would have been true had she been a Bayesian learner.

    Still, Bayesians may not be altogether defenceless against the above cri-tique. In particular, they may be able to get some mileage out of the prag-matic turn that a number of epistemologists are currently taking. Theepistemic status of a belief has traditionally been thought to depend solelyon matters that bear on the truth of the belief, like the quality of ones evi-dence or whether or not one is reliably connected to what the belief isabout. But over the past years, various authors have argued that the episte-mic status of a belief is inextricably bound up with the believers practical sit-uation, in particular, with what is at stake for her in believing correctly.9

    Bayesians wishing to maintain the integrity of the Dutch book defence maynot want to buy into any particular one of the arguments that have been

    5 We should actually speak of putative vulnerability to dynamic Dutch books: that anon-Bayesian updater is bound to regard all bets in a dynamic Dutch book as fair has beendisputed in I. Douven, Inference to the Best Explanation Made Coherent, Philosophy ofScience, 66, 1999, pp. S42435; see also M. Tregear, Utilising Explanatory Factors in Induc-tion?, British Journal for the Philosophy of Science, 55, 2004, pp. 50519.

    6 See J. Joyce, A Nonpragmatic Vindication of Probabilism, Philosophy of Science, 65,1998, pp. 575603, Sect. 2, and references given there.

    7 B. Skyrms, Coherence in N. Rescher (ed.), Scientific Inquiry in Philosophical Perspective(University Press of America, 1987), pp. 22541.

    8 But see Joyce, A Nonpragmatic Vindication of Probabilism, p. 585 f, for a critique.9 See, for example, J. Fantl and M. McGrath, Evidence, Pragmatics, and Justification,

    Philosophical Review, 111, 2002, pp. 6794, and J. Stanley, Knowledge and Practical Interests(Oxford UP, 2005).


    2013 The Author The Philosophical Quarterly 2013 The Editors of The Philosophical Quarterly

  • advanced in favour of this pragmatic encroachment view (as it has beencalled). However, it may suffice for them to argue that the mere prominencein current epistemology of the debate on pragmatic encroachment is enoughto call into question the existence of the clear-cut divide between the episte-mic and the pragmatic that the critics of the Dutch book arguments are pre-supposing. Forit may be saidif there were such a clear-cut divide,contributions to this debate should have gone down like lead balloons. AndBayesians may concludeif there is no such divide, then little is left of thecharge that Dutch book arguments address the wrong type of rationality.

    Be this as it may, there is a deeper problem with the dynamic Dutchbook argument, one that remains even if the pragmatic encroachmentview is endorsed. For note that Dutch book invulnerability is only oneamong an in principle indefinite number of practical interests that peoplemay have. What if IBE, or some other non-Bayesian rule, serves otherpractical interests better than Bayes rule? Could the Bayesian in that casestill maintain that, in view of the dynamic Dutch book argument, Bayesrule is the only rational update rule? Surely there exist practical goalswhose achievement would more than make up for running the risk ofbeing fleeced by a Dutch bookieespecially in view of the fact thatDutch bookies only occur as fictional characters in philosophers tales!

    To show that IBE may indeed have compensating practical advantages,we use a particular probabilistic version of IBE that we apply in the contextof a simple statistical model. Let fHigiOn be a set of self-consistent, mutuallyexclusive, and jointly exhaustive hypotheses, and Pr ones probability functionprior to learning E. Then, according to the version of IBE to be considered,ones new probability for Hi after learning E (and nothing stronger) equals

    PrHiPrEjHi f Hi;EPnj1 PrHjPrEjHj f Hj ;E


    where f is a function that assigns a bonus point (> 0) to the hypothesis (orhypotheses, in case of a tie) that explain(s) E best in light of the back-ground knowledge, and assigns nothing to the other hypotheses.10

    For present concerns, all we need is a definition of best explanation forthe following statistical model. Let fHig0OiO10 be a set of bias hypothesesconcerning a given coin, where Hi is the hypothesis that the bias for heads

    10 A number of authors, including Peter Lipton and Jonathan Weisberg, have proposedversions of IBE that are compatible with Bayesianism; see P. Lipton, Inference to the BestExplanation (Routledge, 2nd ed., 1004), Ch. 7, and J. Weisberg, Locating IBE in the Bayes-ian Framework, Synthese, 167, 2009, pp. 12543. Note that the present version is equivalentto Bayes rule if, and only if, there is no best explanation among the hypotheses andf assigns 0 to all of them.


    2013 The Author The Philosophical Quarterly 2013 The Editors of The Philosophical Quarterly

  • is i/10. These hypotheses are supposed to be jointly exhaustive. Let Ej indi-cate that the outcome of the j-th of a series of tosses with the given coin is E.Then we say that Hi best explains Ej iff i/10 is closer to the actual frequencyof heads in the first j tosses than k/10, for all k {0,,10} different from i.For definiteness, in this case let f Hi;Ej :1. If, for some i and k, i/10 andk/10 are equally close to the actual frequency of heads, and closer than l/10for all l different from i and k, the bonus is split: f Hi;Ej f Hk ;Ej :05. No other hypotheses receive bonuses. So, for instance, if681 heads have been observed in a series of 999 tosses and the 1000th toss isagain heads, then in this model, and absent any further information aboutthe coin, H7 provides the best explanation of that last outcome and thusreceives a bonus if updating proceeds by the above rule.

    We will continue to use IBE as a label for the broad idea that explana-tory considerations have confirmation-theoretic import and use IBE (in sansserif font) to designate our probabilistic explication of that idea. It is to beemphasised that nothing in the following will hinge on whether IBE is thebest or even a satisfactory explication of IBE. The main role of IBE will be inbringing into relief hidden, problematic premises in the dynamic Dutchbook and inaccuracy-minimisation arguments. If we have failed to capturethe notion of best explanation even for the above simple model, that willnot make those arguments valid without the hidden premises, nor will itmake these premises appear unproblematic.

    Now, suppose the Bayesian and the explanationist (as we shall henceforthcall the IBE-updater) are watching the same sequence of coin tosses to whichthe above model pertains. Both have started with a flat probability distribu-tion over the eleven bias hypotheses, and they update their probabilities forthese hypotheses via their respective update rules. It has been said that aproposition is assertable to the extent that it has high subjective probabilityfor its assertor.11 If so, we may ask whothe Bayesian or the explanationist

    11 F. Jackson, On Assertion and Indicative Conditionals, Philosophical Review, 88, 1979,pp. 56589, at p. 565. It has been argued that a proposition is assertable if it is rationallyacceptable; see, for example, I. Douven, Assertion, Knowledge, and Rational Credibility,Philosophical Review, 115, 2006, pp. 44985 and Assertion, Moore, and Bayes, PhilosophicalStudies, 144, 2009, pp. 36175. This yields Jacksons view cited in the text if high subjectiveprobability suffices for rational acceptability. That might seem too simplistic, in particularin view of so-called lottery propositions (propositions to the effect that a given ticket in alarge fair lottery with only one winner will lose), for, although highly probable, such propo-sitions do not at all seem rationally acceptable. But see I. Douven, The Lottery Paradoxand the Pragmatics of Belief, Dialectica, 66, 2012, pp. 35173, and Putting the Pragmaticsof Belief to Work in A. Capone, F. Lo Piparo, and M. Carapezza (eds), Perspectives on Prag-matics and Philosophy (Springer, in press), for a proposal on which high probability is enoughfor rational acceptability, with the exception of lottery propositions, which are excluded onprincipled grounds.


    2013 The Author The Philosophical Quarterly 2013 The Editors of The Philosophical Quarterly

  • is more likely to first be in a position to assert the truth about the bias ofthe coin, supposing the sequence is long enough for both eventually to be inthat position.

    To answer this question, we ran computer simulations of sequences ofcoin tosses that were long enough fo...