Bayesian Problem-Solving and the Dispensibility of Truth

Ridgeview Publishing Company

Bayesian Problem-Solving and the Dispensibility of Truth Author(s): Paul Horwich Source: Philosophical Issues, Vol. 2, Rationality in Epistemology (1992), pp. 205-214 Published by: Ridgeview Publishing Company Stable URL: http://www.jstor.org/stable/1522863 Accessed: 05/12/2008 06:56 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=rpc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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Paul Horwich

One of the most active and successful research programs in current philosophy of science is based on the observation that belief is not an all-or-nothing matter, and is known as "probabilism" or "Bayesianism". The thesis of Bayesianism is that there are degrees of conviction which can be measured by the numbers between 0 and 1 and which should (ideally) obey the probability calculus.1 This point of view is well represented in the present volume by the contributions of Richard 1The probability calculus consists of the following axioms: 1. The probability of a proposition is a number greater than or equal to 0. 2. The probability of a tautology is 1. 3. The probability of a disjunctions of two logically incompatible propositions is equal to the sum of the probabilities of the disjuncts. 4. The conditional probability of A given B -written P(A/B)is equal to the probability of their conjunction divided by the probability of B.

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Jeffrey and Brian Skyrms; and I also am a keen proponent of it. But there is a general difficulty at the heart of the Bayesian program, and the purpose of this note is to propose a way of avoiding that difficulty. Here is the problem in a nutshell. On the one hand we have, in the thesis of Bayesianism, a cluster of ideas that enable us to solve a striking array of puzzles and paradoxes in the philosophy of science. But on the other hand these ideas are defective: some are plainly false, others patantly unjustified, and taken as a whole their picture of science is glaringly incomplete. Thus the Bayesian theory of science is both fruitful and objectionable. What should we do? Before saying how one might respond to this dilemma, let me be a little more specific about its character. The issues that Bayesianism enables us to treat are puzzles that hinge on the concept, strength of evidence. Consider for example the famous "paradox of the ravens". It is plausible to suppose that any hypothesis of the form 'All F's are G's' would be somewhat confirmed by the observation of an F that is also G. But if this is generally true then the discovery of a non-black non-raven (e.g. a red shoe) would confirm 'All non-black things are non-ravens', and thereby confirm the logically equivalent hypothesis, 'All ravens are black' -a seemingly bizarre conclusion. Here is a Bayesian solution to this problem. Regard the degree of support for hypothesis H provided by evidence E to be the factor by which a rational degree of belief in the truth of H would be increased by the discovery of E. Measure this quantity by the ratio of subjective probabilities (degrees of belief), P(H/E) P(H) Notice that this ratio, given conformity with the probability calculus (and, in particular, with Bayes' theorem2), must be For it gives 2Bayes' theorem follows trivially from axiom 4. P(H & E) = P(H/E) x P(E), and P(E & H) = P(E/H) x P(H). But P(H & E) = P(E & H). Therefore P(H/E) x P(E) = P(E/H) x P(H). = P(E/H)/P(E). Therefore P(H/E)/P(H)

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equal to P(E/H) P(E) Now focus on the hypothesis, 'All ravens are black', and compare the degree of support for it that would be provided by two possible discoveries: first, that something already known to be a raven turns out to be black (R * B); and second, that something already known not to be black turns out not to be a raven (-B * -R). In both cases the numerator, P(E/H), equals 1; for both discoveries are entailed by the truth of the hypothesis in question. But there is a significant difference between the prior probabilities, P(E), of the two items of evidence. Since an investigator would know that there are relatively few ravens in the world, she would antecedently expect a randomly selected object to turn out not to be a raven. Thus P(-B

* -R)=

1- e

(where e < 1).

In contrast, there are many colors a random object might have, and no particular reason to think that it would be black. So P(R*B) < 1. Therefore Degree of support of H by R B =

P(R * B/H) P(R * B) 1 P(R*B)>I

and

Degree of support = ofHby-B*-R

P(-B P(-B

* -R/H) *

-R)

1 P(-B*

-B)=+e'

Thus the support provided by R * B is substantial and the support provided by -B * -R is negligible. Our paradoxical conclusion was in fact correct. It looked bizarre, at first,

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because, having forgotten about degreesof belief, we confused negligible confirmation with no confirmation at all.3 Bayesianism deals with other issues similarly, combining the idea of confirmation as increase of rational degree of belief with the principle that rational degrees of belief must satisfy the probability calculus. Some of the further problems that can be handled is this way are: - why a broad spectrum of facts will confirm a theory more than a narrow data set. - what is wrong with ad hoc hypotheses. -

whether prediction has more evidential value than mere accommodation of data.

-why we ought to base our judgements on as much data as possible. - why surprising data have relatively great confirmation power. - how statistical hypotheses can be tested even though they are unfalsifiable. If all this can really be done, we shall have an impressive body of accomplishments.4 However it cannot be denied that the 3What is wrong with the paradoxical argument is, not only that the initially strange conclusion is in fact true, but also that its main premise -that any hypothesis of the form 'All F's are G's' would be confirmed by the observation of an F that is also G- is an oversimplification. As our analysis indicates, whether or not an observation confirms an hypothesis, and by how much, will typically depend on the investigator's background knowledge; and it is not difficult to devise circumstances in which the observation of an F that is G would reduce the probability that all F's are G's. In the present treatment I have assumed, for simplicity, that the background information includes the following facts: (a) that one of the observed objects of is a raven (its color remains to be seen), (b) that the other is not black (whether it is a raven or not remains to be seen), (c) that there are relatively few ravens around, and (d) that a substantial proportion of things are not black. 4For various attempts to solve these problems via Bayesian analysis see the papers of Janina Hosiasson-Lindenbaum, I.J.Good, Mary Hesse, my own Probability and Evidence, (Cambridge: Cambridge University Press, 1982) and Colin Howson and Frank Urbach's Scientific Reasoning: The Bayesian Approach (La Salle, Ill.: Open Court, 1989).

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Bayesian treatment of these problems rests on highly dubious foundations. Here are some of the main objections. In the first place, it seems farfetched to suppose that people actually have precise, numerical, degrees of belief. Can the mind really contain the state of believing to degree .15296 that quantum mechanics is true? That there are such states is a sophisticated psychological hypothesis without much empirical support. Had we evidence to suggest that Maximization of Expected Utility5 were a law of decision-making, then the existence of the theoretical quantities involved in this law -including numerical degrees of belief- would be supported. Or if we could justifiably suppose that peoples' preferences satisfied the axioms of von Neumann and Morgenstern, or Savage's axioms, then again it could be concluded that numerical degrees of belief are real states of mind. But it is well known that experimental results cast considerable doubt on these suppositions.6 Secondly, although there are various lines of reasoning to the conclusion that any degrees of belief that do exist must, if they are rational, satisfy the probability calculus, these considerations are not compelling.7 The best known of them is the 'dutch book' argument. If one defines a person's degree of belief in a proposition as a function of the odds at which he is prepared to bet on its truth, then it can be proved that 5The Expected Utility of a possible action is defined as the sum of the values of the states that might occur if it is performed, each value weighted by the agent's degree of belief that it will be obtained. Thus, if S1, S2,..., Sn, is an exhaustive set of mutually exclusive possible states; if V(S1), V(S2),..., V(S,), are their desirabilities to the agent; and if P(S1 /A), P(S2/A),..., P(Sn/A), are their subjective probabilities, conditional on the performance of action A; then the Expected Utility of action A is given by V(S1)P(S1/A)

+ V(S2)P(S2/A)

+ ** + V(Sn)P(Sn/A).

6For an overview of these issues see Paul Slovic, "Choice", in Thinking: An Invitation to Cognitive Science Vol.3, edited by D.Osherson and E.Smith (Cambridge, Mass.: MIT Press, 1990). 7For a good, recent discussion of these issues see John Earman, Bayes or Bust: A Critical Examination of Bayesian Confirmation Theory (Cambridge, Mass.: MIT Press, 1992).

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if his degrees of belief don't satisfy the probability calculus then he will be prepared to accept a collection of bets which is guaranteed to lead to a loss. Therefore, since it would surely be irrational for him knowingly to put himself in such a no-win situation, it would be irrational to have a system of degrees of belief that violates the probability calculus, QED. However, the definition of 'degree of belief' that is employed in this argument presupposes that people maximize their expected utility.8 And, as I have just mentioned, there is a lot of room for scepticism about that assumption (and about the preference axioms to which it is equivalent). So the 'dutch book' argument is far from airtight. Worse still, there is positive to think that its conclusion is false. For the probability calculus implies a certain logical omniscience: the probability of any tautology is 1 and of any contradiction is 0. Yet it is surely quite rational to be less than perfectly confident in the truth of some tautologies -those that are especially hard to prove- and quite rational to give non-zero degrees of belief to contradictions that are hard to recognize as such.9 A third problem is that Bayesians can't agree among themselves about the proper way to define 'the degree of confirmation of hypothesis H by evidence E', and the moral appears to be that no probabilistic definition is entirely adequate.10 8The definition says that person S's degree of belief in the proposition that q is equal to the ratio x/y just in case S is indifferent given the choice between receiving z units of value for certain and y units if and only if q is true. The propriety of this definition derives from the assumption that the value of the gamble, get y if and only if q is true, is equal to its Expected Utility, as defined above: equal, that is, to y times the probability of getting it -which is y times P(q). For, with that assumption, and the condition that S is indifferent between the gamble and the certainty of z, we infer x = y x P(q) i.e. P(q) = x/y. 9See Dan Garber, "Old Evidence and Logical Omniscience in Bayesian Confirmation Theory", in Testing Scientific Theories edited by J. Earman (Minneapolis: University of Minnesota Press, 1983). 10For an elaboration of this, and other objections to Bayesianism, see Clark Glymour's "Why I Am not a Bayesian", in his book Theory and Evidence (Princeton: Princeton University Press, 1978). Note, in particular, his alleged "problem of old evidence". There has not been space to discuss this here; but a good response is given by Howson and Urbach (op. cit.)

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Some Bayesians characterize degree of support as the difference between P(H/E) and P(H), some say it is the ratio of these probabilities, and a great variety of other formulas are suggested in the literature. All parties are able to point to cases where they capture our intuitions more successfully than the others, and there seems to be no clear way of telling which, if any, is correct. And fourthly, many interesting questions about science are not answered by Bayesianism.11 One might wonder, for example, whether there are any constraints on rational belief beyond conformity with the probability calculus, and if so what they are. Should we perhaps assign higher prior probabilities to simpler hypotheses? But if so, exactly how and why? Bayesianism leaves us in the dark. Thus the Bayesian theory of science is in various respects false, unjustified, and incomplete. Evidently, it would be irresponsible to employ the Bayesian theory and simply ignore these problems. But it would be a shame to reject the approach and deprive ourselves of the satisfying solutions which it provides. My aim here is to suggest that we can have our cake and eat it too. I want to propose that the Bayesian approach be used as an instrument to solve problems, without thinking of it as a theory of science. So the objections which reveal it to be a badtheory of science will have no force against it. This strategy rests on a metaphilosophical distinction between theory-oriented and problem-oriented philosophy. What I have in mind is that certain philosophical projects are directed towards relieving our ignorance in some domain -towards saying, for example, what our canons of epistemic justification are, when we can suppose that an action is morally right, which are the inference rules governing counterfactual conditionals, etc.- and the result in each case is a theory. On the other hand there are philosophical projects that are provoked by confusion rather than ignorance: we are in the grip of a paradox, and the result, if we are successful in 11Complaints of this sort are made by Glymour and Earman, amongst others.

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solving it, is nothing except an untangling of the conceptual knots that we had tied ourselves into. In the study of scientific reasoning the main theory-oriented project begins with data from the history of methodological practice and attempts to infer deep and general principles describing the underlying nature of science. Such a theory about science, like any theory in science, aspires to truth, breadth, empirical adequacy, explanatory power, and consistency with neighboring branches of knowledge. Approximations are tolerated along the way. But they are nonetheless intended to explain and predict, at least within certain limits; and there is always the hope that a better theory will eventually be found. The goal is a complete and perfectly true account that will penetrate beneath the uncontroversial facts of intuition and practice and thereby explain them. Unfortunately, as the above-mentioned difficulties show, there is little reason to think that Bayesianism is going to be of much use here. Suppose, however, that one's interest in the philosophy of science is primarily in the resolution of the well known puzzles and paradoxes, and not with the development of a systematic theory of science. In that case Bayesianism can be helpful. For many of the problems grow out of confusion arising from a tendency to ignore degrees of belief, and it may well be that the thesis of Bayesianism contains just the right balance of accuracy and simplicity to get a clear view of the issues and enable us to see where we were going wrong. From this problem-oriented perspective it doesn't matter if the Bayesian ideas constitute a poor theory of science, because they arn't intended to provide a theory of science in the first place, but merely a perspicuous representation of uncontroversial intuitions. So the objections I mentioned above will be beside the point. Let us reconsider them in turn. First, perhaps there really are no precise, numerical, degrees of belief. Even so, this would not undermine the program of Bayesian problem solving. For there are certainly are belief gradations of some sort; their representation by numbers should be seen as nothing more than a heuristic device -something we do for the sake of convenience, without committing ourselves to its truth. In dealing, for example,

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with the 'raven paradox' the crucial point is that the datum, -B * -R, is much less surprising than R * B- indeed it is almost completely unsurprising. Therefore, although both data are predicted by the hypothesis, only R* B substantially confirms it. The symbolic formulation of these intuitive ideas helps the solution merely by fostering clarity and rigor. Similarly, the picture of rational degrees of belief obeying the probability calculus should be regarded as an idealization. It is uncontroversial (a) that one ought to be certain of elementary logical truths, and (b) that one ought not be confident in the truth of obviously incompatible hypotheses. The probability calculus provides a sharp, perspicuous way of capturing these trivialities, and to the extent that it goes beyond them it need not be taken as purporting to be true. As for the question of which function we should use to measure "degree of support", the answer again is that we pick one rather than another solely on grounds of utility. The only fact that must be respected is that data provide support for an hypothesis to the extent that they ought to enhance its credibility. As long as this uncontroversial intuition is preserved, none of our applications of Bayesianism will depend essentially on which definition of "degree of support" is used. To see this, suppose we had used the function P(H/E)P(H) to measure "degree of support" in the case of the ravens. This quantity, given Bayes' Theorem, is equal to 1 P(H) P(E/H) P(E) which tends to 0 when E is an unsurprising prediction. So the result that -R * -B will confirm 'All ravens are black' negligibly, whereas R * B will confirm it substantially, does not depend on which measure of support is used, as long as the measure conforms to our crude intuitions. Finally, it should be clear that the thesis of Bayesianism might well illuminate our problems sufficiently to help us resolve them, without offering anything like a complete picture of scientific methodology. Bayesianism does not tell us how to decide whether a hypothesis is projectible. It does not

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specify how a person's state of belief should change under the impact of new data and the recognition of new theoretical possibilities. It doesn't solve the traditional problem of induction. But so what? The desire for a complete and perfectly correct theory of science is not what Bayesianism need be intended to satisfy. We may think of the model, rather, as an idealization whose function is to cast our crude uncontroversial ideas into a clear form where their joint implications can be most easily discerned. This is not at all to say that it can be of no interest to investigate more realistic models of belief, more sensitive epistemological norms, more accurate measures of confirmation, and more extensive areas of methodology. If one is doing "epistemology naturalized", and therefore trying to obtain a theory of science that is true and systematic, then such investigations are entirely appropriate. But if the point is to solve the many problems whose origin is the oversimplification that belief is an all-or-nothing matter, then it is by no means clear that anything more realistic than the Bayesian idealization is needed. It appears to offer the ideal compromise between accuracy and simplicity -enabling us to represent the issues starkly without neglecting their essential ingredients or obscuring them with unnecessary detail.