Abstract
The first step is to distinguish two questions:
-
1.
Given the data, what should we believe, and to what degree?
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2.
What kind of evidence do the data provide for a hypothesis H 1 as against an alternative hypothesis H 2 , and how much?
We call the first the “confirmation”, the second the “evidence” question. Many different answers to each have been given. In order to make the distinction between them as intuitive and precise as possible, we answer the first in a Bayesian way: a hypothesis is confirmed to the extent that the data raise the probability that it is true. We answer the second question in a Likelihoodist way, that is, data constitute evidence for a hypothesis as against any of its rivals to the extent that they are more likely on it than on them. These two simple ideas are very different, but both can be made precise, and each has a great deal of explanatory power. At the same time, they enforce corollary distinctions between “data” and “evidence”, and between different ways in which the concept of “probability” is to be interpreted. An Appendix explains how our likelihoodist account of evidence deals with composite hypotheses.
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Notes
- 1.
See Royall (1997). That the distinction between belief and evidence questions is pre-theoretically intuitive is underlined by the fact that Royall himself is a Likelihoodist who eschews any reference to an agent’s subjective degrees of belief (he is, however, a Bayesian in regard to the decision question). Despite the philosophical differences that one or another of us has with him, our monograph owes a great deal to his work. See in particular Royall (2004).
- 2.
For a Bayesian response to Royall’s three questions, see Bandyopadhyay (2007).
- 3.
Mark Kaplan (1996) is one of the very few philosophers to take note of the confirmation/evidence distinction (p. 25, footnote 32), but his argument for making it seems to involve no more than a reference to Nelson Goodman (quoted on p. 26, footnote 34), to the effect that “[a]ny hypothesis is ‘supported’ by its own positive instances; but support … is only one factor in confirmation.” Kaplan’s own very interesting and extensive account of evidence is itself generally “Bayesian” and makes no use of likelihoods. Goodman thinks that since incompatible and arbitrary hypotheses are “supported by the same evidence,” there must be another “linguistic” (data-independent) factor involved in “confirmation.” As we will see in Chap. 9, Goodman’s well-known “grue paradox,” which he uses to argue for this claim, depends on running “confirmation” and “evidence” together. Others who have made a confirmation/evidence distinction include Ellery Eells and Branden Fitelson in their (2000) and Malcolm Forster and Elliott Sober in their (2004). Forster and Sober are neither Bayesians nor Likelihoodists, which fact underlines our claim that the distinction is pre-theoretical, that is to say, statistical-paradigm independent.
- 4.
Earman (1992).
- 5.
Although most of the hypotheses we will use to illustrate our argument do not take the form of universal conditionals, “All A are B,” it is especially clear in their case that the claims they express typically outrun the inevitably finite data gathered to support them.
- 6.
Understanding probabilities as degrees-of-belief and connecting them to confirmation has a long history. See, for example, (Keynes 1921, pp. 11–12). Carnap, too, thought that inductive probability, i.e., the probability of the conclusion of an inductive argument given its various data premises, is “ascribed to a hypothesis with respect to a body of evidence… To say that the hypothesis h has the probability p (say 3/5) with respect to the evidence e, means that for anyone to whom this evidence and no other relevant knowledge is available, it would be reasonable to believe in h to the degree p, or, more exactly, it would be unreasonable for him to bet on h at odds higher than [p(h)/p(1-h]).… Thus inductive probability measures the strength of support given to h by e or the degree of confirmation of h on the basis of e (Carnap 1950, p. 441) As Skyrms (1986, p. 167) summarizes the situation, the concepts of inductive and epistemic (which, as we saw in Chap. 1, applies to statements rather than arguments) probabilities were introduced … as numerical measures grading degree of rational belief in a statement and degree of support the premises give its conclusion…. Why should epistemic and inductive probabilities obey the mathematical rules laid down for probabilities and conditional probabilities? One reason that can be given is that these mathematical rules are required by the role that epistemic probability plays in rational decision” (our italics). James Hawthorne helped prepare this brief history. See his (2011). For our purposes, it is as important to note that neither Keynes, nor Carnap, nor Skyrms distinguishes between confirmation and evidence (as the title of the selection from Carnap’s work in Achinstein 1983, “The Concept of Confirming Evidence,” makes clear).
- 7.
Although it is not needed for our argument, it is worth mentioning that a confirmation-generalization of logical entailment has been worked out by Crupi et al.(2013).
- 8.
On the traditional account of voluntary action, an action is voluntary just in case it is performed by a rational agent on the basis of her desires and beliefs. For Skyrms and other Bayesians, “rational agency” requires at a minimum that the agent’s beliefs conform to the rules of the theory of probability.
- 9.
There are other ways in which to model belief within the context of a confirmation theory, for example, the Dempster-Shafer belief function. See Shafer (1976). Since the probability-based account is well-known and has a long tradition, we are resorting to it.
- 10.
See Bandyopadhyay and Brittan (2010).
- 11.
Except when it is important, we leave out reference to the background information in what follows.
- 12.
Or as it is sometimes called, “the positive relevance condition.” See Salmon (1983) for an extended argument in behalf of the primacy of this condition in an analysis of confirmation.
- 13.
Following Crupi et al. (2013). The article includes a long list of Bayesians who subscribe to this principle.
- 14.
Clark Glymour (in an e-mail comment to us) and Peter Achinstein (2001, especially Chap. 4) object that this sort of account has a counter-intuitive consequence, that the same data could confirm incompatible hypotheses to different degrees. But so long as our assignments of degrees of belief are consistent, i.e., do not violate the rules of probability theory, it is possible to be rationally justified in believing incompatible hypotheses to different degrees on the basis of the same data.
- 15.
That is to say, we use this as an exemplary measure of degree of confirmation. Many others are possible. See Fitelson (1999), for a discussion of the sensitivity of confirmational values to the measure used. We believe that the choice of a specific confirmation measure depends on the type of question one is asking. The same idea, that the type of question asked determines the measure chosen, applies to the evidence question as well. Although we have adopted the likelihood ratio to weigh evidence, different evidential measures would be required if we were to ask a different set of evidential questions. See Taper et al. (2008) and Chap. 5 for further discussion of alternative measures.
- 16.
In the eyes of many statisticians this notation signals the difference between “probability” and “likelihood,” as two different concepts. More than a simple notational difference is involved. The “│” notation indicates conditioning on a random variable, i.e., in the case of Pr(D│H) the hypothesis is a random variable, while Pr(D; H) indicates that the data are conditioned on a variable that is considered fixed. The first is fundamentally Bayesian, the second is fundamentally evidentialist.
- 17.
For convenience, hypotheses are most often presented as exhaustive pairs, H and ̴H, but in theory the list of hypotheses considered is not limited to such pairs. It is difficult, among other reasons, to compute the probability of the data under the catch-all hypothesis [Pr(D│ ̴ H), and in consequence it is difficult to calculate the posterior probability of the catch-all. This difficulty then extends to comparing the posterior probability of the catch-all with the posterior probabilities of other hypotheses. We avoid such difficulties by confining our discussion to simple hypotheses. It might be added here that on the present account of what is often called “incremental confirmation,” the Special Consequence Condition does not hold see Salmon (1983, p. 106). To handle objections in this connection, Kotzen has produced a principle which he calls “Confirmation of Dragged Consequences:” If [Pr(H│D) > Pr(H), and H 1 entails H 2 , and Pr(H 2 ) < Pr(H 1 │D)] then Pr(H 2 │D) > Pr(H 2 ). See Kotzen (2012).
- 18.
Statisticians prefer to use the term “models,” by which they mean statistical models that allow for quantitative testing vis-à-vis the data, rather than “hypotheses” in this connection. Although we mostly stick to the more general (albeit vaguer) philosophical usage, the difference in meaning between the terms is important. As we emphasize at the end of this chapter in connection with various interpretations of probability, a hypothesis is a verbal statement about a natural state or process, a model is a mathematical abstraction that captures some of the potential of the hypothesis. Although we occasionally use the terms interchangeably, it matters a great deal whether one has models or hypotheses in mind when it comes to the correct characterization of the hypothesis-data and model-data relationships.
- 19.
Again in what follows, and except where it matters, we will not include reference to A or B in our formulations.
- 20.
Robert Boik pointed this out to us.
- 21.
It thus stands in sharp contrast to the well-known position of Williamson (2000), whose summarizing slogan is “a subject’s evidence consists of all and only the propositions that the subject knows.” Williamson is not analyzing the concept of “evidence,” or more precisely “evidence for a hypothesis or theory,” but the concept of “evidence for a subject.” This concept is important in classical epistemology, but has little use, or so we believe, as applied to scientific methodology or theory assessment. For us, evidence is evidence, whether or not the subject knows it, and conversely, whether or not the subject knows something does not thereby qualify it as “evidence” for any particular hypothesis.
- 22.
See Mayo (2014).
- 23.
We use “LR” rather than “BF” in what follows to underline the fact that our account of evidence is not in any narrow sense “Bayesian.”.
- 24.
Royall (1997) points out that the benchmark value = 8, or any value in that neighborhood, is widely shared. In fact, the value 8 is closely related to the Type I error-rate 0.05 in classical statistics and to an information criterion value of 2. See Taper (2004) and Taper and Ponciano (2016) for more on this issue.
- 25.
One might argue that the posterior-prior ratio measure (PPR) is equal to the LR measure and therefore precludes the necessity of a separate account of evidence. But the objection is misguided. The LR is equal to Pr(H│D)/Pr(H) only when Pr(D)/Pr(D│ ̴ H) is close to 1. That is, [Pr(D│H)/Pr(D│ ̴H)] = Pr(H│D)/Pr(H) x [Pr(D) x Pr(D/ ̴H)] ≈ 1. Otherwise, the two measures, LR and PPR, yield different values, over and above the fact that they measure very different things.
- 26.
- 27.
We are indebted to Robert Boik for this clarification.
- 28.
See Lele (2004) for the proof. It is worth noting the comments of a referee on a similar claim in another paper of ours: “…if in the end we hope for an account of evidence in which evidence gives us reason to believe, it is totally unclear why efficiency in the sense described would be taken as a sign of a meritorious measure of evidence.” One of the principal aims of this monograph is to disabuse its readers of what we call the “true-model” assumption, that evidence gives us reasons to believe that a hypothesis is true. In our view, evidence provides information about a hypothesis on the basis of which we can make reliable inferences and not reasons to believe that it is true. Since evidence is a way of identifying and assessing such information, efficiency in the sense indicated is indeed a meritorious property of it.
- 29.
What follows is drawn from Lele (2004). We go into the details, technical as some of them are, simply because evidence functions are so much less familiar than confirmation functions.
- 30.
In fact, evidence functions only use the weaker criterion of divergences (statistical distances are divergences with some additional constraints). A divergence quantifies the dissimilarity between two probability distributions. The statistical literature is quite loose in its use of the terms “divergence,” “disparity,” “discrepancy,” and “distance.” We use “distance” rather than the more general term “divergence” because “distance” is more intuitively evocative. Good discussions of statistical distances can be found in Lindsay (2004) and Basu et al. (2011).
- 31.
Although it does not do so in a linear fashion.
- 32.
The technical definition of evidence functions (Lele 2004) includes a consistency requirement (i.e., the probability of correctly selecting the best approximating model must go to 1 as sample size goes to infinity). Thus only “order consistent” information criteria such as Schwarz’s criterion (variously denoted as the SIC or BIC) can be used to construct evidence functions.
- 33.
(HD): \(HD(P,Q) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {\sqrt 2 }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sqrt 2 }$}}\sqrt {\sum\limits_{i} {\left( {\sqrt {P_{i} } - \sqrt {Q_{i} } } \right)}^{2} }\) for discrete distributions.
- 34.
Two previous readers raised this objection. We are assuming the standard account of absolute confirmation. Please see the following footnote for more on this point.
- 35.
There is a non-standard account of “absolute confirmation” in the literature on which it does admit of degrees; on this account a hypothesis is “absolutely confirmed” if it is “confirmed strongly,” where “confirmed strongly” can have different degrees. See Eells (2006, p. 144). Our argument depends on the standard account, on which D confirm H “absolutely” just in case Pr(H│D) = r, where r is identified as a particular number (or, occasionally, any number greater than 0.5).
- 36.
- 37.
Kyburg himself avoids the paradox by denying the uncritical “conjunction principle” on which it rests, that if each of a set of hypotheses is accepted, then their conjunction must be as well.
- 38.
This is our view, not Sober’s.
- 39.
A well-known example was provided by Popper (1959, p. 390). “Consider the next throw with a homogeneous die. Let x be the statement ‘six will turn up’; let y be its negation…; and let z be the information ‘an even number will turn up’. We have the following absolute probabilities: p(x) = 1/6; p(y) = 5/6; p(z) = 1/2. Moreover, we have the following relative probabilities: p(x, z) = 1/3; p(y, z) = 2/3. We see that x is supported by z, for z raises the probability of x from 1/6 to 2/6 = 1/3. We also see that y is undermined by z, for z lowers the probability of y by the same amount from 5/6 to 4/6 = 2/3. Nevertheless, we have p(x,z) < p(y, z).” Popper mistakenly drew the conclusion that there was a logical inconsistency in Carnap’s confirmation theory. But the inconsistency follows only if we were to take confirmation in its “absolute” sense, i.e., just in case the data raise the probability of a hypothesis beyond a high threshold. There is no inconsistency if confirmation is taken, as we do, in its incremental sense. See also Salmon (1983, pp. 102−03).
- 40.
Edwards et al. (1963).
- 41.
Levi (1967).
- 42.
This is perhaps clearest when D are entailed by H, for in this case Pr(D│H) = 1, regardless of what anyone happens to believe.
- 43.
We are indebted to both John. G. Bennett and Colin Howson for some clarifications about the need to introduce the Principal Principle. For more on the justification of the Principal Principle, see the forthcoming book by Robert Pettigrew, Accuracy and the Laws of Credence, Part II of which is devoted to it. We are indebted to Jason Konek for calling our attention to this important discussion.
- 44.
Given the limited scope of this Monograph, we are not going to evaluate rigorously whether this consideration is well-grounded.
- 45.
As we noted earlier and for the reason given, we use “model” (less often) and “hypothesis” (more often) interchangeably. But there are crucial differences between them that in the present context might give rise to misunderstandings. A hypothesis is a verbal statement about a state of nature or some natural process. A model is a mathematical abstraction that captures some dimensions of the hypothesis. When we say that the likelihood relationship between data and hypothesis is “logical,” this is not in the precise sense in which Keynes or Carnap use the term; it has to do rather with the evaluation of a single hypothesis based on the statistical data. On the other hand, the calculation of the probabilities of the data given two or more models is in our sense of the term “deductive.” The central point is that the relationships between data and hypothesis and data and models must be kept distinct; our use of the words “logical” and “deductive” is intended to do so, whatever other connotations those words might carry.
- 46.
- 47.
A doctor in Molière’s Le Malade Imaginaire attributes a “dormitive power” to opium, becoming the object of ridicule then and for the next 300 years. But to attribute a dispositional property is not to provide a causal explanation but to aggregate natural processes which must then be explained at a deeper, in this case pharmaceutical, level of analysis. Sugar has a propensity to dissolve in water, but it has almost no propensity to dissolve in gasoline. This difference can be understood through the chemical structure of sugar, water, and gasoline.
- 48.
See von Mises (1957).
- 49.
See Reichenbach (1949).
- 50.
See Hàjek (1997).
- 51.
Glymour (1980), p. 102, rightly draws attention to what he calls “misplaced rigor.” But rigor is otherwise indispensable. Error-statisticians have focused their criticisms of the evidential position on an alleged failure to deal with composite hypotheses; see Mayo (2014). This is our reply (following Taper and Lele 2011, Sects. 6 and 7). It is rather technical in character, and does not affect the main line of our argument. We have similarly added technical Appendices to Chap. 6 to reply to another error-statistical criticism of our evidential account, that it needlessly employs multiple models in its account of hypothesis testing, and to Chap. 11, to illustrate some of our main themes in more mathematical detail.
- 52.
This assumption is common to different schools of statistics. Both Royall (1997), who is committed to a likelihoodist (and not, as already noted, Bayesian) approach, and Mayo (1996) who operates within the error-statistical framework, also take the assumption for granted (at least Mayo did in 1996, although she does so no longer; again see Mayo 2014).
- 53.
Royall (1997, pp. 19–20).
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Appendix
Appendix
A Note on the Likelihoodist Treatment of Simple and Composite Hypotheses
Bayesians use the Bayes Factor (BF) to compareFootnote 51 hypotheses (Kass and Rafferty 1995), while others use the likelihood ration (LR) to measure evidence. For simple hypotheses, as in the tuberculosis example discussed near the outset of the next chapter, the Bayes Factor and the Likelihood Ratio are identical; both capture the essential core of our analysis of the concept of evidence. Throughout the monograph we assume that the hypotheses being tested are simple statistical hypotheses, which specifies a single value to a parameter, in contrast to a compound or composite hypothesis which restricts a parameter θ only to a range of values.Footnote 52 Since some philosophers have claimed recently that the likelihood account of evidence cannot deal with composite hypotheses, it is worth our while to argue why they are mistaken.
Here is a test case:Footnote 53
[M]edical researchers are interested in the success probability, θ, associated with a new treatment. They are particularly interested in how θ relates to the old treatment’s success probability, believed to be about 0.2. They have reason to hope θ is considerably greater, perhaps 0.8 or even greater. To obtain evidence about θ, they carry out a study in which the new treatment is given to 17 subjects, and find that it is successful in nine.
How would an evidentialist test the composite hypothesis that the true proportion (θ) is greater than 0.2?
The maximum likelihood or ML 0.5294. The k = 32 support interval for quite strong evidence is [0.233, 0.811]. Royall would say that for any value θ’ outside of this interval, there is quite strong evidence for the maximum likelihood estimate or MLE as opposed to θ’. For an evidentialist, this is sufficient to infer with strong evidence that (θ) > 0.2, even though the likelihood of the MLE is not the likelihood of the composite. The following considerations support this claim.
-
(1)
There is quite strong evidence against any value outside the interval relative to a value inside the interval (i.e. the maximum likelihood estimate).
-
(2)
No two values inside the interval can be quite strongly differentiated.
-
(3)
(1) and (2) together imply that there is quite strong evidence that the true proportion θ is in the support interval [0.233, 0.811].
-
(4)
Since 0.2 is entirely below the support interval, there is therefore quite strong evidence that the 0.2 is less than the true proportion.
It does make explicit that there are limits on how high the true proportion is likely to be.
We will use a notion called “the probability of misleading evidence” which will be discussed in much detail in Chap. 8. The probability for the presence of evidence for a hypothesis is called misleading because although there is probability for the presence of the evidence for the hypothesis, the latter is in fact false. If one had set up k = 32 (quite strong evidence) then the probability of misleading evidence for this statement is MG < 1/32 = 0.031. The ML represents the probability of misleading local evidence after the data have been gathered. The MG represents the probability of misleading global evidence before the data have been gathered. Both ML and MG represent a bound on the probability of misleading evidence for a hypothesis. The post hoc probability of misleading evidence is a little lower MG = 0.011.
Taper and Lele (2011) suggest that since there is an estimated parameter in finding the MLE, the MG is biased high, and that composite intervals should use a biased corrected estimate of the likelihood. We use Akaike’s bias correction (for simplicity). With the bias correction, the quite strong evidence support interval is a little bit wider at [0.201, 0.840]. The inference is still the same. There is still quite strong evidence that the true proportion is greater than 0.2, but now the post hoc probability of misleading evidence is slightly greater at 0.030. Using the more severe Swartz bias correction, we find that there is only fairly strong evidence for 0.2 being less than the true value, with a M of 0.045 (see also Taper and Ponciano 2016).
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Bandyopadhyay, P.S., Brittan, G., Taper, M.L. (2016). Bayesian and Evidential Paradigms. In: Belief, Evidence, and Uncertainty. SpringerBriefs in Philosophy(). Springer, Cham. https://doi.org/10.1007/978-3-319-27772-1_2
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