By Prasanta S. Bandyopadhyay, Gordon Brittan Jr., Mark L. Taper
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Extra resources for Belief, Evidence, and Uncertainty: Problems of Epistemic Inference
1 Whether data constitute evidence, on the other hand, has to do with the ways in which they serve to distinguish and compare competing hypotheses. It is a three-part relation involving data and two hypotheses. Data that cannot tell for or against such hypotheses do not constitute evidence for one or the other. A natural way to express this very basic intuition is through the use of likelihood ratios. Thus, data D constitute (positive) evidence for hypothesis H just in case the ratio of likelihoods, Pr(D│H)/Pr(D│H′) is greater than 1, where H and H′ are not necessarily mutually exclusive.
11, to illustrate some of our main themes in more mathematical detail. 53 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:54 [M]edical researchers are interested in the success probability, θ, associated with a new treatment. 2. 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.
In the lottery case, the likelihoods of all of the competing hypotheses, that is, the likelihood of cashing a winning ticket on the hypothesis that it is not a winning ticket, are the same. In which case, the data fail to distinguish between the competing hypotheses, in which case they are, in context, evidence for none of them. If acceptance requires evidential signiﬁcance, as we will in more detail in Chap. 6, then we should not accept any of the hypotheses. The other difﬁculty with the absolute conception of conﬁrmation is that it entails what is sometimes called the inconsistency condition: the data can never conﬁrm incompatible hypotheses.