Direct Inference from Imprecise Frequencies
It is well known that there are, at least, two sorts of cases where one should not prefer a direct inference based on a narrower reference class, in particular: cases where the narrower reference class is gerrymandered, and cases where one lacks an evidential basis for forming a precise-valued frequency judgment for the narrower reference class. I here propose (1) that the preceding exceptions exhaust the circumstances where one should not prefer direct inference based on a narrower reference class, and (2) that minimal frequency information for a narrower (non-gerrymandered) reference class is sufficient to yield the defeat of a direct inference for a broader reference class. By the application of a method for inferring relatively informative expected frequencies, I argue that the latter claim does not result in an overly incredulous approach to direct inference. The method introduced here permits one to infer a relatively informative expected frequency for a reference class R′, given frequency information for a superset of R′ and/or frequency information for a sample drawn from R′.
KeywordsDirect inference Statistical syllogism Reference class problem Imprecise probability
Work on this paper was supported by DFG Grant SCHU1566/9-1 as part of the priority program “New Frameworks of Rationality” (SPP 1516). For comments that motivated the preparation of this paper, I am thankful to participants at EPSA 2015, including Michael Baumgartner, Martin Bentzen, Seamus Bradley, Bert Leuridan, Jan-Willem Romeijn, Gerhard Schurz, and Jon Williamson. I am also grateful for discussions with Christian Wallmann, which motivated the proposal presented in Sect. 28.4.
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