Branden Fitelson

Charles Stein discovered a paradox in 1955 that many statisticians think is of fundamental importance. Here we explore its philosophical implications. We outline the nature of Stein’s result and of subsequent work on shrinkage estimators; then we describe how these results are related to Bayesianism and to model selection criteria like AIC. We also discuss their bearing on scientific realism and instrumentalism. We argue that results concerning shrinkage estimators underwrite a surprising form of holistic pragmatism.

To the extent that we have reasons to avoid these “bad B -properties”, these arguments provide reasons not to have an incoherent credence function b — and perhaps even reasons to have a coherent one. But, note that these two traditional arguments for probabilism involve what might be called “pragmatic” reasons (not) to be (in)coherent. In the case of the Dutch Book argument, the “bad” property is pragmatically bad (to the extent that one values money). But, it is not clear whether the DBA pinpoints any epistemic defect of incoherent agents. The same can be said for Representation Theorem arguments, since they involve the structure of an agent’s preferences

Let L be a sentential (object) language containing atoms ‘A’, ‘B’, . . . , and two logical connectives ‘&’ and ‘→’. In addition to these two logical connectives, L will also contain another binary connective ‘ ’, which is intended to be interpreted as the English indicative. In the meta-language for L , we will have two meta-linguistic operations: ‘ ’ and ‘ ’. ‘ ’ is a binary relation between individual sentences in L . It will be interpreted as “single premise entailment” (or “single premise deducibility in L ”). ‘ ’ is a monadic predicate on sentences of L . It will be interpreted as “logical truth of the logic of L ” (or “theorem of the logic of L ”). We will not presuppose anything about the relationship between ‘ ’ and ‘ ’. Rather, we will state explicitly all assumptions about these meta-theoretic relations that will be required for Gibbard’s Theorem.