From Selection to Shrinkage: Partial Pooling and the Reference Class Problem

The reference class problem arises because an individual belongs to indefinitely many classes that yield conflicting probability estimates. I argue that the problem decomposes into two questions: given a grouping structure, how should information be combined across its levels to yield a probability for an individual (the combination question); and which grouping structure is the right one (the structure question)? I defend four claims. First, the combination question has an operational answer, developed independently in statistics: partial pooling in multilevel models, with the posterior predictive distribution as the direct-inference probability. Second, the classical answer to that question— Reichenbach’s narrowest-class rule—is not merely vague but decision-theoretically defective. By the Stein inadmissibility results for the canonical normal-means setting, the no-pooling estimator it induces is dominated by shrinkage in total risk, uniformly over the parameter space and without assuming the hierarchical model is correct. This defect is inherited by every selection rule in the literature, since each defers to the narrowest class where its statistics are deemed reliable. Third, the classical proposals of Reichenbach, Carnap, Kyburg, and Pollock are recoverable as limits or special cases of the same pooling apparatus. Fourth, the structure question survives, but as a checkable modeling question rather than an undecidable selection, and the regress it threatens is benign rather than vicious. The reference class problem is thereby reduced, not dissolved.