small-sample
Estimating many means at once
It is 1961, and a startling result is about to unsettle statisticians: when you must estimate several unknown means at once — batting averages, say, each from only a few at-bats — the obvious answer, take each group's own average, is provably not the best, even though each average is individually unbeatable. Deliver estimates for a dozen such small-sample means that, taken together, beat the separate averages by deliberately shrinking each toward the common center. Justify why borrowing strength across unrelated-looking groups wins. Get it wrong and you hand back noisy per-group estimates that overfit their tiny samples, crowning flukes and burying signal — the paradox is that ignoring the ensemble, which feels most honest, is measurably worse.
Who this problem belongs to
The two figures whose methods fit it best, out of 38 in contention.
Efron, working with Carl Morris in a series of papers through the early 1970s, showed that Charles Stein's startling 1956 result — that the maximum-likelihood estimator for several means simultaneously is inadmissible once there are three or more — could be understood as empirical Bayes: each group average shrunk toward the grand mean by an amount the data itself estimates. Efron and Morris's 'Stein's Paradox in Statistics' became the canonical exposition, using exactly the batting-average example this problem poses. His later empirical Bayes and bootstrap work extended the same shrinkage logic across statistics. He does not personally discover the 1961 James-Stein estimator, which keeps him just below a perfect score, but he is the figure who made it usable and understood.
Gelman's hierarchical Bayesian models, and the Stan software ecosystem built around them, are the modern general-purpose engine for exactly this phenomenon: estimating many related quantities — batting averages, school effects, state-level polling — by letting each group's estimate borrow strength from the others through a shared prior, which is precisely the shrinkage mechanism the James-Stein estimator first demonstrated in 1961. His applied Bayesian workflow treats partial pooling as the default correct answer to 'many small-sample means at once,' not a paradox needing special justification. He did not develop the original James-Stein result, so the score sits just below the historical originators of the specific 1961 construction. That closeness to the method's historical origin is what keeps Andrew Gelman among the stronger carriers in this particular batch.
In the mind map
The same ideas, as concepts rather than history — in John's ML knowledge map.
38 figures are scored on this problem. Draw it in a battle to see where you land.