The Dudley Entropy Integral

When choosing between thousands of candidate trading signals, the best-looking one in a backtest is expected to look good partly by luck, simply because it's the maximum over so many candidates — testing more candidates makes the best in-sample result more inflated even if none of them are genuinely predictive. Controlling exactly how inflated that maximum gets requires measuring how "large," or complex, the whole family of candidates really is, and the Dudley entropy integral is a classical tool for turning that complexity into a concrete bound.

The idea is to cover the family of candidates by a much smaller set of representative "typical" members at increasingly fine resolution — a coarse covering with few representative points, then finer coverings needing more points, and so on — and count, at each resolution, how many representative points are needed (the covering number; its logarithm is the "entropy" at that resolution). Dudley's bound says the expected maximum deviation across the whole family is controlled by an integral that sums up this entropy across all resolutions from coarse to fine: families that stay small in entropy even at fine resolution have well-controlled maxima, while families that need exponentially many representative points to describe finely have maxima that can be much larger than any individual member's typical size.

In practice this is the theoretical machinery behind statements like "if you tested $N$ largely independent strategies, expect the best one's backtested Sharpe ratio to be inflated by roughly $\sqrt{2 \ln N}$" — the entropy integral is the general-purpose tool of which that familiar multiple-testing correction is a special case.