Selection after Symmetry · Chae-Yeon Xon

Conceptual map

V-07.01The target claim
V-07.02Fixed and selected statistics
V-07.03An economic counterexample
V-07.04Possible repairs
V-07.05Evidence boundaries

1. Symmetry is preserved for fixed coordinates

Coordinatewise symmetry does not survive a selection rule that favors large realized coordinates. The selected statistic has a different law.

Let $\varepsilon=(\varepsilon _{1},\ldots,\varepsilon _{K})$ have iid Rademacher coordinates, each equal to $$-1$$ or 1 with equal probability. Every predetermined coordinate has mean zero. Let $J(\varepsilon)$ be any measurable maximizer of the realized vector, using a declared tie rule. Then $\varepsilon _{J(\varepsilon )}=\operatorname{max}_{j}\varepsilon _{j}$ .

Definition 1 · Selection-aware target

A selected target includes the selection map in its definition. Its expectation is $\mathbb{E}[T_{J(X)}(X)]$ , which generally differs from evaluating $\mathbb{E}[T_{j}(X)]$ and then substituting a random index.

2. The smallest witness has two coordinates

Proposition 1 · Mean of the selected maximum

For $K\ge1$ iid Rademacher coordinates, $\mathbb{E}[\max_{j}\varepsilon _{j}]=1-2^{1-K}$ . This equals zero only at $$K=1$$ and is positive for every $K\ge2$ .

Proof. The maximum equals $$-1$$ only when every coordinate equals $$-1$$ , an event of probability $2^{-K}$ . It equals one otherwise. Therefore its expectation is $(1-2^{-K})-2^{-K}=1-2^{1-K}$ . ∎

For $$K=2$$ , the four equally likely vectors give selected values 1,1,1, $$-1$$ and mean 1/2. Both fixed-coordinate means remain zero. This exact table refutes any step that transfers coordinate symmetry through the maximization rule.

An economic implementation lets two moment inequalities receive symmetric sampling shocks and reports the most violated moment. At a boundary where both population moments are zero, the reported maximum has positive mean. The phenomenon is a property of the selection operator, independent of the economic label attached to the moments.

3. Repairs change the target or account for selection

A pre-specified index retains the coordinatewise symmetry result. Sample splitting can select on one independent subsample and evaluate on another; conditional on the selection sample, the evaluation noise can remain centered. Simultaneous inference controls the family of coordinates before selection. Selective inference conditions on a suitable selection event and changes the reference distribution.

Subtracting the known selection mean repairs this finite Rademacher example, but a general correction depends on the joint law, tie rule, and selection map. Marginal symmetry alone does not identify it when coordinates are dependent.

4. Ties, dependence, and estimated candidate sets matter

Failure case · Fixed-index lemma cited after data-dependent selection

A lemma quantified over each deterministic $$j$$ does not cover $$J(X)$$ unless it is uniform or conditioned through a valid selection argument.

At $$K=1$$ there is no selection effect. Perfectly identical coordinates also remove it. Negative dependence can enlarge the chance that at least one coordinate is high; perfect positive dependence can collapse the effective candidate count. Atoms create ties and make the selected index distribution tie-rule dependent, even when the selected value is unchanged. A candidate set learned from the same data adds another selection stage. Empty candidate sets need a fallback definition, and growing $$K$$ changes asymptotic approximations.

5. Implementation, exercises, and sources

Write the selection map as executable pseudocode, enumerate a finite witness exactly, and compare fixed-index, selected-index, and conditional distributions. In data analysis, preserve the full candidate list, scores, tie rule, and random seed. Re-run all inferential stages inside every resample or use a theorem that explicitly conditions on the selection event.

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Exercises

Verify the proposition for $$K=1$$ ,2,3 by enumeration.
Compute the selected mean when all coordinates are identical Rademacher variables.
Explain why evaluating a selected moment on an independent holdout restores conditional centering.

Partial solutions

1. The values are 0, 1/2, and 3/4. 2. The maximum equals the common coordinate and has mean zero. 3. Conditional on the selection sample, the chosen index is fixed relative to independent holdout noise; centering then applies to that conditional law.

H. A. David and H. N. Nagaraja, Order Statistics.Distributions of maxima and selected order statistics.
Richard Berk et al. (2013), “Valid Post-Selection Inference,” Annals of Statistics 41, 802–837.Simultaneous guarantees after model selection.
Jason D. Lee, Dennis L. Sun, Yuekai Sun, and Jonathan E. Taylor (2016), “Exact Post-Selection Inference, with Application to the Lasso,” Annals of Statistics 44, 907–927.Conditioning on selection events.

6. Audit checkpoint

Fixed versus random index, selection map, candidate count, joint law, symmetry scope, tie probability and rule, empty set, learned candidates, selected value versus selected identity, dependence, conditioning event, split independence, simultaneous family, rerun stages, growing dimension, and the exact estimand after correction.

7. Scope boundary

The chapter covers finite post-selection mean effects. High-dimensional selective inference and adaptive sequential experimentation require stronger probability tools.