Infer-and-widen, or not?

In recent years, there has been substantial interest in the task of selective inference: inference on a parameter that is selected from the data. Many of the existing proposals fall into what we refer to as the \emph{infer-and-widen} framework: they produce symmetric confidence intervals whose midpoints do not account for selection and therefore are biased; thus, the intervals must be wide enough to account for this bias. In this paper, we investigate infer-and-widen approaches in three vignettes: the winner’s curse, maximal contrasts, and inference after the lasso. In each of these examples, we show that a state-of-the-art infer-and-widen proposal leads to confidence intervals that are wider than a non-infer-and-widen alternative. Furthermore, even an ``oracle’’ infer-and-widen confidence interval – the narrowest possible interval that could be theoretically attained via infer-and-widen – can be wider than the alternative.

💡 Research Summary

The paper investigates the performance of the “infer‑and‑widen” (I&W) framework for selective inference, a setting where the parameter of interest is chosen after looking at the data. In the classical fixed‑parameter setting a (1‑α) confidence interval (CI) satisfies P(θ_i∈CI_i)≥1‑α for a predetermined index i. When the index is data‑driven via a selection rule S(Y), the usual CI centered at the naïve estimator becomes biased; to retain unconditional coverage one must widen the interval symmetrically around the biased midpoint. This two‑step procedure—(1) select a target using the data, (2) apply a classical symmetric CI with an inflated critical value—is what the authors call the infer‑and‑widen framework. Traditional multiple‑testing corrections such as Bonferroni, Holm‑Bonferroni, Šidák, Scheffé, and the PoSI method are special cases. Recent work has attempted to make the inflation factor adaptive to the stability of the selection algorithm, but the fundamental issue—ignoring the bias of the midpoint—remains.

The authors evaluate I&W in three concrete vignette settings and compare it to alternative selective‑inference methods that either produce asymmetric intervals or exploit data splitting (sample splitting) to obtain conditional coverage.

Vignette 1 – Winner’s Curse.
The model is Y∼N_n(μ,σ²I) and the parameter of interest is the mean of the largest observed component, μ_{S(Y)} where S(Y)=argmax_i Y_i. To obtain a “stable” selection rule the authors add independent Laplace noise ζ_i∼Laplace(c) and define S^c_{Laplace}(Y)=argmax_i (Y_i+ζ_i). An existing I&W interval (Proposition 1, equation 3.4) uses two tuning parameters η and ν to inflate the critical value. The authors construct a data‑fission interval (Proposition 2, equation 3.7) that conditions on the signs of the Laplace noises, yielding a truncated‑normal distribution for each coordinate and thus an asymmetric interval that attains conditional coverage. Extensive simulations show that, across a wide range of sample sizes n and noise scales c, the I&W interval is on average 1.5–2 times wider than the data‑fission interval. Even an “oracle” I&W interval—defined as the narrowest possible symmetric interval centered at Y_{S(Y)} that still achieves the nominal coverage—remains wider than the data‑fission interval. The conclusion is that the bias in the midpoint forces any symmetric I&W interval to be overly conservative.

Vignette 2 – Maximal Contrasts.
Here a fixed design matrix X with normalized columns is given, and the selection rule picks the column whose inner product with Y is maximal: S(Y)=argmax_j X_j^T Y. Again a Laplace‑perturbed version S^c_{Laplace}(Y) is used for stability. Proposition 3 (equation 4.3) provides the I&W interval. The authors compare it to alternative methods that adjust asymmetrically or condition on the selection event. The results mirror Vignette 1: when the contrast is large, the I&W interval’s width inflates dramatically, whereas the alternative intervals remain substantially narrower.

Vignette 3 – Inference after the Lasso.
The third vignette examines post‑selection inference for coefficients chosen by the Lasso. Existing I&W‑type approaches (PoSI, locally simultaneous inference, stability‑adjusted corrections) produce confidence intervals that become extremely wide as the number of selected variables grows. In contrast, methods based on data splitting, polyhedral selective inference, or other conditional‑coverage techniques achieve the same unconditional (and often conditional) coverage with far smaller average widths. Simulations demonstrate that I&W intervals can be 30–50 % wider than the alternatives, especially in high‑dimensional regimes.

Theoretical Insight (Section 6).
The authors prove that any I&W interval, even the oracle version, must suffer from a “bias‑inflated” midpoint. If the selection rule induces a non‑zero bias, the minimal symmetric interval that guarantees coverage inevitably expands, sometimes dramatically. Thus the “widen” step is a blunt instrument that compensates for bias by sacrificing efficiency.

Overall Conclusions.
While the I&W framework is attractive for its simplicity—requiring no bespoke algorithm beyond a standard symmetric CI—it is fundamentally limited: it ignores the direction and magnitude of selection bias, leading to overly conservative intervals. Alternative strategies that either (i) condition on additional information (signs of randomization, auxiliary variables), (ii) split the data into training and inference sets, or (iii) construct asymmetric intervals tailored to the selection event, consistently produce shorter intervals with the same coverage guarantees. The paper therefore recommends that practitioners favor these more refined selective‑inference methods over naïve infer‑and‑widen approaches, especially in settings where interval width is a critical factor.