Post-Selection Inference via Algorithmic Stability
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Modern approaches to data analysis make extensive use of data-driven model selection. The resulting dependencies between the selected model and data used for inference invalidate statistical guarantees derived from classical theories. The framework of post-selection inference (PoSI) has formalized this problem and proposed corrections which ensure valid inferences. Yet, obtaining general principles that enable computationally-efficient, powerful PoSI methodology with formal guarantees remains a challenge. With this goal in mind, we revisit the PoSI problem through the lens of algorithmic stability. Under an appropriate formulation of stability—one that captures closure under post-processing and compositionality properties—we show that stability parameters of a selection method alone suffice to provide non-trivial corrections to classical z-test and t-test intervals. Then, for several popular model selection methods, including the LASSO, we show how stability can be achieved through simple, computationally efficient randomization schemes. Our algorithms offer provable unconditional simultaneous coverage and are computationally efficient; in particular, they do not rely on MCMC sampling. Importantly, our proposal explicitly relates the magnitude of randomization to the resulting confidence interval width, allowing the analyst to tune interval width to the loss in utility due to randomizing selection.
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