# Optimal Inference After Model Selection

arXiv: https://arxiv.org/abs/1410.2597 · HTML: https://mlsanity.com/paper/1410.2597/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2014.

## Built on by (most cited first, 30 of 67)

- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [More Powerful Conditional Selective Inference for Generalized Lasso by Parametric Programming](https://mlsanity.com/paper/2105.04920/) (2021)
- [Selective inference for k-means clustering](https://mlsanity.com/paper/2203.15267/) (2022)
- [Fast and More Powerful Selective Inference for Sparse High-order Interaction Model](https://mlsanity.com/paper/2106.04929/) (2021)
- [Tree-Values: selective inference for regression trees](https://mlsanity.com/paper/2106.07816/) (2021)
- [Data fission: splitting a single data point](https://mlsanity.com/paper/2112.11079/) (2021)
- [Data thinning for convolution-closed distributions](https://mlsanity.com/paper/2301.07276/) (2023)
- [The Terminating-Random Experiments Selector: Fast High-Dimensional Variable Selection with False Discovery Rate Control](https://mlsanity.com/paper/2110.06048/) (2021)
- [Valid and Exact Statistical Inference for Multi-dimensional Multiple Change-Points by Selective Inference](https://mlsanity.com/paper/2110.08989/) (2021)
- [Valid Inference After Causal Discovery](https://mlsanity.com/paper/2208.05949/) (2022)
- [CAD-DA: Controllable Anomaly Detection after Domain Adaptation by Statistical Inference](https://mlsanity.com/paper/2310.14608/) (2023)
- [PCS-UQ: Uncertainty Quantification via the Predictability-Computability-Stability Framework](https://mlsanity.com/paper/2505.08784/) (2025)
- [Generalized Data Thinning Using Sufficient Statistics](https://mlsanity.com/paper/2303.12931/) (2023)
- [Exact Statistical Inference for the Wasserstein Distance by Selective Inference](https://mlsanity.com/paper/2109.14206/) (2021)
- [Certified Policy Smoothing for Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2212.11746/) (2022)
- [A Flexible Defense Against the Winner's Curse](https://mlsanity.com/paper/2411.18569/) (2024)
- [Towards Reliable LLM Evaluation: Correcting the Winner's Curse in Adaptive Benchmarking](https://mlsanity.com/paper/2605.05973/) (2026)
- [Exact Selective Inference with Randomization](https://mlsanity.com/paper/2212.12940/) (2022)
- [Statistical Test for Attention Map in Vision Transformer](https://mlsanity.com/paper/2401.08169/) (2024)
- [A Statistical Case Against Empirical Human-AI Alignment](https://mlsanity.com/paper/2502.14581/) (2025)
- [Calibrating Decision Robustness via Inverse Conformal Risk Control](https://mlsanity.com/paper/2510.07750/) (2025)
- [On the Validity of Conformal Prediction for Network Data Under Non-Uniform Sampling](https://mlsanity.com/paper/2306.07252/) (2023)
- [Selective inference using randomized group lasso estimators for general models](https://mlsanity.com/paper/2306.13829/) (2023)
- [Decomposing Gaussians with Unknown Covariance](https://mlsanity.com/paper/2409.11497/) (2024)
- [Quantifying Statistical Significance of Deep Nearest Neighbor Anomaly Detection via Selective Inference](https://mlsanity.com/paper/2502.12978/) (2025)
- [Do We Really Even Need Data? A Modern Look at Drawing Inference with Predicted Data](https://mlsanity.com/paper/2512.05456/) (2025)
- [Tree-based Subgroup Discovery In Electronic Health Records: Heterogeneity of Treatment Effects for DTG-containing Therapies](https://mlsanity.com/paper/2208.14329/) (2022)
- [Statistical Inference for Feature Selection after Optimal Transport-based Domain Adaptation](https://mlsanity.com/paper/2410.15022/) (2024)
- [Controllable RANSAC-based Anomaly Detection via Hypothesis Testing](https://mlsanity.com/paper/2410.15133/) (2024)

## Abstract (arXiv)

To perform inference after model selection, we propose controlling the selective type I error; i.e., the error rate of a test given that it was performed. By doing so, we recover long-run frequency properties among selected hypotheses analogous to those that apply in the classical (non-adaptive) context. Our proposal is closely related to data splitting and has a similar intuitive justification, but is more powerful. Exploiting the classical theory of Lehmann and Scheff'e (1955), we derive most powerful unbiased selective tests and confidence intervals for inference in exponential family models after arbitrary selection procedures. For linear regression, we derive new selective z-tests that generalize recent proposals for inference after model selection and improve on their power, and new selective t-tests that do not require knowledge of the error variance.