# Selective Sequential Model Selection

arXiv: https://arxiv.org/abs/1512.02565 · HTML: https://mlsanity.com/paper/1512.02565/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2015.

## Built on by (most cited first, 17 of 17)

- [More Powerful Conditional Selective Inference for Generalized Lasso by Parametric Programming](https://mlsanity.com/paper/2105.04920/) (2021)
- [Valid P-Value for Deep Learning-Driven Salient Region](https://mlsanity.com/paper/2301.02437/) (2023)
- [Conditional Selective Inference for Robust Regression and Outlier Detection using Piecewise-Linear Homotopy Continuation](https://mlsanity.com/paper/2104.10840/) (2021)
- [Fast and More Powerful Selective Inference for Sparse High-order Interaction Model](https://mlsanity.com/paper/2106.04929/) (2021)
- [CAD-DA: Controllable Anomaly Detection after Domain Adaptation by Statistical Inference](https://mlsanity.com/paper/2310.14608/) (2023)
- [Exact Statistical Inference for the Wasserstein Distance by Selective Inference](https://mlsanity.com/paper/2109.14206/) (2021)
- [Adaptive debiased machine learning using data-driven model selection techniques](https://mlsanity.com/paper/2307.12544/) (2023)
- [Statistical Test for Anomaly Detections by Variational Auto-Encoders](https://mlsanity.com/paper/2402.03724/) (2024)
- [Statistical Test for Attention Map in Vision Transformer](https://mlsanity.com/paper/2401.08169/) (2024)
- [Statistical Test for Feature Selection Pipelines by Selective Inference](https://mlsanity.com/paper/2406.18902/) (2024)
- [Statistical Test for Auto Feature Engineering by Selective Inference](https://mlsanity.com/paper/2410.19768/) (2024)
- [Change Point Detection in the Frequency Domain with Statistical Reliability](https://mlsanity.com/paper/2502.03062/) (2025)
- [Statistical Inference for Autoencoder-based Anomaly Detection after Representation Learning-based Domain Adaptation](https://mlsanity.com/paper/2508.07049/) (2025)
- [Bounded P-values in Parametric Programming-based Selective Inference](https://mlsanity.com/paper/2307.11351/) (2023)
- [High Dimensional Time Series Regression Models: Applications to Statistical Learning Methods](https://mlsanity.com/paper/2308.16192/) (2023)
- [Statistical Inference for Clustering-based Anomaly Detection](https://mlsanity.com/paper/2504.18633/) (2025)
- [Post-ADC Inference: Valid Inference After Active Data Collection](https://mlsanity.com/paper/2605.11511/) (2026)

## Abstract (arXiv)

Many model selection algorithms produce a path of fits specifying a sequence of increasingly complex models. Given such a sequence and the data used to produce them, we consider the problem of choosing the least complex model that is not falsified by the data. Extending the selected-model tests of Fithian et al. (2014), we construct p-values for each step in the path which account for the adaptive selection of the model path using the data. In the case of linear regression, we propose two specific tests, the max-t test for forward stepwise regression (generalizing a proposal of Buja and Brown (2014)), and the next-entry test for the lasso. These tests improve on the power of the saturated-model test of Tibshirani et al. (2014), sometimes dramatically. In addition, our framework extends beyond linear regression to a much more general class of parametric and nonparametric model selection problems. To select a model, we can feed our single-step p-values as inputs into sequential stopping rules such as those proposed by G'Sell et al. (2013) and Li and Barber (2015), achieving control of the familywise error rate or false discovery rate (FDR) as desired. The FDR-controlling rules require the null p-values to be independent of each other and of the non-null p-values, a condition not satisfied by the saturated-model p-values of Tibshirani et al. (2014). We derive intuitive and general sufficient conditions for independence, and show that our proposed constructions yield independent p-values.