# Tree-Values: selective inference for regression trees

arXiv: https://arxiv.org/abs/2106.07816 · HTML: https://mlsanity.com/paper/2106.07816/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 36 references that are in the map.
- First posted on arXiv 15 Jun 2021.

## Builds on (most cited first, 10 of 10)

- [Recursive Partitioning for Heterogeneous Causal Effects](https://mlsanity.com/paper/1504.01132/) (2015)
- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Optimal Inference After Model Selection](https://mlsanity.com/paper/1410.2597/) (2014)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [Selective Inference for Hierarchical Clustering](https://mlsanity.com/paper/2012.02936/) (2020)
- [Valid Inference Corrected for Outlier Removal](https://mlsanity.com/paper/1711.10635/) (2017)
- [Testing for a Change in Mean After Changepoint Detection](https://mlsanity.com/paper/1910.04291/) (2019)
- [More powerful post-selection inference, with application to the Lasso](https://mlsanity.com/paper/1801.09037/) (2018)
- [Post-Selection Inference for Changepoint Detection Algorithms with Application to Copy Number Variation Data](https://mlsanity.com/paper/1812.03644/) (2018)
- [Selective inference with unknown variance via the square-root LASSO](https://mlsanity.com/paper/1504.08031/) (2015)

## Built on by (most cited first, 15 of 15)

- [CAD-DA: Controllable Anomaly Detection after Domain Adaptation by Statistical Inference](https://mlsanity.com/paper/2310.14608/) (2023)
- [Selective Inference for Changepoint detection by Recurrent Neural Network](https://mlsanity.com/paper/2311.14964/) (2023)
- [Statistical Test for Auto Feature Engineering by Selective Inference](https://mlsanity.com/paper/2410.19768/) (2024)
- [Do We Really Even Need Data? A Modern Look at Drawing Inference with Predicted Data](https://mlsanity.com/paper/2512.05456/) (2025)
- [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)
- [Controlling the False Split Rate in Tree-Based Aggregation](https://mlsanity.com/paper/2108.05350/) (2021)
- [Bounded P-values in Parametric Programming-based Selective Inference](https://mlsanity.com/paper/2307.11351/) (2023)
- [Statistical Test for Saliency Maps of Graph Neural Networks via Selective Inference](https://mlsanity.com/paper/2505.16893/) (2025)
- [Statistical Inference for the Dynamic Time Warping Distance, with Application to Abnormal Time-Series Detection](https://mlsanity.com/paper/2202.06593/) (2022)
- [Inferring independent sets of Gaussian variables after thresholding correlations](https://mlsanity.com/paper/2211.01521/) (2022)
- [Statistical Inference for Clustering-based Anomaly Detection](https://mlsanity.com/paper/2504.18633/) (2025)
- [Enhanced Survival Trees](https://mlsanity.com/paper/2509.18494/) (2025)
- [A novel Information-Driven Strategy for Optimal Regression Assessment](https://mlsanity.com/paper/2510.14222/) (2025)
- [Selective Inference for CART with Binary Outcomes](https://mlsanity.com/paper/2609.24949/) (2026)

## Abstract (arXiv)

We consider conducting inference on the output of the Classification and Regression Tree (CART) [Breiman et al., 1984] algorithm. A naive approach to inference that does not account for the fact that the tree was estimated from the data will not achieve standard guarantees, such as Type 1 error rate control and nominal coverage. Thus, we propose a selective inference framework for conducting inference on a fitted CART tree. In a nutshell, we condition on the fact that the tree was estimated from the data. We propose a test for the difference in the mean response between a pair of terminal nodes that controls the selective Type 1 error rate, and a confidence interval for the mean response within a single terminal node that attains the nominal selective coverage. Efficient algorithms for computing the necessary conditioning sets are provided. We apply these methods in simulation and to a dataset involving the association between portion control interventions and caloric intake.