# Inferring independent sets of Gaussian variables after thresholding correlations

arXiv: https://arxiv.org/abs/2211.01521 · HTML: https://mlsanity.com/paper/2211.01521/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 66 references that are in the map.
- First posted on arXiv 2 Nov 2022.

## Builds on (most cited first, 16 of 16)

- [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)
- [Selective inference for k-means clustering](https://mlsanity.com/paper/2203.15267/) (2022)
- [Selective Inference for Group-Sparse Linear Models](https://mlsanity.com/paper/1607.08211/) (2016)
- [Post-Selection Inference for Changepoint Detection Algorithms with Application to Copy Number Variation Data](https://mlsanity.com/paper/1812.03644/) (2018)
- [Tree-Values: selective inference for regression trees](https://mlsanity.com/paper/2106.07816/) (2021)
- [Exact Post Model Selection Inference for Marginal Screening](https://mlsanity.com/paper/1402.5596/) (2014)
- [Selective inference in regression models with groups of variables](https://mlsanity.com/paper/1511.01478/) (2015)
- [The huge Package for High-dimensional Undirected Graph Estimation in R](https://mlsanity.com/paper/2006.14781/) (2020)
- [A whitening approach to probabilistic canonical correlation analysis for omics data integration](https://mlsanity.com/paper/1802.03490/) (2018)
- [Selective inference with unknown variance via the square-root LASSO](https://mlsanity.com/paper/1504.08031/) (2015)
- [Graph analysis and modularity of brain functional connectivity networks: searching for the optimal threshold](https://mlsanity.com/paper/1705.06481/) (2017)

## Abstract (arXiv)

We consider testing whether a set of Gaussian variables, selected from the data, is independent of the remaining variables. We assume that this set is selected via a very simple approach that is commonly used across scientific disciplines: we select a set of variables for which the correlation with all variables outside the set falls below some threshold. Unlike other settings in selective inference, failure to account for the selection step leads, in this setting, to excessively conservative (as opposed to anti-conservative) results. Our proposed test properly accounts for the fact that the set of variables is selected from the data, and thus is not overly conservative. To develop our test, we condition on the event that the selection resulted in the set of variables in question. To achieve computational tractability, we develop a new characterization of the conditioning event in terms of the canonical correlation between the groups of random variables. In simulation studies and in the analysis of gene co-expression networks, we show that our approach has much higher power than a ``naive'' approach that ignores the effect of selection.