Selective inference for latent block models

Abstract

Model selection in latent block models has been a challenging but important task in the field of statistics. Specifically, a major challenge is encountered when constructing a test on a block structure obtained by applying a specific clustering algorithm to a finite size matrix. In this case, it becomes crucial to consider the selective bias in the block structure, that is, the block structure is selected from all the possible cluster memberships based on some criterion by the clustering algorithm. To cope with this problem, this study provides a selective inference method for latent block models. Specifically, we construct a statistical test on a set of row and column cluster memberships of a latent block model, which is given by a squared residue minimization algorithm. The proposed test, by its nature, includes and thus can also be used as the test on the set of row and column cluster numbers. We also propose an approximated version of the test based on simulated annealing to avoid combinatorial explosion in searching the optimal block structure. The results show that the proposed exact and approximated tests work effectively, compared to the naive test that did not take the selective bias into account.

Selective inference for clustering with unknown variance

“…There have also been relevant work on data with structural assumptions. For example, the work of Watanabe and Suzuki [2021] provides a method for doing inference on a data matrix represented by a latent block model after choosing the cluster membership of each entry of the data matrix using the same data matrix, and conditions on this selection event to do a valid inference.…”

Selective inference for k-means clustering

“…To develop the test, we leverage the selective inference framework, which has been applied extensively in high-dimensional linear modeling (Lee et al, 2016;Tibshirani et al, 2016;Fithian et al, 2014;Rügamer et al, 2022;Schultheiss et al, 2021;Taylor & Tibshirani, 2018;Charkhi & Claeskens, 2018;Yang et al, 2016;Loftus & Taylor, 2014), changepoint detection (Jewell et al, 2022;Hyun et al, 2021Hyun et al, , 2018Chen et al, 2021b;Le Duy & Takeuchi, 2021;Duy et al, 2020;Benjamini et al, 2019), and clustering (Zhang et al, 2019;Gao et al, 2020;Watanabe & Suzuki, 2021). The key insight behind selective inference is as follows: to obtain a valid test of H 0 , we need to condition on the aspect of the data that led us to test it.…”