For high-dimensional hierarchical models, consider exchangeability of effects across covariates instead of across datasets

• URL: http://arxiv.org/abs/2107.06428v1
• Date: Tue, 13 Jul 2021 23:23:06 GMT
• Title: For high-dimensional hierarchical models, consider exchangeability of effects across covariates instead of across datasets
• Authors: Brian L. Trippe, Hilary K. Finucane, Tamara Broderick
• Abstract summary: We show that standard practice exhibits poor statistical performance when the number of covariates exceeds the number of datasets. In statistical genetics, we might regress dozens of traits (defining datasets) for thousands of individuals (responses) on up to millions of genetic variants. We propose a hierarchical model expressing our alternative perspective.
• Score: 18.74167116981788
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Hierarchical Bayesian methods enable information sharing across multiple related regression problems. While standard practice is to model regression parameters (effects) as (1) exchangeable across datasets and (2) correlated to differing degrees across covariates, we show that this approach exhibits poor statistical performance when the number of covariates exceeds the number of datasets. For instance, in statistical genetics, we might regress dozens of traits (defining datasets) for thousands of individuals (responses) on up to millions of genetic variants (covariates). When an analyst has more covariates than datasets, we argue that it is often more natural to instead model effects as (1) exchangeable across covariates and (2) correlated to differing degrees across datasets. To this end, we propose a hierarchical model expressing our alternative perspective. We devise an empirical Bayes estimator for learning the degree of correlation between datasets. We develop theory that demonstrates that our method outperforms the classic approach when the number of covariates dominates the number of datasets, and corroborate this result empirically on several high-dimensional multiple regression and classification problems.

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