# Bayesian Function-on-Scalars Regression for High Dimensional Data

arXiv: https://arxiv.org/abs/1808.06689 · HTML: https://mlsanity.com/paper/1808.06689/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2018.

## Built on by (most cited first, 4 of 4)

- [Bayesian subset selection and variable importance for interpretable prediction and classification](https://mlsanity.com/paper/2104.10150/) (2021)
- [Subset selection for linear mixed models](https://mlsanity.com/paper/2107.12890/) (2021)
- [Bayesian adaptive and interpretable functional regression for exposure profiles](https://mlsanity.com/paper/2203.00784/) (2022)
- [Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression](https://mlsanity.com/paper/2502.16888/) (2025)

## Abstract (arXiv)

We develop a fully Bayesian framework for function-on-scalars regression with many predictors. The functional data response is modeled nonparametrically using unknown basis functions, which produces a flexible and data-adaptive functional basis. We incorporate shrinkage priors that effectively remove unimportant scalar covariates from the model and reduce sensitivity to the number of (unknown) basis functions. For variable selection in functional regression, we propose a decision theoretic posterior summarization technique, which identifies a subset of covariates that retains nearly the predictive accuracy of the full model. Our approach is broadly applicable for Bayesian functional regression models, and unlike existing methods provides joint rather than marginal selection of important predictor variables. Computationally scalable posterior inference is achieved using a Gibbs sampler with linear time complexity in the number of predictors. The resulting algorithm is empirically faster than existing frequentist and Bayesian techniques, and provides joint estimation of model parameters, prediction and imputation of functional trajectories, and uncertainty quantification via the posterior distribution. A simulation study demonstrates improvements in estimation accuracy, uncertainty quantification, and variable selection relative to existing alternatives. The methodology is applied to actigraphy data to investigate the association between intraday physical activity and responses to a sleep questionnaire.