Bayesian model selection in additive partial linear models via locally adaptive splines

TL;DR

This paper introduces a Bayesian model selection framework for additive partial linear models that effectively distinguishes between linear, nonlinear, or no effects of variables, using adaptive splines and MCMC sampling.

Contribution

It proposes a novel Bayesian approach with latent variables and pseudo-priors for flexible model selection and fitting in additive partial linear models.

Findings

Outperforms existing methods in effective sample sizes.

Reduces misclassification rates in variable effect determination.

Demonstrates effectiveness on numerical studies and epidemiology data.

Abstract

We provide a flexible framework for selecting among a class of additive partial linear models that allows both linear and nonlinear additive components. In practice, it is challenging to determine which additive components should be excluded from the model while simultaneously determining whether nonzero additive components should be represented as linear or non-linear components in the final model. In this paper, we propose a Bayesian model selection method that is facilitated by a carefully specified class of models, including the choice of a prior distribution and the nonparametric model used for the nonlinear additive components. We employ a series of latent variables that determine the effect of each variable among the three possibilities (no effect, linear effect, and nonlinear effect) and that simultaneously determine the knots of each spline for a suitable penalization of smooth functions. The use of a pseudo-prior distribution along with a collapsing scheme enables us to deploy well-behaved Markov chain Monte Carlo samplers, both for model selection and for fitting the preferred model. Our method and algorithm are deployed on a suite of numerical studies and are applied to a nutritional epidemiology study. The numerical results show that the proposed methodology outperforms previously available methods in terms of effective sample sizes of the Markov chain samplers and the overall misclassification rates.