Semiparametric Regression using Variational Approximations

TL;DR

This paper introduces a variational approximation framework for semiparametric regression, combining the stability of mixed models with the computational efficiency of penalized likelihood methods, and demonstrates its effectiveness through simulations.

Contribution

It develops a novel variational approximation approach for generalized additive models that is both computationally efficient and stable, with new inference tools and theoretical guarantees.

Findings

Performs similarly or better than existing software in simulations

Provides fully tractable variational likelihoods for common response types

Offers a variational information matrix and closed-form smoothing parameter updates

Abstract

Semiparametric regression offers a flexible framework for modeling non-linear relationships between a response and covariates. A prime example are generalized additive models where splines (say) are used to approximate non-linear functional components in conjunction with a quadratic penalty to control for overfitting. Estimation and inference are then generally performed based on the penalized likelihood, or under a mixed model framework. The penalized likelihood framework is fast but potentially unstable, and choosing the smoothing parameters needs to be done externally using cross-validation, for instance. The mixed model framework tends to be more stable and offers a natural way for choosing the smoothing parameters, but for non-normal responses involves an intractable integral. In this article, we introduce a new framework for semiparametric regression based on variational approximations. The approach possesses the stability and natural inference tools of the mixed model framework, while achieving computation times comparable to using penalized likelihood. Focusing on generalized additive models, we derive fully tractable variational likelihoods for some common response types. We present several features of the variational approximation framework for inference, including a variational information matrix for inference on parametric components, and a closed-form update for estimating the smoothing parameter. We demonstrate the consistency of the variational approximation estimates, and an asymptotic normality result for the parametric component of the model. Simulation studies show the variational approximation framework performs similarly to and sometimes better than currently available software for fitting generalized additive models.