Semiparametric Regression Using Variational Approximations

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.

Generalized functional linear model with a point process predictor

“…42 We adopt the Gaussian variational approximation approach in this article since it is computationally efficient and usually has superior or similar performance compared to the other approximation methods in different settings. 8,43 Like other approximation methods, the Gaussian variational approximation method is an approximate inference approach and may lead to biased estimates in certain scenarios. Based on our simulation studies, our approximate inference algorithm seems to work well, and the biases in the estimates are negligible.…”

“…Other approximation approaches may also be used for model estimation, such as Laplace approximation, 41 which has been used for fitting generalized linear mixed‐effects models 42 . We adopt the Gaussian variational approximation approach in this article since it is computationally efficient and usually has superior or similar performance compared to the other approximation methods in different settings 8,43 . Like other approximation methods, the Gaussian variational approximation method is an approximate inference approach and may lead to biased estimates in certain scenarios.…”

Sufficient dimension reduction for clustered data via finite mixture modelling

“…Such a modification however may increase the computation time substantially, and given the proposed approach is already relatively computationally burdensome compared to other SDR approaches (which is not surprising given the inclusion of random effects means we employed an Monte Carlo EM-algorithm to fit the model, as seen Section 3), then perhaps research should first focus on more efficient methods of estimation such as variational and expectation-propagation approaches (e.g. Kim & Wand 2018;Hui et al 2019).…”

Variational Inference for Bayesian Bridge Regression

“…A key point in variational inference concerns the choice of the variational family of distributions, which needs to be tractable while at the same time flexible enough to approximate the posterior distribution well. From a classical perspective, Hui, You, Shang, and M üller (2019) proposed coordinate ascending variational inference for generalized aditive models while, Armagan (2009) describe a mean-field variational inference method for Bayesian bridge regression model with approximate inference for the bridge parameter. Alves, Dias, and Migon (2021) also prooses a variational approach based on mean-field assumption for Bayesian inference in regression models with splines, however it is restricted to Lasso penalization.…”