Variational inference: uncertainty quantification in additive models

Markov chain Monte Carlo (MCMC)-based simulation approaches are by far the most common method in Bayesian inference to access the posterior distribution. Recently, motivated by successes in machine learning, variational inference (VI) has gained in interest in statistics since it promises a computationally efficient alternative to MCMC enabling approximate access to the posterior. Classical approaches such as mean-field VI (MFVI), however, are based on the strong mean-field assumption for the approximate posterior where parameters or parameter blocks are assumed to be mutually independent. As a consequence, parameter uncertainties are often underestimated and alternatives such as semi-implicit VI (SIVI) have been suggested to avoid the mean-field assumption and to improve uncertainty estimates. SIVI uses a hierarchical construction of the variational parameters to restore parameter dependencies and relies on a highly flexible implicit mixing distribution whose probability density function is not analytic but samples can be taken via a stochastic procedure. With this paper, we investigate how different forms of VI perform in semiparametric additive regression models as one of the most important fields of application of Bayesian inference in statistics. A particular focus is on the ability of the rivalling approaches to quantify uncertainty, especially with correlated covariates that are likely to aggravate the difficulties of simplifying VI assumptions. Moreover, we propose a method, where we combine both advantages of MFVI and SIVI and compare its performance. The different VI approaches are studied in comparison with MCMC in simulations and an application to tree height models of douglas fir based on a large-scale forestry data set.