Fast and universal estimation of latent variable models using extended variational approximations

Generalized linear latent variable models (GLLVMs) are a class of methods for analyzing multi-response data which has gained considerable popularity in recent years, e.g., in the analysis of multivariate abundance data in ecology. One of the main features of GLLVMs is their capacity to handle a variety of responses types, such as (overdispersed) counts, binomial and (semi-)continuous responses, and proportions data. On the other hand, the inclusion of unobserved latent variables poses a major computational challenge, as the resulting marginal likelihood function involves an intractable integral for non-normally distributed responses. This has spurred research into a number of approximation methods to overcome this integral, with a recent and particularly computationally scalable one being that of variational approximations (VA). However, research into the use of VA for GLLVMs has been hampered by the fact that fully closed-form variational lower bounds have only been obtained for certain combinations of response distributions and link functions. In this article, we propose an extended variational approximations (EVA) approach which widens the set of VA-applicable GLLVMs dramatically. EVA draws inspiration from the underlying idea behind the Laplace approximation: by replacing the complete-data likelihood function with its second order Taylor approximation about the mean of the variational distribution, we can obtain a fully closed-form approximation to the marginal likelihood of the GLLVM for any response type and link function. Through simulation studies and an application to a species community of testate amoebae, we demonstrate how EVA results in a “universal” approach to fitting GLLVMs, which remains competitive in terms of estimation and inferential performance relative to both standard VA (where any intractable integrals are either overcome through reparametrization or quadrature) and a Laplace approximation approach, while being computationally more scalable than both methods in practice.