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Generalized linear latent variable models (GLLVMs) are a class of methods foranalyzing multi-response data which has garnered considerable popularity inrecent years, for example, in the analysis of multivariate abundance data inecology. One of the main features of GLLVMs is their capacity to handle avariety of responses types, such as (overdispersed) counts, binomial responses,(semi-)continuous, and proportions data. On the other hand, the introduction ofunderlying latent variables presents some major computational challenges, asthe resulting marginal likelihood function involves an intractable integral fornon-normally distributed responses. This has spurred research intoapproximation methods to overcome this integral, with a recent and particularlycomputationally scalable one being that of variational approximations (VA).However, research into the use of VA of GLLVMs and related models has beenhampered by the fact that closed-form approximations have only been obtainedfor certain pairs 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 drastically. EVA drawsinspiration from the underlying idea of Laplace approximations: by replacingthe complete-data likelihood function with its second order Taylorapproximation about the mean of the variational distribution, we can obtain aclosed-form approximation to the marginal likelihood of the GLLVM for anyresponse type and link function. Through simulation studies and an applicationto testate amoebae data set in ecology, we demonstrate how EVA results in auniversal approach to fitting GLLVMs, which remains competitive in terms ofestimation and inferential performance relative to both standard VA and aLaplace approximation approach, while being computationally more scalable thanboth in practice.

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