  Given a reference model that includes all the available variables, projection
predictive inference replaces its posterior with a constrained projection
including only a subset of all variables. We extend projection predictive
inference to enable computationally efficient variable and structure selection
in models outside the exponential family. By adopting a latent space projection
predictive perspective we are able to: 1) propose a unified and general
framework to do variable selection in complex models while fully honouring the
original model structure, 2) properly identify relevant structure and retain
posterior uncertainties from the original model, and 3) provide an improved
approach also for non-Gaussian models in the exponential family. We demonstrate
the superior performance of our approach by thoroughly testing and comparing it
against popular variable selection approaches in a wide range of settings,
including realistic data sets. Our results show that our approach successfully
recovers relevant terms and model structure in complex models, selecting less
variables than competing approaches for realistic datasets.
