  Projection predictive inference is a decision theoretic Bayesian approach
that decouples model estimation from decision making. Given a reference model
previously built including all variables present in the data, projection
predictive inference projects its posterior onto a constrained space of a
subset of variables. Variable selection is then performed by sequentially
adding relevant variables until predictive performance is satisfactory.
Previously, projection predictive inference has been demonstrated only for
generalized linear models (GLMs) and Gaussian processes (GPs) where it showed
superior performance to competing variable selection procedures. In this work,
we extend projection predictive inference to support variable and structure
selection for generalized linear multilevel models (GLMMs) and generalized
additive multilevel models (GAMMs). Our simulative and real-word experiments
demonstrate that our method can drastically reduce the model complexity
required to reach reference predictive performance and achieve good frequency
properties.
