Amortized Bayesian Multilevel Models

Obtaining accurate inference and faithful uncertainty quantification in reasonable time is a frontier of today's statistical research (Cranmer et al., 2020). One major difficulty arising in most experimental and almost all observational data is the presence of complex dependency structures, for example, due to natural groupings (e.g., data gathered in different countries) or repeated measurements of the same observational units over time (e.g., particles, bacteria, or people; Gelman and Hill, 2006). To leverage these dependency structures, multilevel models (MLMs), also referred to as latent variable, hierarchical, random, or mixed effects models, have become an integral part of modern Bayesian statistics (Goldstein, 2011; Gelman et al., 2013; McGlothlin and Viele, 2018; Finch et al., 2019; Yao et al., 2022). Despite the wide success of Bayesian MLMs across the quantitative sciences, a major challenge is their limited efficiency and scalability when dealing with large and complex data. This is because estimating the full posterior distribution of all parameters of interest can be very costly (Gelman et al., 2013).