The experience sampling method (ESM) is a longitudinal research design where participants report their thoughts, emotional states and behaviours multiple times a day. Our work is motivated by such data collected by the GrowIt! app, which was released to investigate daily emotions among adolescents during the COVID-19 pandemic. Current procedures to analyse ESM data face various challenges. While standard statistical techniques may not scale well to a high-dimensional setting, machine learning procedures can give biased results due to selection bias introduced by missingness. In our motivating dataset, adolescents dropped out due to previous strong feelings of negative emotions. Hence, the implied missing data are of the missing-at-random type that standard machine learning procedures cannot accommodate. We develop a novel neural network architecture that generalises mixed effects models to deep learning to overcome these challenges. It allows semi-parametric and flexible modelling of data's mean and correlation structure through fixed and random effects. For estimation, we use an adaptation of variational auto-encoders and a Bayesian data augmentation algorithm. Through this approach, the model can accommodate longitudinal outcomes following generic distributions, scale well to high-dimensional settings and provide valid inference when data are missing-at-random. We applied the Deep Generalised Mixed Model to the GrowIt! study and various simulations. The results show potential for the Deep Generalised Mixed Model, yet suboptimal performance due to model instability.
Scalable Bayesian inference for generalized linear mixed models (GLMMs) provides uncertainty-aware analysis of correlated longitudinal data, but existing scalable approaches largely assume low-dimensional tabular predictors and do not directly accommodate high-dimensional modalities such as images and text. We address this limitation by learning one or more modality-specific neural encoders jointly with a GLMM objective, then performing variance-corrected stochasticgradient MCMC for the GLMM parameters conditional on the learned representation. This conditional-Bayes design combines supervised representation learning with posterior uncertainty quantification for population-level effects, subjectspecific heterogeneity, and modality-level random slopes. The resulting model preserves interpretable fixed and random effects for structured covariates and learned modalities while scaling gracefully to large longitudinal datasets. In simulation studies, our method recovers posterior means and variance estimates from full-data MCMC benchmarks after covariance correction. We further evaluate uncertainty through parameter-level interval coverage in simulations and predictive calibration on held-out data. Applications to glaucoma progression and adolescent mental health demonstrate that the framework allows nuanced assessment of the relative importance of each modality on both individual and population levels without sacrificing predictive performance.
Yuankang Zhao, Youngsoo Baek, Felipe A. Medeiros +2
In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural network trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional Hölder class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the partition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural networks without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.
Neural network mixed-effects models (NMMs) have gained traction by combining the strong representation and predictive power of artificial neural networks with the capacity of mixed-effects modeling to capture complex correlation structures. However, existing estimation approaches rely heavily on manual derivations of objective functions and gradients, which inherently forces simplifying approximations and severely constrains the complexity and accuracy of NMMs. In this work, we introduce a general framework for implementing NMMs using Template Model Builder (TMB). By leveraging automatic differentiation and Laplace approximation, TMB requires users to specify only the negative joint log-likelihood and any regularization terms. The framework automatically integrates out random effects and evaluates the marginal objective function alongside its exact gradients, eliminating the need for manual derivations or ad hoc approximations. We demonstrate the efficiency, flexibility, and statistical performance of TMB-based NMMs across two numerical examples, including an application to monotonic NMMs. Reproducible code is provided to facilitate broader adoption.