In variety testing, multi-environment trials (MET) are essential for evaluating the genotypic performance of crop plants. A persistent challenge in the statistical analysis of MET data is the estimation of variance components, which are often still inaccurately estimated or shrunk to exactly zero when using residual (restricted) maximum likelihood (REML) approaches. At the same time, institutions conducting MET typically possess extensive historical data that can, in principle, be leveraged to improve variance component estimation. However, these data are rarely incorporated sufficiently. The purpose of this paper is to address this gap by proposing a Bayesian framework that systematically integrates historical information to stabilize variance component estimation and better quantify uncertainty. Our Bayesian linear mixed model (BLMM) reformulation uses priors and Markov chain Monte Carlo (MCMC) methods to maintain the variance components as positive, yielding more realistic distributional estimates. Furthermore, our model incorporates historical prior information by managing MET data in successive historical data windows. Variance component prior and posterior distributions are shown to be conjugate and belong to the inverse gamma and inverse Wishart families. While Bayesian methodology is increasingly being used for analyzing MET data, to the best of our knowledge, this study comprises one of the first serious attempts to objectively inform priors in the context of MET data. This refers to the proposed Bayesian updating approach. To demonstrate the framework, we consider an application where posterior variance component samples are plugged into an A-optimality experimental design criterion to determine the average optimal allocations of trials to agro-ecological zones in a sub-divided target population of environments (TPE).

Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.

This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.

Predicting Parkinson's Disease (PD) progression is crucial, and voice biomarkers offer a non-invasive method for tracking symptom severity (UPDRS scores) through telemonitoring. Analyzing this longitudinal data is challenging due to within-subject correlations and complex, nonlinear patient-specific progression patterns. This study benchmarks LMMs against two advanced hybrid approaches: the Generalized Neural Network Mixed Model (GNMM) (Mandel 2021), which embeds a neural network within a GLMM structure, and the Neural Mixed Effects (NME) model (Wortwein 2023), allowing nonlinear subject-specific parameters throughout the network. Using the Oxford Parkinson's telemonitoring voice dataset, we evaluate these models' performance in predicting Total UPDRS to offer practical guidance for PD research and clinical applications.

High-cardinality categorical features are a major challenge for machine learning methods in general and for deep learning in particular. Existing solutions such as one-hot encoding and entity embeddings can be hard to scale when the cardinality is very high, require much space, are hard to interpret or may overfit the data. A special scenario of interest is that of repeated measures, where the categorical feature is the identity of the individual or object, and each object is measured several times, possibly under different conditions (values of the other features). We propose accounting for high-cardinality categorical features as random effects variables in a regression setting, and consequently adopt the corresponding negative log likelihood loss from the linear mixed models (LMM) statistical literature and integrate it in a deep learning framework. We test our model which we call LMMNN on simulated as well as real datasets with a single categorical feature with high cardinality, using various baseline neural networks architectures such as convolutional networks and LSTM, and various applications in e-commerce, healthcare and computer vision. Our results show that treating high-cardinality categorical features as random effects leads to a significant improvement in prediction performance compared to state of the art alternatives. Potential extensions such as accounting for multiple categorical features and classification settings are discussed. Our code and simulations are available at https://github.com/gsimchoni/lmmnn.

In this work, inspired by machine learning techniques, we propose a new Bayesian model for Small Area Estimation (SAE), the Fay-Herriot model with Spectral Clustering (FH-SC). Unlike traditional approaches, clustering in FH-SC is based on spectral clustering algorithms that utilize external covariates, rather than geographical or administrative criteria. A major advantage of the FH-SC model is its flexibility in integrating existing SAE approaches, with or without clustering random effects. To enable benchmarking, we leverage the theoretical framework of posterior projections for constrained Bayesian inference and derive closed form expressions for the new Rao-Blackwell (RB) estimators of the posterior mean under the FH-SC model. Additionally, we introduce a novel measure of uncertainty for the benchmarked estimator, the Conditional Posterior Mean Square Error (CPMSE), which is generalizable to other Bayesian SAE estimators. We conduct model-based and data-based simulation studies to evaluate the frequentist properties of the CPMSE. The proposed methodology is motivated by a real case study involving the estimation of the proportion of households with internet access in the municipalities of Colombia. Finally, we also illustrate the advantages of FH-SC over existing Bayesian and frequentist approaches through our case study.

Linear mixed models are widely used for clustered data, but their reliance on parametric forms limits flexibility in complex and high-dimensional settings. In contrast, gradient boosting methods achieve high predictive accuracy through nonparametric estimation, but do not accommodate clustered data structures or provide uncertainty quantification. We introduce Gradient Boosted Mixed Models (GBMixed), a framework and algorithm that extends boosting to jointly estimate mean and variance components via likelihood-based gradients. In addition to nonparametric mean estimation, the method models both random effects and residual variances as potentially covariate-dependent functions using flexible base learners such as regression trees or splines, enabling nonparametric estimation while maintaining interpretability. Simulations and real-world applications demonstrate accurate recovery of variance components, calibrated prediction intervals, and improved predictive accuracy relative to standard linear mixed models and nonparametric methods. GBMixed provides heteroscedastic uncertainty quantification and introduces boosting for heterogeneous random effects. This enables covariate-dependent shrinkage for cluster-specific predictions to adapt between population and cluster-level data. Under standard causal assumptions, the framework enables estimation of heterogeneous treatment effects with reliable uncertainty quantification.