Mixing (or prior) density estimation is an important problem in machine learning and statistics, especially in empirical Bayes $g$-modeling where accurately estimating the prior is necessary for making good posterior inferences. In this paper, we propose neural-$g$, a new neural network-based estimator for $g$-modeling. Neural-$g$ uses a softmax output layer to ensure that the estimated prior is a valid probability density. Under default hyperparameters, we show that neural-$g$ is very flexible and capable of capturing many unknown densities, including those with flat regions, heavy tails, and/or discontinuities. In contrast, existing methods struggle to capture all of these prior shapes. We provide justification for neural-$g$ by establishing a new universal approximation theorem regarding the capability of neural networks to learn arbitrary probability mass functions. To accelerate convergence of our numerical implementation, we utilize a weighted average gradient descent approach to update the network parameters. Finally, we extend neural-$g$ to multivariate prior density estimation. We illustrate the efficacy of our approach through simulations and analyses of real datasets. A software package to implement neural-$g$ is publicly available at https://github.com/shijiew97/neuralG.

While high-dimensional data has been ubiquitous for some time, the use of longitudinal high-dimensional data or grouped (clustered) high-dimensional data has been recently increasing in research. For example, some genetic studies gather gene expression levels for an individual on multiple occasions in response to an exposure over time (Banchereau et al., 2016). Other ongoing studies - like the UK Biobank and the Adolescent Brain Cognitive Development Study - collect high-dimensional genetic/imaging information longitudinally to learn how individual changes in these markers are related to outcomes (Cole, 2020; Saragosa-Harris et al., 2022). Such data usually violates the traditional linear regression assumption that observations are independently and identically distributed. Data analysis should account for the dependence between observations belonging to the same individual. For the low dimensional setting where n p, extensive methodology is available for handling such data structures, e.g., linear mixed models (LMMs). The fields of LMMs and high-dimensional linear regression have extensive bodies of literature. However, they are largely separate, with a very narrow body of literature existing at the intersection of LMMs and high-dimensional longitudinal data (where p n). Unlike low-dimensional (p n) LMMs for which restricted maximum likelihood (REML) methods are readily available, fitting high-dimensional LMMs is considerably more challenging due to the non-convexity of the optimization function, which requires the inversion of large matrices in addition to iterative approaches. The few available methods for highdimensional LMMs rely on sparsity-inducing penalizations (e.g.

Sparse linear regression methods for high-dimensional data often assume that residuals have constant variance. When this assumption is violated, it can lead to bias in estimated coefficients, prediction intervals (PI) with improper length, and increased type I errors. We propose a heteroscedastic high-dimensional linear regression model through a partitioned empirical Bayes Expectation Conditional Maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient maximum a posteriori estimation approach based on a Parameter-Expanded Expectation-Conditional-Maximization algorithm. It requires minimal prior assumptions on the regression parameters through plug-in empirical Bayes estimates of hyperparameters. The variance model uses a multivariate log-Gamma prior on coefficients that can incorporate covariates hypothesized to impact heterogeneity. The motivation of our approach is a study relating Aphasia Quotient (AQ) to high-resolution T2 neuroimages of brain damage in stroke patients. AQ is a vital measure of language impairment and informs treatment decisions, but it is challenging to measure and subject to heteroscedastic errors. It is, therefore, of clinical importance -- and the goal of this paper -- to use high-dimensional neuroimages to predict and provide PIs for AQ that accurately reflect the heterogeneity in residual variance. Our analysis demonstrates that H-PROBE can use markers of heterogeneity to provide narrower PI widths than standard methods without sacrificing coverage. Through extensive simulation studies, we exhibit that H-PROBE results in superior prediction, variable selection, and predictive inference than competing methods.