H-Likelihood Approach to Deep Neural Networks with Temporal-Spatial Random Effects for High-Cardinality Categorical Features

Deep Neural Networks (DNNs) are one of the most powerful tools for prediction, but many of them implicitly assume that the data are statistically independent. However, in the real world, it is common for large-scale data to be clustered with temporal-spatial correlation structures. Variational approaches and integrated likelihood approaches have been proposed to obtain approximate maximum likelihood estimators (MLEs) for correlated data. However, due to the large size of data, they cannot provide exact MLEs. In this study, we propose a new hierarchical likelihood approach to DNNs with correlated random effects for clustered data. By jointly optimizing the the negative h-likelihood loss, we can provide exact MLEs for both mean and dispersion parameters, as well as the best linear unbiased predictors for the random effects. Moreover, the hierarchical likelihood allows a computable procedure for restricted maximum likelihood estimators of dispersion parameters. The proposed two-step algorithm enables online learning for the neural networks, whereas the integrated likelihood cannot decompose like a widely-used loss function in DNNs. The proposed h-likelihood approach offers several advantages, which we demonstrate through numerical studies and real data analyses.