To improve the uncertainty quantification of variance networks, we propose a novel tree-structured local neural network model that partitions the feature space into multiple regions based on uncertainty heterogeneity. A tree is built upon giving the training data, whose leaf nodes represent different regions where region-specific neural networks are trained to predict both the mean and the variance for quantifying uncertainty. The proposed uncertainty-splitting neural regression tree (USNRT) employs novel splitting criteria. At each node, a neural network is trained on the full data first, and a statistical test for the residuals is conducted to find the best split, corresponding to the two subregions with the most significant uncertainty heterogeneity between them. USNRT is computationally friendly, because very few leaf nodes are sufficient and pruning is unnecessary. Furthermore, an ensemble version can be easily constructed to estimate the total uncertainty, including the aleatory and epistemic. On extensive UCI datasets, USNRT or its ensemble shows superior performance compared to some recent popular methods for quantifying uncertainty with variances. Through comprehensive visualization and analysis, we uncover how USNRT works and show its merits, revealing that uncertainty heterogeneity does exist in many datasets and can be learned by USNRT.

We propose a novel, succinct, and effective approach for distribution prediction to quantify uncertainty in machine learning. It incorporates adaptively flexible distribution prediction of [Formula: see text] in regression tasks. This conditional distribution's quantiles of probability levels spreading the interval (0,1) are boosted by additive models which are designed by us with intuitions and interpretability. We seek an adaptive balance between the structural integrity and the flexibility for [Formula: see text], while Gaussian assumption results in a lack of flexibility for real data and highly flexible approaches (e.g., estimating the quantiles separately without a distribution structure) inevitably have drawbacks and may not lead to good generalization. This ensemble multi-quantiles approach called EMQ proposed by us is totally data-driven, and can gradually depart from Gaussian and discover the optimal conditional distribution in the boosting. On extensive regression tasks from UCI datasets, we show that EMQ achieves state-of-the-art performance comparing to many recent uncertainty quantification methods. Visualization results further illustrate the necessity and the merits of such an ensemble model.