Ensemble Multi-Quantile: Adaptively Flexible Distribution Prediction for Uncertainty Quantification

We propose a novel, succinct, and effective approach to quantify uncertainty in machine learning. It incorporates adaptively flexible distribution prediction for $\mathbb{P}(\mathbf{y}|\mathbf{X}=x)$ in regression tasks. For predicting this conditional distribution, its 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 $\mathbb{P}(\mathbf{y}|\mathbf{X}=x)$, 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 including Gaussian assumption-based, Bayesian methods, quantile regression-based, and traditional tree models, under the metrics of calibration, sharpness, and tail-side calibration. Visualization results show what we actually learn from the real data and how, illustrating the necessity and the merits of such an ensemble model.