Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals

Abstract

Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NFs) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTMs) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTMs with state-of-the-art and autoregressive NFs to leverage the transparency of MCTMs for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NFs techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method’s versatility in various numerical experiments and compare it with MCTMs and other NF models on both simulated and real-world data.