Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals
This work addresses the trade-off between interpretability and flexibility in multivariate density regression for data analysis applications, representing an incremental improvement by hybridizing existing techniques.
The paper tackled the problem of interpreting multivariate density regression by combining multivariate conditional transformation models (MCTM) with normalizing flows (NF) to achieve interpretable marginals and flexible joint distributions, demonstrating improved performance in numerical experiments compared to existing methods.
Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NF) 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 (MCTM) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTM with state-of-the-art and autoregressive NF to leverage the transparency of MCTM for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NF 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 MCTM and other NF models on both simulated and real-world data.