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\title{Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals}

% The standard author block has changed for UAI 2025 to provide
% more space for long author lists and allow for complex affiliations
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% All author information is authomatically removed by the class for the
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% Add authors
\author[1]{\href{mailto:<marcel.arpogaus@htwg-konstanz.de>?Subject=Your UAI 2025 paper}{Marcel Arpogaus}{}}
\author[2]{Thomas Kneib}
\author[3,4]{Thomas Nagler}
\author[3,4]{David R\"ugamer}

% Add affiliations after the authors
\affil[1]{%
    Institute for Applied Research \\
    HTWG - University of Applied Sciences \\
    Konstanz, GERMANY
}
\affil[2]{%
    Chair of Statistics and Campus Institute Data Science (CIDAS) \\
    University of Göttingen \\
    Göttingen, GERMANY
}
\affil[3]{%
    Department of Statistics \\
    LMU Munich\\
    Munich, GERMANY
}
\affil[4]{%
    Munich Center for Machine Learning (MCML) \\
    Munich, GERMANY
}
\begin{document}
\maketitle

\begin{abstract}
  Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution.
  While flexible estimation approaches such as~\glspl{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~\glspl{MCTM} are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions.
  In this paper, we combine~\glspl{MCTM} with state-of-the-art and autoregressive~\glspl{NF} to leverage the transparency of~\glspl{MCTM} for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based~\gls{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~\glspl{MCTM} and other \gls{NF} models on both simulated and real-world data.
\end{abstract}

\input{content}

\begin{acknowledgements} 
During the finalization of this paper, large language models (LLMs) were used to optimize language, and grammar, and restructure some parts of the document.

This research was funded by the Carl-Zeiss-Stiftung in the project ”DeepCarbPlanner” (grant no. P2021-08-007).

We thank the DACHS data analysis cluster, hosted at Hochschule Esslingen and co-funded by the MWK within the DFG's ,,Großgeräte der Länder'' program, for providing the computational resources necessary for this research\footnote{https://wiki.bwhpc.de/e/DACHS}.
\end{acknowledgements}

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\title{Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density
Regression with Interpretable Marginal\\(Supplementary Material)}
\maketitle

\appendix

\input{appendix}

\end{document}

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