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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
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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}
\onecolumn

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

\section{Model Details}
\label{sec:model_details}

In the following section, we provide an overview of the most important hyper parameters used to produce our results.
All models are implemented using \texttt{TensorFlow} (v2.15.1) and \texttt{TensorFlow Probability} (v0.23.0) and are trained using Adam~\citep{Tensorflow,TensorflowProbability,Kingma2017a}.
The implementation is available on GitHub\footnote{\url{https://github.com/MArpogaus/hybrid_flows}}.

\subsection{Simulation Data}
The following tables contain all hyper parameters of the models trained on the bivariate simulation data.

% \begin{table}[h!]
%   \centering
%   \begin{tabular}{lll}
%     \toprule
%     Hyperparameter                                                                      & Type        & Search Space              \\
%     \midrule
%     \texttt{fit\_kwargs.0.learning\_rate}                                               & float32     & $[0.0001, 0.1]$           \\
%     \texttt{fit\_kwargs.1.learning\_rate.scheduler\_kwargs.initial\_learning\_rate}     & float32     & $[0.00001, 0.01]$         \\
%     \texttt{fit\_kwargs.0.batch\_size}                                                  & categorical & $\{32, 512, 1024, 4096\}$ \\
%     \texttt{fit\_kwargs.1.batch\_size}                                                  & categorical & $\{32, 512, 1024, 4096\}$ \\
%     \texttt{fit\_kwargs.1.epochs}                                                       & categorical & $\{200, 400, 1000\}$      \\
%     \texttt{model\_kwargs.joint\_bijectors.0.parameters\_fn\_kwargs.hidden\_units.0}    & categorical & $\{32, 128, 512, 1024\}$  \\
%     \texttt{model\_kwargs.joint\_bijectors.0.parameters\_fn\_kwargs.hidden\_units.1}    & categorical & $\{32, 128, 512, 1024\}$  \\
%     \texttt{model\_kwargs.joint\_bijectors.0.parameters\_fn\_kwargs.hidden\_units.2}    & categorical & $\{32, 128, 512, 1024\}$  \\
%     \texttt{model\_kwargs.joint\_bijectors.0.parameters\_fn\_kwargs.parameter\_shape.1} & int         & $[8, 4096]$               \\
%     \bottomrule
%   \end{tabular}
%   \caption{Hyperparameter search space for the \gls{HCF}(B) model variants.}
%   \label{tab:hcfb_hyperparameter_search}
% \end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_cf_b.tex}
  \caption{Common Hyperparameters of \gls{CF} (B) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_model_cf_b.tex}
  \caption{Specific Hyperparameters of \gls{CF} (B) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_cf_s.tex}
  \caption{Common Hyperparameters of \gls{CF} (S) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_model_cf_s.tex}
  \caption{Specific Hyperparameters of \gls{CF} (S) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_hcf_b.tex}
  \caption{Common Hyperparameters of \gls{HCF} (B) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \resizebox{\linewidth}{!}{\input{hpos/hpos_sim_model_hcf_b.tex}}
  \caption{Specific Hyperparameters of \gls{HCF} (B) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_hcf_s.tex}
  \caption{Common Hyperparameters of \gls{HCF} (S) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \resizebox{\linewidth}{!}{\input{hpos/hpos_sim_model_hcf_s.tex}}
  \caption{Specific Hyperparameters of \gls{HCF} (S) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_maf_b.tex}
  \caption{Common Hyperparameters of \gls{MAF} (B) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_model_maf_b.tex}
  \caption{Specific Hyperparameters of \gls{MAF} (B) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_maf_s.tex}
  \caption{Common Hyperparameters of \gls{MAF} (S) models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_model_maf_s.tex}
  \caption{Specific Hyperparameters of \gls{MAF} (S) models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_mctm.tex}
  \caption{Common Hyperparameters of \gls{MCTM} models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \resizebox{\linewidth}{!}{\input{hpos/hpos_sim_model_mctm.tex}}
  \caption{Specific Hyperparameters of \gls{MCTM} models trained on simulated 2D data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_common_mvn.tex}
  \caption{Common Hyperparameters of \gls{MVN} models trained on simulated 2D data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_sim_model_mvn.tex}
  \caption{Specific Hyperparameters of \gls{MVN} models trained on simulated 2D data.}
\end{table}

\clearpage
\subsection{Benchmark Data}

The following tables contain all hyper parameters of the models traind on the multivariate benchmark data.

\begin{table}[h!]
  \centering
  \input{hpos/hpos_benchmark_common_hmaf_s.tex}
  \caption{Common Hyperparameters of \gls{HMAF} (S) models trained on benchmark data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_benchmark_model_hmaf_s.tex}
  \caption{Specific Hyperparameters of \gls{HMAF} (S) models trained on benchmark data.}
\end{table}

\begin{table}[h!]
  \centering
  \input{hpos/hpos_benchmark_common_maf_s.tex}
  \caption{Common Hyperparameters of \gls{MAF} (S) models trained on benchmark data.}
\end{table}
\begin{table}[h!]
  \centering
  \input{hpos/hpos_benchmark_model_maf_s.tex}
  \caption{Specific Hyperparameters of \gls{MAF} (S) models trained on benchmark data.}
\end{table}

\clearpage
\section{Malnutrition Data}

The following tables contain all hyper parameters of the models traind on the trivariate malnutrition data.

\begin{table}[h!]
  \centering
  \input{hpos/hpos_malnutrition_common.tex}
  \caption{Common Hyperparameters of models trained on malnutrition data.}
\end{table}

\begin{table}[h!]
  \centering
  \resizebox{\linewidth}{!}{\input{hpos/hpos_malnutrition_model.tex}}
  \caption{Specific Hyperparameters of models trained on malnutrition data.}
\end{table}

\clearpage

\begin{landscape}
  \section{Description of Hyperparameters}
  The following table gives a brief description of the hyperparameters and maps their table names to the canonical path inside the configurations files in the git repository\footnote{\url{https://github.com/MArpogaus/hybrid_flows}}.
  %For further details, please refer to the README.
  \input{hpos/hpos_description}
\end{landscape}

% References
\printbibliography

\end{document}
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