\documentclass{article}


% if you need to pass options to natbib, use, e.g.:
%     \PassOptionsToPackage{numbers, compress}{natbib}
% before loading neurips_2022


% ready for submission
% \usepackage{include/neurips_2022}


% to compile a preprint version, e.g., for submission to arXiv, add add the
% [preprint] option:
% \usepackage[preprint]{include/neurips_2022}


% to compile a camera-ready version, add the [final] option, e.g.:
    % \usepackage[final]{neurips_2022}


% to avoid loading the natbib package, add option nonatbib:
%    \usepackage[nonatbib]{neurips_2022}

\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc}    % use 8-bit T1 fonts   % hyperlinks
\usepackage{url}            % simple URL typesetting
\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}
\usepackage{hyperref}  
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage{wrapfig}
\usepackage{tikz}
\usepackage{natbib}
% \usepackage{include/titletoc}
% \usetikzlibrary{bayesnet}
\usepackage[a4paper, total={6in, 8in}]{geometry}


\newcommand{\fxb}[1]{\textcolor{orange}{[FXB: #1]}}
\newcommand{\masha}[1]{\textcolor{teal}{[Masha: #1]}}
\newcommand{\hudson}[1]{\textcolor{teal}{[Hudson: #1]}}

\renewcommand{\thefigure}{S\arabic{figure}}

% \linespread{0.97} \selectfont
% \setlength{\parskip}{1pt}
% \usepackage{enumitem}
% \setlist{nolistsep,leftmargin=*}

%\title{Bayesian Estimation of Conditional Expectations}

% The \author macro works with any number of authors. There are two commands
% used to separate the names and addresses of multiple authors: \And and \AND.
%
% Using \And between authors leaves it to LaTeX to determine where to break the
% lines. Using \AND forces a line break at that point. So, if LaTeX puts 3 of 4
% authors names on the first line, and the last on the second line, try using
% \AND instead of \And before the third author name.


% \author{Zonghao Chen \\ Department of Computer Science \\ University College London 
% \And Masha Naslidnyk \\ Department of Computer Science \\ University College London
% \AND Arthur Gretton \\ Gatsby Computational Neuroscience Unit \\University College London
% \And Fran\c{c}ois-Xavier Briol\\ Department of Statistical Science \\ University College London}

% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma}

\begin{document}
\begin{figure}[t]
\vspace{-10pt}
    \centering
    \begin{minipage}{\textwidth}
    \centering
    \includegraphics[width=300pt]{neurips_2023/rebuttal_figures/legend.pdf}
    \vspace{-5pt}
    \end{minipage}
    
    \centering
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/bayes_sensitivity_T_50.pdf}
        % \caption{RMSE with fixed $N$}
        \label{fig:bayes_sensitivity_1}
    \end{subfigure}%
    \hfill % Add horizontal space between the subfigures
    % Second plot
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/bayes_sensitivity_time_T_50.pdf}
        % \caption{RMSE with fixed $T$}
        \label{fig:bayes_sensitivity_2}
    \end{subfigure}%
    \hfill % Add horizontal space between the subfigures
    % Third plot
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/sensitivity_nystrom.pdf}
        % \caption{RMSE with increasing $D$}
        \label{fig:bayes_sensitivity_3}
    \end{subfigure}
    \vspace{-5pt}
    \caption{\emph{Bayesian sensitivity analysis.} \textbf{Left:} RMSE of all methods when $d=2$ and $T=50$. \textbf{Middle:} Computational time for MOBQ and CBQ when $d=2$ and $T=50$. \textbf{Right:} RMSE of CBQ, MOBQ, MOBQ (Nystr\"om) when $d=2$ and $T=50$.}
    \label{fig:bayes_sensitivity}
\end{figure}

\begin{figure}[t]
\vspace{-10pt}
    \centering
    \begin{minipage}{\textwidth}
    \centering
    \includegraphics[width=300pt]
    {neurips_2023/rebuttal_figures/legend.pdf}
    \vspace{-5pt}
    \end{minipage}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/finance_T_20.pdf}
        % \caption{RMSE with fixed $N$}
        \label{fig:finance_1}
    \end{subfigure}%
    \hfill % Add horizontal space between the subfigures
    % Second plot
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/finance_time_T_20.pdf}
        % \caption{RMSE with fixed $T$}
        \label{fig:finance_2}
    \end{subfigure}%
    \hfill % Add horizontal space between the subfigures
    % Third plot
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/finance_nystrom.pdf}
        % \caption{RMSE with increasing $D$}
        \label{fig:finance_3}
    \end{subfigure}
    \vspace{-5pt}
    \caption{\emph{Black-Scholes model.} \textbf{Left:} RMSE of all methods when $T=20$. \textbf{Middle:} Computational time for MOBQ and CBQ when $T=20$. \textbf{Right:} RMSE of CBQ, MOBQ, MOBQ (Nystr\"om) when $T=20$.}
    \label{fig:bayes_sensitivity}
\end{figure}

\begin{figure}[t]
\vspace{-10pt}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/rates_N_1000.pdf}
        % \caption{RMSE with fixed $N$}
        \label{fig:rates_1}
    \end{subfigure}%
    \hfill % Add horizontal space between the subfigures
    % Second plot
    \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/rates_T_1000.pdf}
        % \caption{RMSE with fixed $T$}
        \label{fig:rates_2}
    \end{subfigure}%
        \begin{subfigure}{0.33\textwidth}
        \centering
        \hspace{-10pt}
        \includegraphics[width=\linewidth]{neurips_2023/rebuttal_figures/ablation_noise.pdf}
        % \caption{RMSE with fixed $T$}
        \label{fig:noise}
    \end{subfigure}%
    \caption{\emph{Numerical verification of convergence rate in Theorem 1.} \textbf{Left:} RMSE of \textrm{CBQ} when $N=1000$. \textbf{Middle:} RMSE of \textrm{CBQ} when $T=1000$. \textbf{Right:} RMSE of CBQ with different observation noises $\frac{1}{2} \sigma^2_{\textrm{BQ}}$, $\sigma^2_{\textrm{BQ}}$, 
    $2 \sigma^2_{\textrm{BQ}}$.}
    \label{fig:bayes_sensitivity}
\end{figure}

\end{document}