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\begin{poster}

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\newtheorem{theorem}{Theorem}
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\begin{minipage}[c][9cm][c]{0.9\textwidth}
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  \begin{center}
    {\sffamily \veryHuge \textsc{Conditional Bayesian Quadrature} }\\[6mm]
    {\huge \sffamily Zonghao Chen${}^1$ $\quad$ Masha Naslidnyk${}^1$ $\quad$ Arthur Gretton${}^1$ $\quad$ Fran\c{c}ois-Xavier Briol${}^1$
    \\[4mm]
    \LARGE
    ${}^1$\,University College London
    }
    \LARGE
    Correspondence to: \texttt{zonghao.chen.22@ucl.ac.uk}. \,
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%%% Beginning of Document
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\begin{multicols}{3}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mysection{\color{white}{TL;DR}}
% \vspace{0.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{itemize}
	\item
	We propose conditional Bayesian quadrature (CBQ): a numerical integration algorithm for conditional expectations. 
    \begin{align*}
        I(\theta)=\mathbb{E}_{X \sim \mathbb{P}_\theta}[f(X, \theta)]=\int_{\mathcal{X}} f(x, \theta) \mathbb{P}_\theta(\mathrm{d} x),
    \end{align*}
    \begin{itemize}
        \item $\mathbb{P}_\theta$ could be conditional distribution $p(x \mid \theta)$ and could also be any distribution parameterized by $\theta$.
    \end{itemize}
        \item The proposed CBQ algorithm can be expressed in two stages:
        \begin{itemize}
            \item \textbf{Stage 1}: Bayesian quadrature is employed to obtain posteriors on $I(\theta_1), \cdots, I(\theta_T)$.
            \item \textbf{Stage 2}: Heteroscedastic GP regression is performed to obtain the CBQ estimator $\hat{I}_{\text{CBQ}}(\theta)$.
        \end{itemize} 
	\item 
	CBQ provides a way of quantifying uncertainty, and leads to a fast convergence rate confirmed both theoretically and empirically on challenging tasks in mathematical finance, Bayesian sensitivity analysis and health economics.
\end{itemize}
\vspace*{0.5cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mysection{\color{white}{Notations}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{itemize}
    \item Our observations are parameters, points, and function values: $\theta_{1: T}:=\left[\theta_1, \cdots, \theta_T \right]^{\top}$, $x_{1: N}^t:=\left[x_1^t,  \cdots, x_N^t\right]^{\top}$, and $f\left(x_{1: N}^t, \theta_t\right)=\left[f\left(x_1^t, \theta_t\right) \cdots f\left(x_N^t, \theta_t\right)\right]^{\top} \in \mathbb{R}^N$ for all $t \in\{1, \ldots, T\}$.
\end{itemize}

\vspace*{0.5cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mysection{\color{white}{Background}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
	\textbf{Bayesian Quadrature}
\end{center}

\begin{itemize}
\item Bayesian quadrature (BQ) aims at computing the (unconditional) expectation $I=\mathbb{E}_{X \sim \mathbb{P}}[f(X)]$.
\item Suppose a Gaussian process (GP) prior on $f$ with zero mean and covariance $k_\calX: \calX \times \calX \to \R$ of smoothness $s_\calX$.
\item After conditioning on data $f\left(x_{1: N}\right)$, the posterior on $f$ will induce a univariate Gaussian posterior distribution on $I$.

{
\fontsize{35pt}{12pt}\selectfont
\begin{align*}
\begin{aligned}
\hat{I}_{\mathrm{BQ}} & = \mu\left(x_{1: N}\right)^{\top} k_{\mathcal{X}} \left(x_{1: N}, x_{1: N}\right)^{-1} f\left(x_{1: N}\right), \\
\sigma_{\mathrm{BQ}}^2 & =\mathbb{E}_{X, X^{\prime} \sim \mathbb{P}}\left[k_{\mathcal{X}}(X, X^{\prime})\right] -  \mu\left(x_{1: N}\right)^{\top} k_{\mathcal{X}}\left(x_{1: N}, x_{1: N}\right)^{-1} \mu\left(x_{1: N}\right)
\end{aligned}
\end{align*}
}

\item $\mu(x)=\mathbb{E}_{X \sim \mathbb{P}}\left[k_{\mathcal{X}}(X, x)\right]$ is the kernel mean embedding.

\item BQ has a faster rate of convergence $\calO(N^{-\frac{s_ \mathcal{X}}{d}})$ compared to the rate $\calO(N^{-\frac{1}{2}})$ of standard Monte Carlo.
\end{itemize}
\vspace*{-10cm}


\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mysection{\color{white}{{Conditional Bayesian Quadrature}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\centering
\vspace{10pt}

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every node/.style={scale=1.0, font=\LARGE, line width=1mm}, latent/.style={circle,fill=white, draw=black, minimum size=5pt}, 
my arrow/.style={->, >=latex, line width=1mm, scale=2}]
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  \node[latent, right=2.5cm of fx] (IBQ) {$\hat{I}_{\mathrm{BQ}}(\theta_t)$};
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\node[latent, above=2.9cm of ICBQ] (thetanew) {$\theta^\ast$};
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\end{tikzpicture}
    \setcounter{figure}{0}
    \centering
    \captionof{figure}{Directed acyclic graph representation of CBQ. Circle nodes indicate random variables and large rectangles correspond to independent replications over indices.}
    
% \vspace*{-0.5cm}
\begin{itemize}

\item CBQ is a two-stage approach
\begin{itemize}
    \item \textbf{Stage 1}: Bayesian quadrature is employed to obtain BQ posteriors $\hat{I}_{\mathrm{BQ}}\left(\theta_{1: T}\right), \sigma_{\mathrm{BQ}}^2\left(\theta_{1: T}\right)$ on $I(\theta_1), \cdots, I(\theta_T)$.
    \item \textbf{Stage 2}: Heteroscedastic GP regression is performed over $I(\theta)$ using the outputs from stage one. The posterior mean $\hat{I}_{\mathrm{CBQ}}(\theta^\ast)$ is the CBQ estimator for $I(\theta^\ast)$, and the variance $k_{\mathrm{CBQ}}(\theta^\ast, \theta^\ast)$ quantifies the uncertainty.
\end{itemize}

\item The CBQ estimator with uncertainty
\def\Id{\mathrm{Id}}
\vspace{-5pt}
\begin{align*}
    \hat{I}_{\mathrm{CBQ}}(\theta) &:= k_\Theta(\theta, \theta_{1:T})^\top \big( k_\Theta(\theta_{1:T}, \theta_{1:T}) \\ & \qquad  \qquad \qquad \quad + \sigma^2_\mathrm{BQ}(\theta_{1:T}) \Id_T \big)^{-1} \hat{I}_\mathrm{BQ}(\theta_{1:T}), \\
    k_{\mathrm{CBQ}}(\theta,\theta') &:= k_{\Theta}(\theta,\theta') - k_\Theta(\theta,\theta_{1:T})^\top \big( k_{\Theta}(\theta_{1:T}, \theta_{1:T}) \\ & \qquad  \qquad \qquad \quad + \sigma^2_\mathrm{BQ}(\theta_{1:T}) \Id_T \big)^{-1} k_\Theta(\theta_{1:T},\theta')
\end{align*}
\end{itemize}

\begin{theorem}\label{thm:convergence}
Suppose the following assumptions hold:
\vspace{-2pt}
\begin{enumerate}[itemsep=0.1pt,topsep=0pt,leftmargin=*]
\item $\Pb_\theta$ has density $p_\theta$ and $\sup_{\theta \in \Theta}\|p_{\theta}\|_{\calL^2(\calX)}<\infty$.
~\label{as:densities}
\item The kernels $k_\calX$ and $k_\Theta$ are Mat\'ern kernels of smoothness $s_\calX>d/2$ and $s_\Theta>p/2$ respectively.
~\label{as:kernels}
\item The function $x \mapsto f(x,\theta)$ is of smoothness at least $s_\calX$, $\theta \mapsto I(\theta)$ is of smoothness at least $s_\Theta$.
~\label{as:true_f_smoothness}
\item The source condition for $I(\theta)$ and $k_\Theta$ holds with $r \in [1/2,1]$.
~\label{as:src}
\end{enumerate}
\vspace{-55pt}
Then, we have that for any $\delta \in (0, 1)$ there is a $T_0(\delta)>0$ and an $N_0>0$ such that for any $N \geq N_0$ and $T \geq T_0$, with probability at least $1-\delta$ it holds that
\hspace{-50pt}
\begin{align*}
    \| \hat I_\mathrm{CBQ}(\theta) - I(\theta) \|_{\calL^2(\Theta)} \leq  C_1(\delta) T^{-\frac{r}{2r+1}} + C_2(\delta) T^{-\frac{r+1}{2r+1}} N^{-\frac{s_\calX}{d}} 
\end{align*}
with $C_1(\delta)=\calO(\log(1/\delta))$ and $C_2(\delta)=\calO((1/\delta)\log(1/\delta))$ that are independent of $N, T$.
\end{theorem}


\vspace*{0.5cm}


\vspace*{-3cm}



\newpage

\mysection{\color{white}{\mbox{Empirical Evaluations}}}

\begin{center}
	\textbf{Bayesian Sensitivity Analysis}
\end{center}
\begin{itemize}
    \item \colorbox{black!20}{Figure 2 \textbf{Left:}} shows the result of running Bayesian sensitivity analysis for linear models. The target is to analyze the sensitivity of the posterior to $\theta$ by computing $I(\theta) = \int_{\R^d} f(w)p_\theta(w|\calD) dw$.
\end{itemize}
\begin{minipage}{0.55\columnwidth}
\centering
\includegraphics[width=1.0\linewidth]{neurips_2023/figures/legend.pdf}\vspace{-20pt}
\end{minipage}

\begin{minipage}[c]{\columnwidth}
\centering
\hspace*{-20pt}\includegraphics[width=0.48\linewidth]{neurips_2023/figures/bayes_sensitivity_N_50.pdf}
\includegraphics[width=0.48\linewidth]{neurips_2023/figures/SIR_15.pdf}
\captionof{figure}{\colorbox{black!20}{\textbf{Left:}} Bayesian sensitivity analysis for linear model. \colorbox{black!20}{\textbf{Right:}} Bayesian sensitivity analysis for SIR model.}
\end{minipage}

\vspace*{0.5cm}

\begin{itemize}
    \item \colorbox{black!20}{Figure 2 \textbf{Right}} shows the result of running Bayesian sensitivity analysis for Susceptible-Infectious-Recovered (SIR) Model. The target is to analyze the sensitivity of the infection rate $\theta$ to the estimate of peak infection number $I_{\text{max}}(\theta)$.
\end{itemize}

\begin{minipage}{0.55\columnwidth}
\centering
\includegraphics[width=1.0\linewidth]{neurips_2023/figures/legend_finance.pdf}\vspace{-20pt}
\end{minipage}
\begin{minipage}[c]{\columnwidth}
\centering
\includegraphics[width=0.47\linewidth]{neurips_2023/figures/finance_T_20.pdf}
\includegraphics[width=0.47\linewidth]{neurips_2023/figures/decision_T_10_no_legend.pdf}
\captionof{figure}{\colorbox{black!20}{\textbf{Left:}} Butterfly Call Option with the Black-Scholes Model. \colorbox{black!20}{\textbf{Right:}} Uncertain Decision Making in Health Economics.}
\end{minipage}

\vspace*{0.5cm}
\begin{itemize}
    \item \colorbox{black!20}{Figure 3 \textbf{Left}} shows the result of computing the expected portfolio loss under a shock to the economy. \colorbox{black!20}{Figure 3 \textbf{Right}} shows the result of computing the expected value of partial perfect information (EVPPI) given precise knowledge of extra variables.
\end{itemize}

\vspace*{0.5cm}
\textbox{\flushleft{
\vspace*{-0.2cm}
\fontsize{40}{40}\selectfont
\begin{center}
    \textbf{\color{red}{Full paper}}:
    \href{https://hudsonchen.github.io/papers/CBQ.pdf}{\texttt{hudsonchen.github.io/papers/CBQ.pdf}}
\end{center}
}}

\vspace*{-6cm}


\raggedright

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\end{multicols}
\end{poster}

\end{document}
