text:
\subsubsection*{\bibname}}

\begin{document}
\runningtitle{$\gamma$-ABC: Outlier-Robust Approximate Bayesian Computation Based on a Robust Divergence Estimator}

\twocolumn[

\aistatstitle{$\gamma$-ABC: Outlier-Robust Approximate Bayesian Computation\newline{} Based on a Robust Divergence Estimator}

\aistatsauthor{ Masahiro Fujisawa \And Takeshi Teshima \And  Issei Sato \And Masashi Sugiyama }

\aistatsaddress{ The University of Tokyo \\ RIKEN AIP \\ fujisawa@ms.k.u-tokyo.ac.jp \And The University of Tokyo \\ RIKEN AIP \\ teshima@ms.k.u-tokyo.ac.jp \And The University of Tokyo \\ RIKEN AIP \\ sato@k.u-tokyo.ac.jp \And RIKEN AIP \\ The University of Tokyo \\ sugi@k.u-tokyo.ac.jp} ]

\begin{abstract}
Approximate Bayesian computation (ABC) is a likelihood-free inference
method that has been employed in various applications.
\diff{However, ABC can be sensitive to outliers if a data discrepancy measure is chosen inappropriately.}
In this paper, we propose to use a
nearest-neighbor-based $\gamma$-divergence estimator as a data
discrepancy measure. We show that our estimator possesses a suitable theoretical
robustness property called the \emph{redescending} property. In
addition, our estimator enjoys various desirable properties such as
high flexibility, asymptotic unbiasedness, almost sure convergence,
and linear-time computational complexity.
Through experiments, we demonstrate that our method achieves
significantly higher robustness than existing discrepancy measures.
\end{abstract}

\section{Introduction}
\label{intro}
Approximate Bayesian computation (ABC) has been proposed as a ``likelihood-free'' inference scheme to approximately perform Bayesian inference when a complex model is used and it is impossible or difficult to compute its likelihood (see \cite{Marin12} for a general overview).
Instead of investigating the explicit form of the likelihood function, \diff{ABC seeks parameters of a simulator-based model that can generate data that is close to the observed data under some discrepancy measure.}
ABC has been applied to many research fields, e.g., evolutionary biology~\cite{Tavare97}, dynamic systems~\cite{Simon10}, economics~\cite{Peters12}, epidemiology~\cite{Blum20}. aeronautics~\cite{Jason17}, and astronomy~\cite{Cameron12}.

Rejection ABC~\cite{Tavare97,Pritchard99,Jiang18}, the basic ABC algorithm, proceeds as follows: (i) we draw an independent sample of the parameter $\theta$ from some prior $\pi$,
(ii) we simulate data $Y^m = \{Y_{j}\}_{1:m}$ for each value of $\theta$,
(iii) the parameter $\theta$ is discarded if the discrepancy $D(X^{n},Y^{m})$ between the observed data $X^{n} = \{X_{i}\}_{1:n}$ and the simulated data $Y^m = \{Y_{j}\}_{1:m}$ exceeds a tolerance threshold $\epsilon$.
The accepted $\theta$ is used in the subsequent inference as a sample from an approximation to the posterior distribution called the \emph{ABC posterior distribution}.
Many studies have been performed to enhance the computational efficiency of the rejection ABC scheme, e.g., applying Markov Chain Monte Carlo (MCMC)~\cite{Marjoram03,Wegmann09} or sequential Monte Carlo (SMC)~\cite{Sisson09,Drovandi11,Pierre12}.
\begin{table*}[t]
\centering
\caption{Relationship between previous work and our work (\emph{OR}: Outlier robustness, \emph{RP}: Redescending property~\cite{MaronnaRobust2019}, \emph{AU}: Asymptotically unbiasedness, \emph{ASC}: Almost sure convergence, \emph{QA}: ABC posterior analysis).
In MONK~\cite{Lerasle19}, the number of blocks that divide the data is denoted by $Q$.
The order of time costs for the $q$-Wasserstein distance is based on approximate optimization algorithms~\cite{Cuturi13,cuturi14}.
The order of time costs for the CAD is based on logistic regression, where $d$ is the dimension of the observed and synthesized data.
The symbol $(n \lor m)$ denotes $\max \{n, m \}$.
}
\label{Related work
\scalebox{1}{
\begin{tabular}{l|cccccc}
\hline
\multicolumn{1}{c}{Discrepancy measure}                & \begin{tabular}[c]{@{}c@{}}OR\end{tabular} & \begin{tabular}[c]{@{}c@{}}RP\end{tabular} & \begin{tabular}[c]{@{}c@{}}AU\end{tabular} & \begin{tabular}[c]{@{}c@{}}ASC\end{tabular} & \begin{tabular}[c]{@{}c@{}}QA\end{tabular} & \begin{tabular}[c]{@{}c@{}}Time cost\end{tabular} \\ \hline \hline
MMD~\cite{Park16,smola07}                                      & -                                                            & -                                                               & -                                                                 & -                                                                 & -                                                                  & $\mathcal{O}((n+m)^{2})$                            \\
$q$-Wasserstein
distance~\cite{Bernton17}                 & -                                                            & -                                                               & -                                                                 & -                                                                 & -                                                                  & $\mathcal{O}((n+m)^{2})$                            \\
CAD~\cite{Gutmann18}                                & -                                                            & -                                                               & -                                                                 & -                                                                 & $\checkmark$                                                       & $\mathcal{O}((n+m)d)$                                  \\
MONK BCD-Fast
~\cite{Lerasle19}               & $\checkmark$                                                 & -                                                               & -                                                                 & -                                                                 & -                                                                  & $\mathcal{O}
\bigg(\frac{(n+m)^{3}}{Q^{2}}\bigg)$              \\
KL-divergence estimator~\cite{PrezCruz08,Jiang18}                  & -                                                            & -                                                               & $\checkmark$                                                      & $\checkmark$                                                      & $\checkmark$                                                       & $\mathcal{O}((n \lor m) \log (n \lor m))$           \\
\textbf{$\gamma$-divergence estimator (ours)} & $\checkmark$                                                 & $\checkmark$                                                    & $\checkmark$                                                      & $\checkmark$                                                      & $\checkmark$                                                       & $\mathcal{O}((n \lor m) \log (n \lor m))$
\end{tabular}
}
\end{table*}

The core element of ABC is the data discrepancy measure $D(X^{n},Y^{m})$ and the \diff{accuracy of parameters} from the ABC posterior distribution crucially depends on its choice.
While many discrepancies for ABC have been proposed, such as the distance between summary statistics~\cite{blum13,Drovandi15,Wegmann09}, the maximum mean discrepancy (MMD)~\cite{Park16}, the Wasserstein distance~\cite{Bernton17}, a Kullback-Leibler (KL) divergence estimator~\cite{Jiang18}, and the classification accuracy discrepancy (CAD)~\cite{Gutmann18}, these are often not robust to severe contamination of data~\cite{Ruli20,Lerasle19,staerman20}.
Recently, two outlier-robust discrepancies have been proposed:
one is a robust discrepancy based on MMD~\cite{Lerasle19}, and the other is using a robust M-estimator, e.g., Huber's estimation function~\cite{Ruli20}.
However, these methods do not possess an ideal robust property for an extreme outlier, called the \textit{redescending property}~\cite{MaronnaRobust2019}.
In addition, the former method has the cubic time cost $\mathcal{O}((n + m)^{3}/Q^{2})$, where $Q$ is the number of blocks that divide the data.
When ABC is applied to astronomy~\cite{Kremer17,Cameron12}, for example, we have to deal with noisy large-scale datasets, and the cubic time cost can be intractable.
For the CAD, we may improve the robustness of the CAD by employing a robust classifier, such as robust LDA~\cite{croux01}, robust FDA~\cite{kim06}, or robust logistic regression~\cite{feng14}; however, its performance depends on the choice of the classifier, and its validity and robustness for heavy contamination data are not guaranteed in the ABC framework.
\diff{Outside the ABC framework, recently, many robust inference schemes have been proposed, e.g., using robust divergences for \emph{parametric} model inference~\cite{Basu98,Fujisawa08}, Bayesian inference~\cite{Knoblauch18,Jewson18,Nakagawa20}, variational inference~\cite{futami18}, \diff{and} constructing Bayesian inference through a pseudo-likelihood via MMD~\cite{Cherief20}.
Unfortunately, these methods cannot be used as a data discrepancy measure for ABC because these assume a \emph{tractable} likelihood or \emph{parametric} models.}
Therefore, there is no discrepancy measure that has both \emph{well-guaranteed} robustness for an extreme outlier and reasonable time costs.

In this paper, we propose a novel outlier-robust and computationally-efficient discrepancy measure based on the $\gamma$-divergence~\cite{Fujisawa08}.
Our discrepancy measure results in a robustness property of the ABC posterior called the \emph{redescending property}~\cite{MaronnaRobust2019}, i.e., it automatically ignores extreme outliers in the observed data.
Furthermore, we show that the $\gamma$-divergence estimation using a naive $k$-nearest neighbor density estimate has desirable asymptotic properties, which is not straightforward to prove unlike divergence estimators in $f$-divergence class, such as $\alpha$-divergence estimator~\cite{poczos11a}.
Table~\ref{Related work} summarizes the relations among our method and major existing discrepancy measures.
Our contributions are as follows.
\begin{itemize}
\setlength{\parskip}{-0.2cm}
\setlength{\itemsep}{0.15cm}
    \item We construct a non-parametric and robust divergence estimator based on the $\gamma$-divergence (Section \ref{subsec:gamma_div_estimator_derivation}).
    \item We show that our method theoretically enjoys the robustness and validity (Sections \ref{sec:robust_analysis} and \ref{sec:asymp_quasi_posterior}).
    \item \diff{We show that our method has the asymptotic unbiasedness and the almost sure convergence property, which are mathematically much harder to show than those for the divergence estimators belonging to the $f$-divergence family (Section \ref{subsec:asymp_analysis}).}
    \item Through experiments, we show that our estimator can significantly reduce the influence of the outliers even when the observed data have a large number of outliers (Section \ref{sec:experiments}).
\end{itemize}

The rest of this paper is organized as follows.
We summarize the ABC framework in Section \ref{sec:ABC}.
In Sections~\ref{sec:robust_div_est} and \ref{sec:asm_anals}, we introduce the $k$-nearest neighborhood ($k$-NN) based density estimation, explain how to derive our divergence estimator based on $k$-NN, and conduct the theoretical analyses.
Finally, we show the experimental results and the conclusion in Sections~\ref{sec:experiments} and \ref{sec:conclusion}.

\section{Preliminaries}
\label{sec:ABC}
In this section, we give an overview of ABC.
More detailed descriptions of the discrepancy measures which are often used in ABC can be found in Appendix~\ref{app:detail_discrepancy}.

\subsection{Approximate Bayesian Computation}
\diff{We define $\mathcal{X} \subset \mathbb{R}^{d}$ as the data space and $\Theta$ as the parameter space.}
The model $\{p_{\theta}: \theta \in \Theta \}$ is a family of probability distributions on $\mathcal{X}$ and has no explicit formula, but we assume that we can generate i.i.d.\  random samples from $p_\theta$ given the value of $\theta$.
The purpose of ABC is to seek the model parameter $\theta$ by comparing the observed data $ X_{1},\ldots,X_{n}\overset{\textrm{i.i.d.}}{\sim} p_{\theta^{*}}$ and the synthetic data $ Y_{1},\ldots,Y_{m} \overset{\textrm{i.i.d.}}{\sim} p_{\theta}$, where $\theta^*\in\Theta$ is the true parameter.
The criterion used to compare these datasets $X^n=\{X_1,\ldots,X_n\}$ and $Y^m=\{Y_1,\ldots,Y_m\}$ is the data discrepancy measure $D(X^{n},Y^{m})$ \diff{defined over $\mathcal{X}^{n} \times \mathcal{X}^{m}$.}

\begin{algorithm}[t]
                \caption{Rejection ABC Algorithm~\cite{Tavare97,Pritchard99}}
                \label{alg:reject_abc}
                \footnotesize
                \begin{algorithmic}[1]
                        \REQUIRE {Observed data $\{X_{i}\}_{i=1}^{n}$, prior $\pi(\theta)$ on the parameter space $\Theta$, tolerance threshold $\epsilon$, data discrepancy measure $D$}

                        \STATE \textbf{Initialize:} $\epsilon$
                        \FOR {$t=1$ to $T$}
                        \STATE \textbf{repeat:} propose $\theta \sim \pi(\theta)$ and draw $Y_{1} \ldots,Y_{m} \overset{\textrm{i.i.d.}}{\sim} p_{\theta}$
                        \STATE \textbf{until:} $D(X^{n}, Y^{m}) < \epsilon$
                        \STATE Obtain $\theta^{(t)} = \theta$
                        \ENDFOR
                        \RETURN $\{ \theta^{(t)} \}_{t=1}^{T}$
                \end{algorithmic}
\end{algorithm}

\diff{A well known algorithm of ABC is the rejection ABC~\cite{Tavare97,Pritchard99,Jiang18}, which proceeds as follows: (i) we draw an independent sample of the parameter $\theta$ from some prior $\pi$,
(ii) we simulate data $Y^m = \{Y_{j}\}_{1:m}$ for each value of $\theta$,
(iii) the parameter $\theta$ is discarded if the discrepancy $D(X^{n},Y^{m})$ between the observed data $X^{n} = \{X_{i}\}_{1:n}$ and the simulated data $Y^m = \{Y_{j}\}_{1:m}$ exceeds a tolerance threshold $\epsilon$.
}
The rejection ABC algorithm is shown in Algorithm~\ref{alg:reject_abc}.
It enables us to obtain i.i.d.\  random samples $\{\theta^{(t)} \}_{t=1}^{T}$ from the ABC posterior distribution defined as follows.
\begin{definition}[ABC posterior distribution]
\diff{Let $\epsilon$ be fixed.}
Then, the ABC posterior distribution is defined by

\footnotesize
\begin{align}
\label{eq:quasi_posterior}
    \pi(\theta&|X^{n},D, \epsilon)

    \propto \int \pi(\theta) \Ind{D(X^{n},Y^{m}) < \epsilon}p_{\theta}(Y^{m}) \mathrm{d}Y^{m},
\end{align}
\normalsize
where $\pi(\theta)$ is a prior over the parameter space $\Theta$, $\epsilon > 0$ is a tolerance threshold, and $p_{\theta}(Y^{m}) = \prod_{j=1}^{m}p_{\theta}(Y_{j})$.
\end{definition}

We only focus on the rejection ABC~\cite{Tavare97,Pritchard99,Jiang18} throughout this paper.
\diff{The main reason is two-fold: (i) to make a fair comparison and followed the experimental setting of the recent paper~\cite{Jiang18} proposing discrepancy measures for ABC, and (ii) to give theoretical guarantees to the ABC posterior explicitly, e.g., Theorem~\ref{thm:redescending-sensitivity-curve} and Corollary~\ref{cor:asymp_quasi_posterior_gamma} in this paper.
The rejection ABC is a reasonable choice for this purpose.}
\diff{While there are many sophisticated ABC algorithms, they are often extensions of the rejection ABC~\cite{Marjoram03,Wegmann09,Sisson09,Drovandi11,Pierre12}; therefore, we can easily combine our development of the rejection ABC with these algorithms.}

\subsection{Model of Data Contamination}
\label{subsec:def_contami}
In this paper, \diff{we assume that the \emph{observed} data are contaminated with outliers} and focus on Huber's contamination-by-outlier case~\cite{Huber64}, where observed data are sampled i.i.d.\  from the following distribution:
\begin{align}
\label{def_huber_contami}
    (1-\eta)G(x) + \eta H(x),
\end{align}
where $G(x)$ is a distribution we are interested in, $H(x)$ is an arbitrary contamination distribution, and $\eta \in [0, 1]$ is the proportion of contamination.
If $\eta$ is relatively high, e.g., $\eta=0.2$, the observed data are highly contaminated by $H$.

Due to distribution contamination described above, severe bias occurs in parameter estimation.
Many robust estimation methods have been proposed to reduce the estimation bias caused by outliers~\cite{Huber64,Huber09,windham95,Basu98}.
However, these methods tend to exhibit undesirable behaviors both empirically and theoretically, for heavily contaminated data \cite{Fujisawa08}.
Furthermore, for \emph{non-parametric} inference schemes such as ABC, many of such robust inference frameworks cannot be used because they normally assume that the likelihood is tractable.
Although \citet{Lerasle19} and \citet{Ruli20} have proposed robust discrepancy measures that can be compatible with non-parametric inference, these methods also cannot deal with a heavy contamination and the former method has high time costs.
To conduct a robust non-parametric inference for heavily contaminated data, it is necessary to construct an alternative discrepancy measure with both robustness for an extreme outlier and reasonable time costs.

\section{\texorpdfstring{$\gamma$}{Lg}-ABC and Its Robustness}
\label{sec:robust_div_est}
In this section, we construct a \emph{non-parametric} ``likelihood-free'' inference scheme based on the $\gamma$-divergence that has been used in robust parameter estimation from heavily contaminated data.
In Section \ref{subsec:derivation_gamma}, we introduce the $\gamma$-divergence and explain why we choose a $k$-NN based density estimation to derive our estimator.
Next, we overview a $k$-NN based density estimation in Section \ref{subsec:knn_density_est} and derive our estimator in Section \ref{subsec:gamma_div_estimator_derivation}.
Finally, we guarantee the robustness of the ABC based on our estimator in Section \ref{sec:robust_analysis}.

\subsection{\texorpdfstring{$\gamma$}{Lg}-divergence and Its Estimation}
\label{subsec:derivation_gamma}
To make a robust parameter estimation in the heavily contamination situation described in Section \ref{subsec:def_contami}, \citet{Fujisawa08} proposed the \emph{$\gamma$-divergence}, which possesses strong robustness for heavily contaminated data.
\begin{definition}[$\gamma$-divergence~\cite{Fujisawa08}]
\label{def:gamma_div}
Let $p$ and $q$ be positive measurable functions from
a measurable set $\mathcal{M}_0 \subseteq \mathbb{R}^d$ to $\mathbb{R}$.
Let $\gamma > 0$.
Then, the $\gamma$-divergence is defined as
\small
\begin{align}
\label{gamma_div}
    D_{\gamma}&(p\|q) \notag \\
    &= \frac{1}{\gamma(1+\gamma)} \log \frac{\bigg( \int_{\mathcal{M}_{0}} p^{1+\gamma}(x) \mathrm{d}x \bigg) \bigg( \int_{\mathcal{M}_{0}} q^{1+\gamma}(x) \mathrm{d}x \bigg)^{\gamma} }{\bigg( \int_{\mathcal{M}_{0}} p(x)q^{\gamma}(x)\mathrm{d}x \bigg)^{1+\gamma}},
\end{align}
\normalsize
\end{definition}
To combine the $\gamma$-divergence with ABC, we need to estimate Eq.~\eqref{gamma_div} from observed and synthesized data.
\diff{A potential approach we can consider} to estimating Eq.~\eqref{gamma_div} is extending $f$-divergence estimation frameworks, e.g., kernel density estimation (KDE)~\cite{Hardle06} and direct density ratio estimation methods such as KLIEP~\cite{sugi08} or uLSIF~\cite{kanamori09}.
However, the former method suffers from high time cost due to a kernel function and the necessity to select appropriate kernels and its parameters.
Furthermore, the latter methods need to construct a model directly for a density ratio; therefore, it is hard to use this approach in the estimation of Eq.~\eqref{gamma_div} since $\gamma$-divergence, which is not included in the $f$-divergence class, is not expressed as a functional of the density ratio.

For these reasons, we consider using a $k$-NN based density estimation to estimate $\gamma$-divergence.
This approach has only one hyper-parameter, $k$.
Furthermore, this approach does not depend on any additional models.

\subsection{\texorpdfstring{$k$}{Lg}-Nearest Neighbor based Density Estimation}
\label{subsec:knn_density_est}

Let $X^{n}$ be an i.i.d.\  sample of size $n$ from a probability distribution with density $p$, and $Y^{m}$ be an i.i.d.\  sample of size $m$ from $q$.
Furthermore, we define $\rho_{k}(i)$ as the Euclidean distance between the $i$-th sample $X_i$ of $X^{n}$ and its $k$-th nearest neighbor ($k$-NN) among $X^{n}\setminus X_{i}$.
Similarly, we define $\nu_{k}(i)$ as the Euclidean distance between the $i$-th sample $X_{i}$ and its $k$-NN among $Y^{m}$.
Let $\mathcal{B}(x,R)$ be a closed ball with radius $R$ around $x \in \mathbb{R}^{d}$.
Finally, $\mathcal{V}(\mathcal{B}(x,R)) = \bar{c}R^{d}$ is defined as its volume, where
$\bar{c}$ is the volume of the $d$-dimensional unit ball.

\citet{loftsgaarden65} constructed the density estimators of $p$ and $q$ at the $i$-th sample $X_{i}$ via $k$-NN as follows:
\normalsize
\begin{align}
\label{def:knn_est_p}
\hat{p}_{k}(x_{i}) &= \frac{k}{(n-1)\mathcal{V}(\mathcal{B}(x_i,\rho_{k}(i)))} = \frac{k}{(n-1)\bar{c}\rho_{k}^{d}(i)}, \\
\label{def:knn_est_q}
\hat{q}_{k}(x_{i}) &= \frac{k}{m\mathcal{V}(\mathcal{B}(x_i,\nu_{k}(i))} = \frac{k}{m\bar{c}\nu_{k}^{d}(i)}.
\end{align}
\normalsize

These density estimators, Eqs.~\eqref{def:knn_est_p} and \eqref{def:knn_est_q}, are consistent estimators of the density only when the number of neighbors $k$ goes to infinity as the sample size $n$ goes to infinity.
We use these estimators for constructing our robust divergence estimator.
Hereafter, we fix the value of $k$ and show that our divergence estimator still has desirable asymptotic properties, including consistency.

\subsection{Robust Divergence Estimator on \texorpdfstring{$\gamma$}{Lg}-divergence}
\label{subsec:gamma_div_estimator_derivation}
Now we derive a \emph{non-parametric} $\gamma$-divergence estimator based on $k$-NN density estimation.
In ABC settings, outliers could be included in the observed data $X^{n}$.
To reduce the influence of outliers, we rewrite the term $\int_{\mathcal{M}_{0}} q^{1+\gamma}(x) \mathrm{d}x$ in Eq.~\eqref{gamma_div} to be $k$-NN estimatable from $\mathcal{M}' \subseteq \mathbb{R}^d$, where $\mathcal{M}'$ is the support of $q$,
i.e.,
\begin{align*}
    \int_{\mathcal{M}'} q^{1+\gamma}(y) \mathrm{d}y.
\end{align*}
We can use the same notion of Eq.~\eqref{def:knn_est_p} when we focus on the synthetic data $Y^{m}$; therefore, the density estimation for $q(y)$ based on $k$-NN can be written as
\begin{align}
\label{def:knn_est_q_y}
\hat{q}_{k}(y_{j}) &= \frac{k}{(m-1)\mathcal{V}(\mathcal{B}(y_j,\bar{\rho}_{k}(j))} = \frac{k}{(m-1)\bar{c}\bar{\rho}_{k}^{d}(j)},
\end{align}
where $\bar{\rho}_{k}(j)$ is the Euclidean distance between the $j$-th sample $Y_j$ of $Y^{m}$ and its $k$-NN among $Y^{m}\setminus Y_{j}$.

By plugging in the $k$-NN density estimator into Eqs.~\eqref{def:knn_est_p}, \eqref{def:knn_est_q}, and \eqref{def:knn_est_q_y}, we derive the $k$-NN based $\gamma$-divergence estimator as
\begin{align}
\label{eq:default_gamma_est}
    &\widehat{D}_{\gamma}(X^{n}\|Y^{m}) =  \frac{1}{\gamma(1+\gamma)} \notag \\
    &\times\left(
    \log \frac{\bigg(\displaystyle{\frac{1}{n} \sum_{i=1}^{n} \left(\frac{\bar{c}}{k}\hat{p}_{k}(x_{i})\right)^{\gamma}\bigg) } \bigg(\frac{1}{m} \displaystyle{\sum_{j=1}^{m} \left(\frac{\bar{c}}{k}\hat{q}_{k}(y_{j})\right)^{\gamma} \bigg)^{\gamma}}}
    {\bigg(\displaystyle \frac{1}{n}\sum_{i=1}^{n} \left(\frac{\bar{c}}{k}\hat{q}_{k}(x_{i})\right)^{\gamma} \bigg)^{1+\gamma}}
    \right).
\end{align}
The details of the derivation is in Appendix \ref{proof:gamma_estimator}.

The estimator in Eq.~\eqref{eq:default_gamma_est} involves $2n$ and $2m$ operations of nearest neighbor search. If we implement them by $KD$ trees~\cite{Bentley75,Maneewongvatana01}, the time cost of finding $\widehat{D}_{\gamma}(X^{n}\|Y^{m})$ is $\mathcal{O}((n\lor m)\log (n\lor m))$, where $(n\lor m) = \max \{n,m\}$, which is among the fastest (up to log factors) of the existing robust discrepancy approximators (see Table~\ref{Related work}).
Furthermore, this estimator fortunately enjoys ideal asymptotic properties: \emph{asymptotic unbiasedness} and \emph{almost surely convergence} under mild assumptions.
We will show them in Section \ref{subsec:asymp_analysis}.

\subsection{Robustness Property of \texorpdfstring{$\gamma$}{Lg}-ABC against Outliers}
\label{sec:robust_analysis}
Here, we investigate the behavior of the \emph{sensitivity curve} (SC), which is a finite-sample analogue of the \emph{influence function} (IF), both of which are used in quantifying the robustness of statistics~\cite{futami18,Ruli20}.
We fix the observed data $X^{n}$ and consider a contamination by an outlier $X_{0}$. We define the contaminated data as $X_{[X_{0}]}^{n} \coloneqq (X_{0}, X_{1},\ldots,X_{n})$.
Then, the SC is defined as follows.
\begin{definition}[Sensitivity curve {\citep[2.1e]{HampelRobust2005}}]
Let \(\gamma, \epsilon > 0\).
Let us define the (population) pseudo-posterior as $\hat{\pi}(\theta|X^{n}) \coloneqq \pi(\theta|X^{n},\widehat{D}_\gamma, \epsilon)$.
The sensitivity curve of \({\hat \pi}\) is defined as
\begin{equation*}\begin{split}
\mathrm{SC}_{n+1}^{\theta}(X_{0}) \coloneqq (n+1)\left({\hat \pi}(\theta|X_{[X_{0}]}^{n}) - {\hat \pi}(\theta|X^{n})\right).
\end{split}\end{equation*}
\end{definition}
We consider SC instead of IF for two reasons: (i) we are interested in the pseudo-posterior distribution \(\hat \pi(\theta|X^n)\) with respect to a finite sample \(X^n\),
and (ii) the IF of the quantities based on the considered divergence estimator may not be even defined (a detailed explanation is in Remark \ref{remark2} in Appendix \ref{sec:remarks}).

Under this definition and some additional assumptions, we obtain the following theorem.
Our analysis is a finite-sample analogue of what is called the \emph{redescending property} \citep{MaronnaRobust2019} in the context of IF analysis.
\begin{thm}[Sensitivity curve analysis]
Assume \(k < \min\{n, m\}\).
Also assume that \(F_\theta(\epsilon) \coloneqq \int \Ind{\widehat{D}_\gamma(X^{n}\|Y^\Ynum{}) < \epsilon} p_{\theta}(Y^{m}) \mathrm{d}Y^{m}\) is \(\beta\)-Lipschitz continuous for all \(\theta \in \Theta\).
Then, we have
\begin{equation*}\begin{split}
\lim_{\|X_{0}\| \rightarrow \infty} \mathrm{SC}_{n+1}^{\theta}(X_{0}) \leq -\frac{\beta \pi(\theta)}{\Lambda_n(1+\gamma)}\log\left(1 - \frac{1}{n^2}\right)^{n+1},
\end{split}\end{equation*}
where \(\Lambda_n \coloneqq \int \pi(\theta') F_{\thetap}(\epsilon) \mathrm{d}\thetap\).
Furthermore, if \(\lim_{n \to \infty} \Lambda_n\) exists and is non-zero, then the right-hand side of the above inequality converges to \(0\).
\label{thm:redescending-sensitivity-curve}
\end{thm}
The proof is in Appendix \ref{sec:thm_and_proof_for_sc}.
Through Theorem~\ref{thm:redescending-sensitivity-curve}, we can see that the influence of contamination is reduced when we have enough data, even if the magnitude of the outlier $X_0$ is very large.
\diff{Intuitively, an estimator has the redescending property if its IF first ascends away from zero as outliers become more pronounced, while the IF ``redescends'' towards zero as outliers become increasingly extreme.}
\diff{Since our analysis is a finite-sample analogue of the redescending property in the context of IF,} this result implies the robustness of our method that an extreme outlier is automatically ignored.

\begin{figure*}[t]
    \centering

    \includegraphics[scale=0.32]{fig/res_disc_error2.pdf}
    \caption{\diff{Experimental results for the absolute error of discrepancy.}

    We compared the robustness of the discrepancy measure based on robust MMD (MONK-BCD) and divergence-based discrepancies (KL and ours).
    The mean $\pm$ std. values of each error are plotted (solid lines are mean values and shaded areas represent the range of mean $\pm$ std values).
    Lower values are better. The true discrepancy $D$ was estimated by $10^{4}$ Monte Carlo samples. We conducted experiments on each discrepancy for $100$ times for various contamination rates ($\eta = 0, 0.1, 0.2$) and sample sizes ($200, 400, \ldots, 2000$).
    We set the hyper-parameter candidates of our $\gamma$-divergence estimator as $\gamma = (0.1,0.2,0.25,0.4,0.5,0.6,0.75,0.9)$ and the one with the smallest mean score among them is displayed.}
    \label{fig:disc_error}
\end{figure*}

\subsection{Robustness on Estimation Error of Discrepancy}
\diff{In Section~\ref{sec:robust_analysis}, we showed the theoretical robustness of our method.
Here, we experimentally investigate the robustness based on the estimation error of discrepancy.}
Figure \ref{fig:disc_error} shows the discrepancy estimation error.
In this figure, we show the errors between robust MMD and true MMD, and the KL- or $\gamma$-divergence estimator and true KL-divergence.
We considered the $1$-dimensional standard normal distribution $\mathcal{N}(0,1)$ and the contaminated $1$-dimensional standard normal distribution $(1-\eta)\mathcal{N}(0,1) + \eta \mathcal{N}(10,1)$.
\diff{Whereas the outliers negatively affect the KL divergence estimator~\cite{PrezCruz08}, MONK-BCD~\cite{Lerasle19} and our $\gamma$-divergence estimator are robust to an increase in the contamination rate $\eta$.}
Furthermore, if we choose the hyper-parameter $\gamma$ properly, our discrepancy estimator achieves comparable accuracy to the KL-divergence estimator in the non-contaminated case.

\section{Asymptotic Analysis on ABC}
\label{sec:asm_anals}
In this section, we elucidate essential asymptotic properties, such as asymptotic unbiasedness and almost sure convergence, for our divergence estimator defined by Eq.~\eqref{eq:default_gamma_est}.
Furthermore, we analyze an asymptotic behavior of the ABC posterior distributions built on our divergence estimator.

\subsection{Theoretical Analysis for \texorpdfstring{$\gamma$}{Lg}-divergence Estimator}
\label{subsec:asymp_analysis}
To confirm the validity of the proposed estimator in Eq.~\eqref{eq:default_gamma_est}, we show two properties: the asymptotic unbiasedness and the almost sure convergence.

We show the asymptotic unbiasedness by assuming that $\mathcal{M}$, i.e., the support of $p$, has the following mild regularity condition.
These conditions are commonly used for investigating the asymptotic properties of divergence estimators, e.g., the $\alpha$-divergence estimator in \citet{poczos11a}.
\begin{assump}[Restrictions on the domain $\mathcal{M}$~\cite{poczos11a}]
\label{assmp:restrict_M}
We assume
\begin{align*}
    \inf_{0<\delta<1} \inf_{x \in \mathcal{M}} \frac{\mathcal{V}(\mathcal{B}(x,\delta)\cap\mathcal{M})}{\mathcal{V}(\mathcal{B}(x,\delta))} \coloneqq r_{\mathcal{M}} > 0.
\end{align*}
\end{assump}
\diff{Assumption~\ref{assmp:restrict_M} states that the intersection of $\mathcal{M}$ with an arbitrary small ball having the center in $\mathcal{M}$ has a volume that cannot be arbitrarily small relative to the volume of the ball.}
It intuitively means that almost all points of $\mathcal{M}$ are in its interior.

Furthermore, we define the following function:
\normalsize
\begin{align}
\label{func:powered}
    &H(x,p,\delta,\omega) \coloneqq \sum_{j=0}^{k-1} \bigg( \frac{1}{j!} \bigg)^{\omega} \Gamma(\kappa + j\omega) \bigg( \frac{p(x) + \delta}{p(x) - \delta} \bigg)^{j\omega} \notag \\
    &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times
    (p(x) - \delta)^{-\gamma}((1-\delta)\omega)^{-\kappa-j\omega},
\end{align}
\normalsize
where $\Gamma(\cdot)$ is the gamma function defined as $\Gamma(z) = \int_{0}^{\infty} t^{z-1}\exp(-t) \mathrm{d}t$.
\citet{poczos11a} used Assumption \ref{assmp:restrict_M} to show a uniform variant of Lebesgue's lemma and to show that Definition~\ref{def:uniform_lebesgue_approx_func} below is well-defined.
The function in Eq.~\eqref{func:powered} appears in the upper bound of the moment for the $\omega$-powered conditional distribution function (CDF) when Assumption~\ref{assmp:restrict_M} holds (see Theorem 37 in \citet{poczos11a}).
In addition, we impose some reasonable assumptions, which are also assumed in \citet{poczos11a}.
\begin{definition}[Uniformly Lebesgue-approximable function~\cite{poczos11a}]
\label{def:uniform_lebesgue_approx_func}
\diff{Denote by $L_{1}(\mathcal{M})$ the set of Lebesgue integrable functions on $\mathcal{M}$ and} let $g \in L_{1}(\mathcal{M})$.
The function $g$ is uniformly Lebesgue approximable on $\mathcal{M}$ if for any series $R_{n} \rightarrow 0$ and any $\delta >0$, there exists $n_0=n_{0}(\delta) \in \mathbb{Z}^{+}$
such that if $n > n_{0}$, then for almost all $x \in \mathcal{M}$,
\normalsize
\begin{align}
    g(x) - \delta < \frac{\int_{\mathcal{B}(x,R_{n})\cap\mathcal{M}}g(t)\mathrm{d}t}{\mathcal{V}(\mathcal{B}(x,R_{n})\cap\mathcal{M})} < g(x) + \delta.
\end{align}
\normalsize
\end{definition}
\begin{assump}[Condition for $p$ and $q$ from \citet{poczos11a}]
\label{assmp:p_bounded}
The positive functions $p$ and $q$ are bounded away from zero and uniformly Lebesgue approximable.
Furthermore, the expectations of the $l_{2}$-norm powered by $\kappa$ over $p$ and $q$ are bounded, i.e.,
\normalsize
\begin{align*}
    \int_{\mathcal{M}} \|x - y \|^{\kappa}p(y) \mathrm{d}y < \infty, \ \
    \int_{\mathcal{M}} \|x - y \|^{\kappa}q(y) \mathrm{d}y < \infty,
\end{align*}
\normalsize
for almost all $x \in \mathcal{M}$.
Furthermore, the following conditions hold:
\footnotesize
\begin{align*}
    &\int \int_{\mathcal{M}^{2}} \|x - y \|^{\kappa}p(y)p(x) \mathrm{d}y \mathrm{d}x < \infty, \\
    &\int \int_{\mathcal{M}^{2}} \|x - y \|^{\kappa}q(y)p(x) \mathrm{d}y \mathrm{d}x < \infty.
\end{align*}
\normalsize
\end{assump}

\begin{assump}[Condition for powered CDF in $\mathcal{M}$~\cite{poczos11a}]
\label{assmp:p_bounded_cdf}
\diff{The expectations of $H(x,p,\delta,1)$ and $H(x,q,\delta,1)$ are bounded as follows: $\exists \delta_{0} \ \ \mathrm{s.t.} \ \ \forall \delta \in (0,\delta_{0})$,}

\footnotesize
\begin{align*}
\int_{\mathcal{M}} H(x,q,\delta,1)p(x)\mathrm{d}x < \infty,
\int_{\mathcal{M}} H(x,p,\delta,1)p(x)\mathrm{d}x < \infty.
\end{align*}
\normalsize
\end{assump}
\diff{This assumption indicates that the expectations of the $\omega$-powered CDFs of $p$ and $q$ are bounded, respectively.}
The expectations appear in the upper bound of the moment of the $\omega$-powered CDFs.

Since our method has a term involving an expectation with respect to $q$, we set the following assumption that is almost the same as the condition for the support of $p$.
\begin{assump}[Extra condition for powered CDF in $\mathcal{M}'$]
\label{assmp:p_bounded_extra}
\diff{The expectation of $H(y,q,\delta,1)$ is bounded as}
\begin{align*}
    &\exists \delta_{0} \ \ \mathrm{s.t.} \ \ \forall \delta \in (0,\delta_{0}), \ \ \int_{\mathcal{M}'} H(y,q,\delta,1)q(y)\mathrm{d}y < \infty.
\end{align*}
\end{assump}
\diff{This assumption means that the expectation of the $\omega$-powered CDF with respect to $q$ is bounded.}
The above expectation appears in the upper bound of the moment for the $\omega$-powered CDFs.
Under these assumptions, the following theorem holds.
\begin{thm}[Asymptotic unbiasedness]
\label{thm:asymp_unbiased_others_p}
Let $0 < \gamma < k$ or $-k < \gamma < 0$. Suppose that Assumption~\ref{assmp:p_bounded} holds with $\kappa = \gamma$ and that Assumptions~\ref{assmp:p_bounded_cdf} and \ref{assmp:p_bounded_extra} hold.
Also assume that $q$ is bounded from above. Then, $\widehat{D}_{\gamma}(X^{n}\|Y^{m})$ defined in Eq.~\eqref{eq:default_gamma_est} is asymptotically unbiased, i.e.,
\begin{align*}
    \lim_{n,m \rightarrow \infty} \mathbb{E}\bigg[\widehat{D}_{\gamma}(X^{n}\|Y^{m}) \bigg] = D_{\gamma}(p\|q).
\end{align*}
\end{thm}
\diff{From this result, we can see that the asymptotic unbiasedness holds even if we set $\gamma$ as $-k < \gamma < 0$ (see Theorems \ref{thm:asymp_unbiased_positive} and \ref{thm:asymp_unbiased_q} in Appendix \ref{proof:unbiasedness}).}

Next, we establish the almost sure convergence of our estimator.
\begin{thm}[Almost sure convergence]
\label{thm:almost_surely_conv}
Let $\gamma < k$.
Suppose that Assumption~\ref{assmp:p_bounded} holds with $\kappa = \gamma$ and that Assumptions~\ref{assmp:p_bounded_cdf} and \ref{assmp:p_bounded_extra} hold.
Also assume that $p$ and $q$ are bounded from above.
Let $k(n)$ denote the number of neighbors applied at sample size $n$ such that $\lim_{n \rightarrow \infty} k(n) = \infty$, $\lim_{n \rightarrow \infty} n/k(n) = \infty$, $\lim_{m \rightarrow \infty} k(m) = \infty$ and $\lim_{m \rightarrow \infty} m/k(m) = \infty$. Then, our estimator converges almost surely to $D_{\gamma}(p\|q)$, that is,
\begin{align*}
\widehat{D}_{\gamma}(X^{n}\|Y^{m}) \overset{\textrm{a.s.}}{\rightarrow} D_{\gamma}(p\|q).
\end{align*}
\end{thm}

The proofs for these theorems are in Appendices \ref{proof:unbiasedness} and \ref{proof:almost_surely_convergence}.
Note that in the proofs of Theorems \ref{thm:asymp_unbiased_others_p} and \ref{thm:almost_surely_conv}, we cannot reuse the known theoretical results for the divergence estimators in the $f$-divergence class~\cite{PrezCruz08,poczos11a} and it required us to newly show several asymptotic properties specifically for $\gamma$-divergence estimation, which are given in Appendices \ref{app:asymptotic_analysis} and \ref{app:proof_asymp_anal}.

\subsection{Asymptotic Property of ABC Posterior Distributions with \texorpdfstring{$\gamma$}{Lg}-divergence Estimator}
\label{sec:asymp_quasi_posterior}
Now we analyze whether the ABC posterior based on our robust discrepancy measure can accurately estimate the parameter $\theta$ with small exact $\gamma$-divergence $D_{\gamma}(p_{\theta^{*}}\|p_{\theta})$ asymptotically.

According to Theorem 1 in \cite{Jiang18}, the asymptotic ABC posterior is a restriction of the prior $\pi$ to the region $\{ \theta \in \Theta : D(p_{\theta^{*}}\| p_{\theta}) < \epsilon \}$ with appropriate scaling.
Combining this with the almost sure convergence of $\widehat{D}_{\gamma}(X^{n}\|Y^{m})$ established in Theorem \ref{thm:almost_surely_conv}, we can obtain the following corollary.
\begin{cor}[Asymptotic ABC posterior with \texorpdfstring{$\gamma$}{Lg}-divergence estimator]
\label{cor:asymp_quasi_posterior_gamma}
Suppose that Assumptions \ref{assmp:p_bounded}-\ref{assmp:p_bounded_extra} are satisfied with $\kappa = \gamma$.
Let $n \rightarrow \infty$ and $m/n \rightarrow \alpha > 0$.
\diff{Let $\pi(\theta | D_{\gamma}(p_{\theta^{*}}\|p_{\theta})<\epsilon)$ be the posterior under $D_{\gamma}(p_{\theta^{*}}\|p_{\theta})<\epsilon$.}
If $\widehat{D}_{\gamma}(X^{n}\|Y^{m})$ is used as the data discrepancy measure in Algorithm \ref{alg:reject_abc}, the ABC posterior distribution satisfies
\begin{align*}
   \pi(\theta| X^{n};\widehat{D}_{\gamma},\epsilon) \rightarrow \pi(\theta | D_{\gamma}(p_{\theta^{*}}\|p_{\theta}) < \epsilon),
\end{align*}
almost surely, and therefore
\begin{align*}
    \lim_{n,m \rightarrow \infty} \pi(\theta| X^{n};\widehat{D}_{\gamma},\epsilon)

    \propto \pi(\theta) \Ind{D_{\gamma}(p_{\theta^{*}}\|p_{\theta}) < \epsilon},
\end{align*}
almost surely.
\begin{proof}[Proof sketch]
In the same way as \citet{Jiang18},
we use L\'{e}vy's upward theorem (enabled by Theorem~\ref{thm:almost_surely_conv}; see Theorem \ref{thm:levy_upward} in Appendix \ref{app:asymptotic_analysis}) to $Z_{n} = \Ind{\widehat{D}_{\gamma}(X^{n}\|Y^{\infty}) < \epsilon}$ and apply the dominated convergence theorem~\cite{vaart_1998} to complete the proof.
\end{proof}
\end{cor}
Corollary~\ref{cor:asymp_quasi_posterior_gamma} shows that the ABC posterior based on our estimator converges to \diff{the maximum likelihood estimator minimizes the \emph{exact} $\gamma$-divergence between the empirical distribution of $p_{\theta^{*}}$ and $p_{\theta}$.}
Thus, ABC with our $\gamma$-divergence estimator asymptotically collects the $\theta$ with small $\gamma$-divergence.

\begin{figure*}[th]
    \centering
    \includegraphics[scale=0.28]{fig/summary_energy_mgm.pdf}

    \caption{Simulation error for the GM (left) and the MG1 model (right) experiments.}
    \label{fig:energy_summary_mgm}
\end{figure*}
\section{Experiments}
\label{sec:experiments}
In this section, we report the performance of our estimator through five benchmark experiments of ABC.
Here, we confirm that the ABC with our discrepancy measure has immunity against heavily contaminated data.

\subsection{Settings}
\diff{We set $n=m$ following \citet{Jiang18} to prevent the resulting ABC posterior of the indirect method to be over-precise~\cite{Drovandi15} and to avoid arbitrariness in the experiments.}
The tolerance threshold $\epsilon$ was adaptively initialized so that $0.5\%$ of proposed
parameters $\theta$ were accepted in each experiment by calculating each discrepancy measure $10^{3}$ times.
\diff{Furthermore, we artificially generated the $n$ i.i.d.\  observed data from $G(X_{i})$ and replaced them by some outliers from $\mathcal{N}(10,1)$, where $G(X_{i})$ is an observed data distribution.
In short, the contaminated data can be expressed as $(1-\eta)G(X_{i}) + \eta \mathcal{N}(10,1)$ in each dimension.}
In addition, we varied the contamination level $\eta$ in $\{ 0, 0.1, 0.2\}$ to confirm the robustness.
The hyper-parameter $\gamma$ was selected from $\{0.1, 0.2, 0.25, 0.4, 0.5, 0.6, 0.75, 0.9 \}$ for our method.

\diff{We measured the accuracy by the simulation error based on the \emph{energy distance}, which is a standard metric for distributions in statistics and has been used in the ABC literature, e.g., \citet{kajihara18a}.}
\diff{This allows us to directly compare the distributions between the non-contaminated observed data and the synthesized data simulated with the estimated parameter.}
We approximated the MAP estimator $\hat{\theta}_{\textrm{MAP}}$ of the ABC posterior by kernel density estimation with the Gaussian kernel \diff{with the bandwidth parameter $n^{-1/(d+4)}$, that is known as Scott's Rule~\cite{Scott15}.}

From each of the $10$ different models, we sampled the data and performed the ABC (Algorithm~\ref{alg:reject_abc}) with $T = 10^{5}$.
We repeated the procedure independently for $10$ times, and reported the average results.
The results with the standard errors are reported in Appendix \ref{app:simulation_error}, and the results of the mean-squared error (MSE) between $\hat{\theta}_{\textrm{MAP}}$ and the true parameter are also reported in Appendix \ref{app:full_MSE_scores}.

\diff{For our method, we conducted experiments independently for several $\gamma$ and displayed the one with the smallest mean score of the energy distance and the MSE among them.
The full results are reported in Figures \ref{fig:gamma_gaussian}--\ref{fig:gamma_gk} and \ref{fig:gamma_gm_energy}--\ref{fig:gamma_gk_energy} in Appendix~\ref{app:full_MSE_scores}.}
\diff{Furthermore, we compared the ABC posteriors of our method and that of the second-best method. These results are reported in Figures~\ref{fig:pos_gm}--\ref{fig:pos_gk} in Appendix~\ref{app:abc_post}.
}
\subsection{Models and Results}
Here, we summarize the model settings and the results of each experiment.
The details of the baseline discrepancies and the model architectures are shown in Appendices \ref{app:detail_discrepancy} and \ref{app:detail_models_exp}.

\paragraph{Gaussian Mixture Model (GM):}
The univariate Gaussian mixture model is the most basic benchmark setup in the ABC literature~\cite{Wilkinson13,Jiang18}.
We adopted a bivariate Gaussian mixture model with the true parameters $p^{*} = 0.3$, $\mu_{0}^{*} = (0.7, 0.7)$ and $\mu_{1}^{*} = (-0.7, -0.7)$, where $p^{*}$ is the mixture weight and $\mu_{0}^{*}, \mu_{1}^{*}$ are the means of the component distributions.
The variances are fixed as $0.5I-0.3I^\top$ and $0.25 I$, where $I$ is the identity matrix of size $(2, 2)$.

From the experimental results in Figure~\ref{fig:energy_summary_mgm}, we can see that our method achieves a better performance when the observed data are contaminated, whereas the other methods fail to give good scores.
In addition, in terms of the MSE, our method outperforms the baseline methods (see Appendix~\ref{app:full_MSE_scores}) under contamination.
\diff{From the results in Figure~\ref{fig:posterior_sum1}, we can confirm that the ABC posterior with our method places high density around the ground-truth parameter, whereas the baseline method fails to do so.}

\begin{figure*}[th]
    \centering
    \includegraphics[scale=0.31]{fig/summary_energy2.pdf}
    \caption{Simulation error for the BB (left), the MA2 (center), and the GK model (right) experiments.}
    \label{fig:energy_summary}
\end{figure*}
\paragraph{\textit{M}/\textit{G}/1-queuing Model (MG1):}
Queuing models are an example of stochastic models which are easy to sample
from but have intractable likelihoods~\cite{fearnhead12}.
The $M$/$G$/$1$-queuing model has been often used in the ABC literature~\cite{fearnhead12,Jiang18}.
This model has three parameters: $\theta = (\theta_1, \theta_2, \theta_3)$.
We adopted this model with the true parameter $\theta^{*} = (1,5,0.2)$.

From the experimental results in Figure~\ref{fig:energy_summary_mgm}, we can see that our method outperforms the other methods even if the data has no contamination.
In addition, our method also achieves better performance in terms of the MSE scores than the baseline methods (see Appendix \ref{app:full_MSE_scores}).
\diff{Figure~\ref{fig:posterior_sum2} indicates that the ABC posterior with our method places high density around the ground-truth parameter, e.g., for $\theta_{2}$.
On the other hand, the CAD via boosting places higher density around the wrong parameter than our method, e.g., for $\theta_{3}$.}

\begin{figure*}[th]
    \centering
    \includegraphics[scale=0.29]{fig/pos_res.pdf}
    \caption{ABC posterior distributions of the GM and MA2 model experiments for $\eta = 0.2$ (excerpted).}
    \label{fig:posterior_sum1}
\end{figure*}
\begin{figure*}[th]
    \centering
    \includegraphics[scale=0.22]{fig/pos_res2.pdf}
    \caption{ABC posterior distributions of the MG1, BB, and GK model experiments for $\eta = 0.2$ (excerpted).}
    \label{fig:posterior_sum2}
\end{figure*}
\paragraph{Bivariate Beta Model (BB):}
The bivariate beta model can be used to model data sets exhibiting positive or negative correlation~\cite{Arnold11}.
This model was originally proposed as a model with $8$ parameters $\theta = (\theta_{1},\ldots,\theta_{8})$ by \citet{Arnold11}, and \citet{Crackel17} later reconsidered its $5$-parameter sub-model by restricting to $\theta_{3},\theta_{4},\theta_{5} = 0$. \citet{Jiang18} used the $5$-parameter models for ABC experiments and therefore we also adopted this with the true parameter $\theta^{*} = (3,2.5,2,1.5,1)$ as a benchmark model.

From the experimental results in Figure \ref{fig:energy_summary}, the KL- and the $\gamma$-divergence based methods achieve better performances than those of the baseline methods when the observed data are heavily contaminated.
The Wasserstein method achieves a better performance than the others when the data has no contamination; however, the performance becomes significantly worse when the contamination occurs.
In addition, the CAD with boosting method achieves a better performance in terms of the MSE; however, the simulation error is worse than the KL- and the $\gamma$-divergence based methods (see Appendix~\ref{app:full_MSE_scores}).
\diff{Figure~\ref{fig:posterior_sum2} shows that the ABC posterior with our method places higher density around the ground-truth parameter than the KL method.
In addition, the KL method sometimes places high density around the wrong parameter, e.g., for $\theta_{8}$.
However, in this experiment, the simulation error of the KL method is slightly better than that of our method.
This indicates that carefully tuning the hyperparameter $\gamma$ is important in our method.
}

\paragraph{Moving-average Model of Order 2 (MA2):}
The moving-average model is often used for modeling univariate time series.
\citet{Marin12} used the moving-average model of order $2$ as a benchmark model for ABC.
We adopted this model with $10$-length time series. For the unobserved noise distribution, we used Student's t-distribution with $5$ degrees of freedom.
We set the true parameter $\theta^{*} = (0.6,0.2)$.

From the experimental results in Figure \ref{fig:energy_summary}, our method and the CAD with boosting outperform the other methods.
The CAD with boosting achieves comparable performance to our method even if the observed data are contaminated; however, in terms of the time cost, our method is better than that of this method \diff{because the gradient boosting has $\mathcal{O}(Kd(n+m) + (n+m)\log B)$ time cost, where $K$ is the total number of trees and $B$ is s the maximum number of rows in each block (see \citet{Chen16})}.
In addition, in terms of the MSEs, our method achieves better performance when the observed data are heavily contaminated (see Appendix \ref{app:full_MSE_scores}).
\diff{In Figure~\ref{fig:posterior_sum1}, we found that the ABC posterior with our method places high density around the ground-truth parameter.
On the other hand, the CAD via boosting places higher density around the wrong parameter than our method.}

\paragraph{Multivariate $g$-and-$k$ Distribution (GK):}
The univariate $g$-and-$k$ distribution is a generalization of the standard normal distribution with extra parameters: the \emph{skewness} and the \emph{kurtosis}.
This distribution is known to have no analytical form of the density function, and the numerical evaluation of the likelihood function is costly \cite{rayner02}.
Thus, it is a model for which ABC is specifically suited \cite{fearnhead12,Allingham09}.
Some studies \cite{Drovandi11,Li15} also considered the multivariate $g$-and-$k$ distribution.
We adopted the multivariate model proposed by \citet{Drovandi11} with the true parameters $A^{*} = 3$, $B^{*} = 1$, $g^{*} = 2$, $k^{*} = 0.5$ and $\rho^{*} = -0.3$, where $A^{*},B^{*},g^{*},k^{*}$ control the location, the scale, the skewness and the kurtosis, respectively.

From the experimental results in Figure \ref{fig:energy_summary}, we can see that our method achieves better performance even if the observed data are contaminated, although the other baseline methods fail to give good scores.
In addition, in terms of the MSE, our method outperforms the other baseline methods when the observed data have heavy contamination (see Appendix \ref{app:full_MSE_scores}).
\diff{Figure~\ref{fig:posterior_sum2} shows that the ABC posterior with our method places slightly higher density on the ground-truth parameter than that of the CAD via boosting.
}

\section{Conclusion and Discussion}
\label{sec:conclusion}
We have proposed a $\gamma$-divergence estimator and used it as a robust data discrepancy for ABC.
We theoretically have guaranteed its robustness against outliers and its desirable asymptotic properties, i.e., the asymptotic unbiasedness and the almost sure convergence to the approximate posterior.
In addition, we have shown the redescending property of the ABC posterior of our method, indicating the high robustness of our method against extreme outliers.
Through the experiments on benchmark models, we empirically confirmed that our method is robust against heavy contamination by outliers.

\diff{Our work has two limitations: (i) our method can become statistically inefficient in high-dimensional cases (the curse of dimensionality) due to the $k$-NN based density estimation, and therefore (ii) its performance has only been confirmed in some low-dimensional cases.
To overcome these limitations, in the future, we will consider extending our method to a non-parametric estimation that can handle high dimensions and conduct experiments for more realistic high-dimensional cases.
Furthermore, on the basis of our idea, we plan to develop outlier-robust methods for other ABC approaches, e.g., ABC without discrepancies~\cite{george16,greenberg19a,Thomas20}.}

\diff{It is worth mentioning that there have been several studies on general losses within Bayesian procedures~\cite{Bissiri16,Knoblauch19}, and recent studies have connected the ideas of these studies with ABC~\cite{schmon21}.
Following these studies, it would be interesting to see whether our method stands in the framework of generalized approximate Bayesian inference under the condition of ABC with a general loss function~\cite{schmon21}.
}

\diff{In addition, research on the consistency and robustness of Bayesian estimation against model misspecification has attracted attention recently.
For example, \citet{Cherief20} has proposed Bayesian estimation based on a pseudo-likelihood by using MMD and has theoretically shown that it is effective in this problem setting.
We also explore the potential of our method in the context of model misspecification.
}

\subsubsection*{Acknowledgements}
MF and TT were supported by RIKEN Junior Research Associate Program.
MF and TT were supported by Toyota/Dwango AI scholarship.
MF was supported by JST CREST including AIP challenge program.
TT was supported by Masason Foundation.
MS was supported by KAKENHI 17H00757.
We appreciate Dr. Ikko Yamane, Kento Nozawa, Dr. Yoshihiro Nagano, and Han Bao for their kind effort to maintain the experimental environment.

redpajama_set_name: RedPajamaArXiv


This is a list of seven things (one for each day of the week, maybe? Yeah, I'm poetic like that) highlighting the differences between my first day on the Camino and the last day on the Camino.
Day One: If you start at St. JPDP like I did, your first day is tackling the Pyrenees. St. JPDP is a sweet little town with red roofs and the Route Napoleon takes you through some bucolic landscaping. I was seriously lucky that the Route Napoleon was open the two days I took to go through the pass (it had only just recently opened and it closed again a few days after due to weather issues). Overall a gorgeous walk.
Final Day: Mostly various stages of civilization and barely dry weather. But both things didn't really register as I was so focused on getting to Santiago on time for Noon Mass. Also, by that time, I was used to tramping through civilization and in all sorts of weather ranging from hot to snowy. Overall, not so gorgeous walk.
Day One: I knew nobody but most people were willing to talk, especially if one stays at a place like L'Espirit du Chemain. People are friendly and no one actually looks apprehensive – though I think most really secretly are – and excitement is definitely in the air! I think people start to make connections based on how you each plan to tackle the Pyrenees – going only to Orisson or going all the way to Roncevalles.
Final Day: Social people generally walk into Santiago with friends made along the Way. The rest of us, though, tend to tell our new friends that we'll see each other once in the city and until then, "Buen Camino!" I chose to spend my last walking day by myself. It was how I had always imagined arriving in Santiago and I don't regret it one bit. As selfish as it may sound, it was my time and I didn't want to share my arrival with someone else's tears/hugs/cheers. I also think I was feeling a little sad (and when sad, I'm extra not-social) because this was coming to an end.
Day One: Quite frankly, I wanted to die. Uphill and I have a mutual hatred for each other – I was thinking "oh my god, oh my god, oh my god, oh hey pretty view, oh my god, oh my god, oh my god" the entire multi-hour 8km walk up that mountain to Orisson.
Final Day: I pretty much sped walked from O Pedruzo to the Cathedral in Santiago. I covered 22km in about 5 hours, with only a couple short food breaks. I made it to the Cathedral about 11am, got my Compostela by 11:30, checked into my hotel by 11:45, and collapsed onto a pew just before Noon Mass started.
Day One: No matter that I've used my bag before and broke in my boots, I still didn't trust them fully to not betray me somewhere along the next few weeks. I gave them the stink-eye rather often.
Final Day: They are my bestest friends ever!
Day One: Seriously? You want me to eat ALL of that??
Final Day: Seriously? Is that ALL you're giving me?!
Whoo! Item #53 on my List of Adventures achieved!
What do i do tomorrow……….. i know that feeling very well. That lost feeling when at journeys end.
Yup, exactly. I guess that just means time for a next adventure 🙂 Thanks for coming by, Dave!
Favourites: The Bag & Boots comparison and the People comparison!
i'll have to show you all the sheep photos – i have a few! but yeah, it's only about now that my stomach has realized the feast time is over.
I have to get some of your sheep pictures!
Nice contrasts! The change in calorie consumption is hilarious though!
I like that you finished on your own terms, after such an amazing journey; you deserved it!
thoroughly enjoyed this entry! I love couch-traveling 🙂 Keep it coming, kid; keep it coming!
thanks! it's fun for me to remember my Camino this way as well!


This seminar series explores interactions between Christian faith and scholarship in the natural and applied sciences. A schedule of recent seminars is given below. Seminar topics vary over a range of interdisciplinary issues, drawing insights from religion, philosophy, astronomy, geology, biology, biotechnology, chemistry, physics, engineering, nursing, mathematics, computer science, psychology, sociology, history, and other departments and programs.
Seminars are typically held several Fridays per semester, 3:30-4:45 p.m., at Calvin College in Science Building room 110, unless otherwise noted. See Calvin's Visitor Resources for maps and directions to theScience Building. Faculty, students, staff and off-campus visitors are welcome.If you would like to receive regular email announcements for each week's seminar, or if you have other questions or comments, contact Loren Haarsma.
—lead a discussion about how to teach Christian perspectives on a certain topic in the classroom.
"Evolutionary creation and the problem of animal death and suffering before the Fall"
In recent years, evolutionary creation has become an increasingly popular model for Christians seeking to integrate faith and science. However, this approach, in which evolution is understood as God's method of creation, raises some difficult theological questions. While much recent focus has been on human origins, original sin, and the Fall, there are other more proximate questions related to the pre-human creation that also deserve attention. If evolution is indeed a God-ordained and God-sustained process, then what are we to make of millions of years of competition, predation, and suffering in the animal world before the advent of humans? Is the Fall still somehow to blame, or can organismal death be considered part of God's good creation? What do such biological processes say about God's character? Where might we be able to see redemption in any of this? This seminar will focus on recent work in the area of evolutionary theodicy, looking specifically at non-human animals and exploring an array of possible ways to address these difficult questions.
"Science, Faith and the Dialog of Cultures: Islamic Perspectives"
Dr. Guiderdoni is Director of Research at the Institute of Astrophysics of Lyon, a member of the Center National de la Recherche Scientifique. His main field of research is the formation of galaxies and their evolution. He has published more than 100 articles and has organized several international conferences on these issues. He is director of the Lyon Observatory. He is also one of the leading experts on Islam in France and has published 50 documents on Islamic theology and its mystics. He was in charge of a French television program called "Knowing Islam" ("Connaître l'Islam") from 1993 to 1999, and is now the director of the Islamic Institute of Advanced Studies (Institut des Hautes Études Islamiques).
"Common Ground in Science and Faith: A Theologian and a Scientist in Dialogue"
Do science and Christian faith have something to offer to each other and to the world around us? Or are these pursuits locked in separate worlds, achieving an uneasy truce at best? Join us for a thought-provoking evening featuring respected theologian Richard Mouw and Cornell geneticist Praveen Sethupathy. Together they will give a fresh perspective on the possibilities of finding common ground in a renewed dialogue between science and faith..
"Augustine's beginning: transitioning from organism to mechanism and the idea of a natural world"
A key development in the shaping of early modern science was the conceptualization of nature in mechanistic terms rather than treating it as an organism, possessing a structure that was organized and predictable. The change was not so revolutionary, however, as it was an idea that began evolving much earlier and was deeply embedded in the theology of Augustine. In his largest commentary on Genesis, "de Genesi ad litteram," Augustine decisively set aside the vision of the cosmos that saw all particular phenomena as a result of the particular will of the gods or a God and rejected the enchanted world which dominated Near Eastern, Greek, and Roman cosmologies. A profound revolution in his thought evolved which had major implications for the development of the West and its attitudes towards nature and the natural world (notably shaping early modern sciences and the tradition of thought leading up to the sixteenth- and seventeenth-century developments). The commentary signals a sea change that might be described as a Christianization of the cosmos, as it more thoroughly implemented the doctrine of creatio ex nihilo; it desacralized and demystified nature, making it into a contingent, objective, rational structure subject to coherent and intelligible reason.
Stan Rosenberg founded and directs Scholarship and Christianity in Oxford (SCIO) and its undergraduate visiting student programmes, The Scholars Semester in Oxford and The Oxford Summer Programme. He is a member of Wycliffe Hall and the University of Oxford's Theology and Religion Faculty, teaching early Christian history and doctrine. In addition, he is an associate member of the Ian Ramsey Centre for Science and Religion and a member of the Oxford Centre for Late Antiquity. His research and teaching interests focus on Augustine's works (the commentaries on Genesis and the sermons, in particular), early Christian cosmology and its relationship to Greco-Roman science, culture, and philosophy, and the interplay between intellectual and popular thought during this period; he is also involved in contemporary discussions on the relationship between science and religion. Rosenberg has directed multiple Science and Religion projects in Oxford gathering multinational teams of scholars, funded by the John Templeton Foundation, Templeton Religion Trust, the Blankemeyer Foundation, and the BioLogos Foundation, and is the general editor of the recently published book, Finding Ourselves after Darwin: Conversations on the Image of God, Original Sin, and the Problem of Evil (Baker, 2018) which includes his article, 'Can Nature be "Red in Tooth and Claw" in the Thought of Augustine?' He is on the advisory council of the BioLogos Foundation, the editorial board of the journal, Religions, and the international advisory council of the Museum of the Bible, advising on both science and the Bible, and early Christianity.
After the talk, students may remain for pizza and Q&A conversation with Stan Rosenberg. Q&A organized by the Science and Religion Forum and sponsored by a grant given by Bridging the Two Cultures of Science and the Humanities II, a project run by Scholarship and Christianity in Oxford, the UK subsidiary of the Council for Christian Colleges and Universities, with funding by Templeton Religion Trust and The Blankemeyer Foundation.
"A Critical Appraisal of the Evangelical Pro-Life Movement"
As evangelicals went to the polls in 2016, many embraced a utilitarian ethic: the end of combating abortion justified the means of elevating an unusually controversial candidate to a position of immense power. This logic results largely from a widespread belief that full moral life begins at conception, but Dudley will argue that belief itself is poorly supported by historic Christianity, the Bible, and modern science. While historic Christianity has featured consistent opposition to abortion, the form of that opposition has varied significantly, with most theologians stating that full moral life begins at some point later than conception. The claim that the Bible teaches life begins at conception, meanwhile, is based on exegesis that was not widespread among Christians until the 1980s. In advocating for their goals, pro-life organizations have made inaccurate claims about medical science and pursued policies that increase the abortion rate. Dudley will conclude by advocating an alternative approach to political engagement that doesn't require subordinating other convictions, and that has a stronger claim to represent historic Christianity by pursuing anti-abortion policies that are based on scientific evidence.
"The Liberal Origins (and Conservative Trajectory) of the Pro-Life Movement: A Political and Theological History of the American Campaign against Abortion"
Today American pro-life organizations are usually strong allies of conservative Republicans, but the movement originated as a liberal human rights cause in the mid-20th century, and many of its early leaders were New Deal Democrats who favored an expanded social welfare state to assist women facing crisis pregnancies. This talk will examine the pro-life movement's early grounding in Catholic social theology, and will examine why this theology of human dignity found a home in the Democratic Party for several decades. The talk will then examine how the pro-life movement responded to changes in American liberalism in the 1970s that eventually strained the link between Catholic social teaching and the Democratic Party, and it will explain why the movement eventually allied with the Republican Party. Evangelical Protestants played a critical role in the political reorientation of the Catholic-inspired pro-life movement. The talk will examine why evangelicals belatedly joined the pro-life campaign and how their view of abortion and politics differed from those of their Catholic allies. The talk will conclude by examining what this history means for Christian practice today, and what tools the history of Christian theology might offer for approaching the issue of abortion policy in contemporary American politics.
Dr. Daniel K. Williams is Associate Professor of History at the University of West Georgia. His research focuses on the intersection between politics and religion in modern America. He is the author of "God's Own Party: The Making of the Christian Right" (Oxford University Press, 2010) and "Defenders of the Unborn: The Pro-Life Movement before Roe v. Wade" (Oxford University Press, 2016). He is currently writing a study of cultural conflict in the 1976 presidential election, which is under contract with the University Press of Kansas for publication in the American Presidential Elections series.
"A History of Conflict: Protestantism and the Origins of the 'Conflict Thesis'"
The conflict thesis is typically traced to the popular historical narratives of John W. Draper's "History of Conflict between Religion and Science" (1874) and Andrew D. White's two-volume "History of the Warfare of Science with Theology in Christendom" (1896). I argue, however, that Draper and White, contrary to conventional views, did not posit a conflict between "science and religion." Rather, they envisioned the conflict as one between conservative and liberal theological traditions. This conflict predated Draper and White. Such narratives of theological conflict appeared in Protestant historiography as early as the sixteenth century, among liberal Anglicans in particular. In time, however, such polemics were transformed from Protestant anti-Catholic sentiment to an intra-Protestant self-critique. Nourished in this religious context, by the mid-nineteenth century narratives of conflict between "science and religion" were largely deployed between contending theological schools of thought. However, these narratives were later appropriated by secularists, freethinkers, and atheists as weapons against all religion. I contend that the origins, development, and popularization of the conflict thesis was one of the many unintended consequences of Protestant thought.
Dr. James Ungureanu is an Honorary Research Fellow in the Department of History of the University of Wisconsin-Madison, and an Honorary Post-Thesis Fellow in Religious Studies in the School of Historical and Philosophical Inquiry at the University of Queensland. He studies history of science and religion; intellectual history; history of philosophy; western civilization; world history; history of Christianity; world religions; religious studies; and nineteenth-century religious thought. (https://uq.academia.edu/jamescungureanu). He has a forthcoming book, "Science, Religion and the Protestant Tradition: The Origins of Conflict" (University of Pittsburgh Press, in Bernard Lightman's series "Science and Culture in the Nineteenth Century, Fall 2019).
"Current Neuroscience and the Subject Integration Theory"
Current neuroscience seems to be converging upon several properties required for consciousness: (1) that local and global connections are required for integrating information contents, (2) that events in frontal regions are required for accessing and manipulating information contents, (3) that events in the thalamus and proximal structures, including the insula, are required for enabling specific information contents, and (4) that the common mechanism of the aforementioned (1-3) neurobiological properties is the synchronous activations of neurons (see Dehaene 2014; Engel & Fries 2016; Singer 2017). As Patricia Churchland observes: "The linkages, it is thought, may consist in synchrony in the activities of populations of neurons" (2013, p. 247). While neuronal synchrony might be necessary in some important sense (e.g., when it comes to matters of predictive coding), it is not sufficient for consciousness, including its integration. Recent data strongly indicates that when an animal is rendered unconscious by means of anesthesia, neuronal synchrony is not only present but actually shows signs of strengthening (see Bola et al 2017). Further, Semir Zeki suggests that connections between V4 and V5 probably do not exist on grounds of neuroanatomical, lesion-based, neuropsychological, and chronoarchitectonic mapping evidence. There is now widespread agreement (a) that V4 and V5 "have distinct, and characteristic, anatomical inputs, despite the many anatomical opportunities for them to interact" and (b) that a lesion in one of those processing sites "does not invade and disable the perceptual territory of the other," which underscores the functional autonomy of those processing sites (Zeki 2003, p. 214; Zeki 2015). The neuropsychological evidence has shown (c) that damage to V4 does not impact V5, and vice versa. Perhaps most significantly, it can now be shown (d) that the chronoarchitectonic mapping evidence puts pressure on claims (1) and (4) above: when human subjects are exposed to complex visual scenes, "the time courses of activity in human V4 and V5 are significantly uncorrelated, from which we can infer that there are no direct anatomical links between them." In fact, the time courses between V4 and V5 (and other visual processing sites) can be as great as 80 milliseconds (Zeki 2003, p. 216; Zeki 2015). In sum, the current evidence reveals an asynchronous relation between processing sites, an observable datum that cuts directly against the purported common neurobiological mechanism of consciousness (LaRock 2018). Zeki ultimately concludes that the only entity that counts as truly unitary is the subject that "sits at the apex" of the processing hierarchy (2003, p. 217; see also Zeki 2015). Unfortunately, Zeki does not develop this suggestion into a testable theory. To bridge this theoretical gulf, I propose a new theory called the Subject Integration Theory (SIT), whereby the subject plays a (top-down) role of integrating information contents across cortices of its brain, and develop the testability of SIT in light of recent advances in neuroanesthesia.
audio_recording not available, slides not available.
"Beyond Objectivism and Relativism: The Scientific Enterprise in a Polanyian Framework"
I consider here the structure, functions, and purpose of modern natural science within the frame of Michael Polanyi's theory of tacit knowing (TTK). Since its birth, modern natural science is de facto epistemic authority of the liberal and post-liberal order. Hence, misrepresentation of how science achieves new knowledge distorts science, knowledge in general and falsifies our outlook. I place TTK in an emergent Enlightenment reform tradition which seeks to overcome the epistemic difficulties of modern objectivism and post-modern subjectivism and/or relativism. By reference to actual practice of science, TTK gives a truer understanding of knowing by disclosing the tacit nature of all knowledge. Its triadic structure derives from a thorough consideration of scientific discovery. TTK presupposes the fiduciary, tacit, and social roots of all knowledge but overcomes their constraints by the creative exercise of tacit knowing in the pursuit of truth of which discovery is the realization. TTK re-forges the bond between the independence of thought and the obligation to transcendent reality (or truth) which animates science. Science, in Polanyi's purview, is the pursuit of truth for its own sake. Hence, science divides: pure science is driven by discovery and the ideal of truth while applied science (or technology) by invention and pragmatic or human ends; TTK focuses on pure science. While TTK re-establishes more adequate grounds for and justification of the scientific enterprise, Polanyi admits its quest does not totally sate the human need for a purpose that bears on eternity. Here, TTK re-opens the door to its religious solution.
"Discerning a Christian Approach to AI"
AI techniques employing "deep learning" have recently achieved remarkable strides in tackling difficult problems and spurring applications in many new areas. Responses to these developments have ranged from existential fear to unbridled optimism. These discussions open up a plethora of ethical considerations and ontological questions about what it means to be human. The approach one takes to questions rising in AI is largely shaped by our philosophical presuppositions and our worldview. As Christians who care about God's world, we must do more than wax eloquently about the issues or critique them from the sidelines. Christians need to actively join this conversation bringing insights from Scripture and from Christian philosophy and theology to inform a responsible approach that contributes to the common good. In particular, as we wrestle with these new developments, we must remember what Scripture teaches about what it means to be human, the meaning of work, and the kind of world God would have us unfold.
audio_recording, text of paper presented.
"Beyond the Free Will Defense: natural evil, theodicy, and sacrificial love"
Atheists sometimes point to features of the natural world as arguments against Theism (e.g. age and immensity of the universe, hiddenness of divine action, randomness, suffering caused by natural events and moral evil, evolution, the neuroscience of belief). In response, numerous Christians have developed "free will" or "soul-making" accounts. A recent book by Christian Barrigar ("Freedom All the Way Up", Friesen Press) affirms these accounts but advocates a shift of emphasis, arguing for free will as only a necessary pre-condition for God's ultimate purpose of creating beings capable of understanding and living in relationships of self-sacrificial love towards each other and God. Self-sacrificial love is especially central to God's Trinitarian nature and revealed in Christ's redeeming work. This "agape" account for these features of the world can be appealing to many Christians and powerfully inviting for non-Christians. It also has some implications regarding the subtlety of divine action in the natural world, and the (perhaps) inevitability of human sin, which some Christians might find theologically troubling, and are worth further discussion.
"Growing Faith in the Field of Geology"
Field opportunities are abundant in geology: We travel to Montana to teach Big Sky Geology or journey to Hawaii to study volcanoes. We traverse Scotland to study the development of geological sciences within the context of Earth history. These places are awe-inspiring in their landscapes, let alone the tectonic forces, chemical reactions, and time necessary to produce them. How can we help our students use these awe-inspiring experiences to grow in their Christian faith? God is revealing His character through His Good Creation and of course, His character is revealed in Scripture as well. Teaching at Calvin College, we have the privilege of recognizing both and sharing that with our students. In this presentation, we will consider passages in Psalms, Job, Ecclesiastes, and the Gospels to see how Scripture complements what we see manifested in Creation as geology. It is important to recognize that growing in the faith not only occurs because of knowledge and understanding. We will also explore opportunities for growth in the field through purposeful prayer, intentional fellowship around the dinner tables, and reserved time for praise and worship.
Transhumanism is a movement that seeks to better human existence by employing various technological enhancements, some of which exist now while some are being sought. Among these are genetic modifications, forms of artificial intelligence, life extension, and brain uploading to computers. This seminar will examine these and other such technologies from a Christian perspective. It will discuss theological aspects as well as ethical and policy issues.
audio_recording unavailable due to technical glitch, slides.
"The Faith of a Scientist"
Henry F. Schaefer III received his undergraduate and doctoral degrees from MIT and Stanford, respectively, both in Chemical Physics. He served as a professor at University of California, Berkeley, for 18 years. He is currently Graham Perdue Professor of Chemistry and Director of the Center for Computational Quantum Chemistry at the University of Georgia. Thomson-Reuters shows Professor Schaefer to be one of the most highly cited physical scientists in the world. He is the recipient of 30 honorary degrees. Professor Schaefer has also given lectures on the interface between science and the Christian faith at more than 200 universities.
"Do we really live in a non-deterministic world?"
Most of classic Christian theology denied the reality of chance, asserting rather that God controls all that happens. Furthermore, prior to the mid-19th century, scientists generally saw the physical world as deterministic. However, since that time, evolutionary biology and quantum physics have led a shift toward the view that the natural world is non-deterministic. But such a view has been problematic for Christian theology, especially for the Reformed tradition. In this talk, I will attempt to clarify what scientists mean by chance and how that differs from popular views. I will also argue (without using quantum mechanics) that the physical world really is non-deterministic. And I will conclude by engaging in some shameless speculations about why God might have made it that way.
"Science and Faith: A Global Conversation"
"There is one question that troubled me when I was in college: can science and faith be brought on the same boat?" -Adrianus Yosia, a student at South East Asia Bible Seminary in Indonesia.
As conversations and interactions between science and faith continue to grow and deepen in the US, there are similar conversations going on in the Majority world involving varying faith traditions and convictions. How are Christians and other people of faith engaging issues of science and religion? What answers do institutions and individuals have for Adrianus and others who are asking these difficult questions? The Theological Book Network has been listening to and observing these conversations and asking questions about how best to address them with the appropriate theological resources. Many important insights have been learned through this exercise. Wayne Bornholdt will share examples of the multiple conversations on science and faith that the Network has had over many years.
"Patristics, Genomics, and Finding God in the Cell"
Christians who confront what genomic science and evolution are telling us about human origins face the same existential crisis that has disturbed Christians for 2000 years – the veracity of truth learned from studying the world that seems at first to contradict scripture. The Patristics, the ancient Church Fathers who wrote during the first centuries of Christianity, faced similar struggles as they wrestled with their understanding of how the world seemed to function, Platonism, and the claims of scripture. Today we can learn from their examples and, with reverence and humility, interpret genomic science of human origins through the lens of the biblical narrative. Even as we are persuaded to modify dearly held interpretations of scripture, we can be comforted by scientific observations that seem to be letters to make us ponder the invisible things of God.
"The Lost World of Adam and Eve"
Dr. Walton will evaluate Genesis 2 relative to human origins. Based on a close reading of the Hebrew text and correlation with what is known from the ancient Near East, conclusions will be drawn concerning what the biblical claims are and how they help us navigate the modern conversations between science and faith.
Science and Religion Forum at Calvin College.
"Managing Stormwater for Quantity and Quality"
Stormwater pollution is one of the greatest threats to Great Lakes water quality. Bill Byl (Calvin '74) recently retired Kent County Drain Commissioner, will explain the causes and potential solutions to the problem from both a technical and political perspective. He will also identify two projects that affect the Calvin campus and how Calvin students and alumni are contributing to the solutions.
"You Look Like a Movie: Why science still needs its critics"
Institutions of modern science have always aimed to serve human wellbeing, and no one can deny that they have delivered countless social goods. But should scientists themselves expect to make decisions about what social goods are and how they should be pursued? During the Cold War era, moral questions were generally considered to be outside the scope of science. Science fiction and noir films of that era explored various frightful human consequences to which amoral science might lead. Now, in the 21st century, we can see that some of the underlying fears were justified. This talk will revisit Cold War science and cinema, in order to suggest why we should expect—and even welcome—a renewed debate about the roles that scientists should play in society.
The presentation is intended to appeal both to STEM students and to general audiences with interests in history, philosophy, political science, sociology, and/or film studies.
"The remarkable tale of the whale: fossils, DNA, isotopes, and the many facets of cetacean evolution"
Cetaceans (including modern whales, dolphins, and porpoises) have become one of the most frequently cited examples of macroevolution. The fossil record documenting their transition from terrestrial ancestors has exploded in recent decades, providing a series of transitional fossils that demonstrates how the earliest cetaceans adapted to life in water. But the fossils are only part of the story. Lines of evidence from other fields (including comparative anatomy, development, genetics, biogeography, and stable isotope analysis) provide additional details about the evolution of cetaceans. While these varied disciplines highlight distinct facets of this remarkable transition, together they create a compelling and remarkable case of large-scale evolutionary change.
"Thinking About Human Creation Through the Complementary Lenses of Faith and Science"
In this lecture I review some key aspects of the Wesleyan Quadrilateral that complement the scientific method. Then I discuss the human fossil record, beginning with early evidence for bipedalism and continuing through to Homo sapiens.


text:
<?php

declare(strict_types=1);

namespace OpenTelemetry\Tests\Unit\SDK\Metrics\Exemplar;

use OpenTelemetry\Context\Context;
use OpenTelemetry\SDK\Common\Attribute\Attributes;
use OpenTelemetry\SDK\Metrics\Data\Exemplar;
use OpenTelemetry\SDK\Metrics\Exemplar\FixedSizeReservoir;
use PHPUnit\Framework\TestCase;

/**
 * @covers \OpenTelemetry\SDK\Metrics\Exemplar\FixedSizeReservoir
 */
final class FixedSizeReservoirTest extends TestCase
{
    public function test_reservoir_returns_exemplars(): void
    {
        $reservoir = new FixedSizeReservoir(Attributes::factory(), 4);
        $reservoir->offer(0, 5, Attributes::create([]), Context::getRoot(), 7, 0);

        $this->assertEquals([
            0 => [
                new Exemplar(5, 7, Attributes::create([]), null, null),
            ],
        ], $reservoir->collect([0 => Attributes::create([])], 0, 1));
    }
}

redpajama_set_name: RedPajamaGithub


BISS 072414 - DJ LOGIC & EARFUL / KERMIT RUFFINS / FAR WEST FEST / SLY & ROBBIE //NXT WK::GUITARFISH MUSIC FESTIVAL AUG 1, 2, & 3!!!!
I am headed up there on Tuesday to set up the fest so you will only hear about it this one last BISS List and then you are "off the hook" until next year's festival! Get it "off the hook", just like Guitarfish is gonna be!!?
So last chance for me to "school" you on the fabulousness of GUITARFISH MUSIC FESTIVAL. In case you haven't been reading your weekly BISS, it takes place on August 1st, 2nd and 3rd at Cisco Grove Campground which is just 20 miles before Truckee off of Highway 80 (Near Tahoe). Our lineup is more stellar than ever with: NICKI BLUHM & THE GRAMBLERS, PIMPS OF JOYTIME, EVERYONE ORCHESTRA CONDUCTED BY MATT BUTLER, MOTHER HIPS, EDDIE ROBERTS WEST COAST SOUNDS, LEBO & FRIENDS, SAMBADA, ZONGO JUNCTION, GUITARMAGEDDON , YO MAMAS BIG FAT BOOTY BAND, JOY & MADNESS, WHEELAND BROTHERS, TRACORUM, WILL MAGID 5, LECHE DE TIGRE, CON BRIO, JIMMY LESLIE & THE FLOW, THE GREASE TRAPS, TIM SNIDER TRIO, PETER JOSEPH BURTT & THE KINGTIDE, SAMBA STILT CIRCUS, ANDREA BROOK "SONIC BUTTERFLY", THE SPACE COWBOYS!!
Then, we'll roll on over to The Boom Boom Room for THE EARFUL with DJ LOGIC (*WIN). If you get there early they will be streaming the Widespread Panic TRI show starting at 5pm. Other shows going on around the bay tonight? IRMA THOMAS is at Yoshi's Oakland, ERIKA SUNSHINE LEE with THE GO AHEAD is making it radiate at The Great American Music Hall (*WIN) and WE THE FOLK & ROYAL JELLY are at Hopmonk in Sebastopol (*WIN).
Friday night, it's night one in the bay for KERMIT RUFFINS & THE BBQ SWINGERS at the New Parish in Oaktown (*WIN). Staying in the East Bay for the evening? AFROLICIOUS is at Ashkenaz in Berkeley (*WIN). In the city, NAIVE MELODIES is Burning Down The House, Talking Heads style at The Boom Boom Room (*WIN). Down south, PAT BENATAR & RICK SPRINGFIELD are at the Saratoga Mountain Winery.
Saturday is pretty BISSing crazy! First of all, it is time for one of BISS List's favorite local festivals, The FAR WEST FEST! This year's line-up features Honey Island Swamp Band, New Monsoon, LoCura, The Seshen, Grateful Bluegrass Boys, The Grease Traps, San Geronimo and Many MORE!! Voted BEST MUSIC FESTIVAL IN MARIN year after year, the 9th Annual Far West Fest is happening July 26th in Point Reyes. With gates opening at 10am, for acoustic Sunrise Café Sessions, you'll find three stages of top-notch, original music from national touring acts like the Honey Island Swamp Band and New Monsoon, and Bay Area up-and coming talent like The Seshen, in a variety of genres from Bluegrass to Funky R&B.
Did I say that Saturday is BISSing crazy?! Yeah, there's also ZIGABOO MODELISTE at the newest venue to the BISS List & Bay, Leo's in Oakland (*WIN), as well as JOURNEY, STEVE MILLER BAND & TOWER OF POWER at Shoreline,THOMAS MAPUFMO at Askenzaz (*WIN), HOT BUTTERED RUM at Hopmonk in Sebastopol (*WIN), THE DUSTBOWL REVIVAL at The Great American Music Hall (*WIN), KERMIT RUFFINS comes into the city for night two in the bay at Brick & Mortar (*WIN) and... VINYL is at The Boom Boom Room taking you late night (*WIN).....whoa, what a night of music...you can tell this city is getting geared up (for Guitarfish maybe?)!!!
Clearly Shoreline has already kicked into high gear with SANTANA playing on Sunday to follow up the Saturday show over there. Here in the city, SLY & ROBBIE will be smokin up the Indy (*WIN). And of course Sunday would not be Sunday without Dub Mission at Elbo Room and ROTC at Boom Boom Room.
Continuing into next week cause BISS can't leave you hanging for a whole week. Besides wait til you hear what's coming!
Monday 7/28, WOLFMOTHER is at The Fillmore.
Tuesday, JOURNEY AND STEVE MILLER BAND + TOWER OF POWER are at Concord Pavilion.
Wednesday, THE BUDOS BAND is at The Independent (*WIN), STOOGES BRASS BAND is at The Chapel (*WIN), & TERRAPIN FAMILY BAND FT. LEBO is at Terrapin.
Thursday, get ready for GUITARFISH MUSIC FESTIVAL with *YO MAMA'S BIG FAT BOOTY BAND* at Boom Boom Room (*WIN) or EVERYONE ORCHESTRA : : Featuring MATT BUTLER, EDDIE ROBERTS (The New Mastersounds), DAN 'LEBO' LEBOWITZ (ALO), JENNIFER HARTSWICK (Trey Anastasio Band), NATALIE CRESSMAN (Trey Anastasio Band), TREVOR GARROD (Tea Leaf Green), MIKE SUGAR (Jambay), JERMAL WATSON (Dirty Dozen Brass Band) are at Terrapin.
Also next Thursday, BLACKALICIOUS is at Sweetwater (*WIN), & CHICAGO is with REO SPEEDWAGON at Concord Pavilion, or TAB BENOIT is at Yoshi's Oakland.
Friday August 1st, 2nd, & 3rd...sorry nothing on this BISS List girl's calendar except...GUITARFISH MUSIC FESTIVAL featuring NICKI BLUHM & THE GRAMBLERS, PIMPS OF JOYTIME, EVERYONE ORCHESTRA CONDUCTED BY MATT BUTLER, MOTHER HIPS, EDDIE ROBERTS WEST COAST SOUNDS, LEBO & FRIENDS, SAMBADA, ZONGO JUNCTION, GUITARMAGEDDON , YO MAMAS BIG FAT BOOTY BAND, JOY & MADNESS, WHEELAND BROTHERS, TRACORUM, WILL MAGID 5, LECHE DE TIGRE, CON BRIO, JIMMY LESLIE & THE FLOW, THE GREASE TRAPS, TIM SNIDER TRIO, PETER JOSEPH BURTT & THE KINGTIDE, SAMBA STILT CIRCUS, ANDREA BROOK "SONIC BUTTERFLY", THE SPACE COWBOYS!!
There are plenty of other shows to find out about if you just scroll down and check out the calendar!!
AfroSolo Jazz in the Gardens (AfroSolo Arts Festival presents its 13th annual free concert at Yerba Buena Gardens).