text:
\section{Introduction}
\label{sec:intro}
Single index regression model is defined by
\begin{equation*}

	Y=m(\bm{X})+\epsilon=g(\bm{X}^T\bm{\beta})+\epsilon,
\end{equation*}
where $\bm{X} \in R^p$ is the covariate vector, the coefficient vector $\bm{\beta} \in\mathbb{R}^p$ is called single index, $g:\mathbb{R}\rightarrow\mathbb{R}$ is the unknown link function and $\epsilon$ is the noise term.  The constraint that $\|\bm{\beta}\|_2=1$ with the first non-zero element being positive is usually put to ensure the identifiability of $\bm{\beta}$. By assuming the response only relates to a single linear combination of the covariates,  single index model provides an efficient way for dimensional reduction in high-dimensional problems, which is often used in econometrics.

Based on observations $\{(y_i, \bm{x}_i): i=1, 2, \cdots, n)\}$, the interest is to estimate the single index parameter $\bm{\beta}$. The difficulty of this aim arises from the unknown link function $g$, and there are some different strategies to overcome the difficulty. Firstly, one may approximate the link function $g$ by base function expansion (e.g. splines, B-splines, kernel expansion)  or its local linear expansion \citep{hardle1993,Ichimura1993,xia2002,wang2009spline,radchenko2015, kuchibhotla2020,feng2020}, and then obtain its estimator $\hat{\bm{\beta}}$ as $\hat{\bm{\beta}}= \arg _{\bm{\beta}}  \min \sum_{i=1}^{n}\psi\left(y_{i}, \hat{g}(\bm{x}_{i}^{T}\bm{\beta})\right)$ where $\hat{g}$ is an estimator of the unknown link function. Secondly, the relationship that $m^{\prime}(\bm{X}) = \partial g(\bm{X}^{T}\bm{\beta})/\partial \bm{X} = g^{\prime}(\bm{X}^{T}\bm{\beta})\bm{\beta}$  leads to the average derivative estimator \citep{hardle1989,horowitz1996,hristache2001}.  Although the estimator is proved to have some nice property, additional parameters such as bandwidths are difficult to choose for real problems. In addition, Bayesian estimation methods for single index regression are proposed to estimate simultaneously the single index parameter and unknown link function. For example, by putting a Gaussian process prior on the unknown link function $g$, estimating $\bm{\beta}$ follows naturally into the Bayesian framework, see for example \citet{choi2011}.  However, when the sample size gets larger, it is very time-consuming to get the estimates. Differently, one may also approximate the unknown link function by B-splines and estimates all parameters in Bayesian way \citep{antoniadis2004, wang2009}. However, one should be careful in selecting parameters for B-splines. There are also other methods proposed for estimating parameters in single index models or extended single index models; see for example, \citet{cui2011,wang2015,li2017penalized}.

Variable selection for single index model is more difficult than that for traditional models such as linear model and generalized linear model due to the unknown link function $g$.  Classical model selection methods such as AIC  \citep{naik2001}, cross validation \citep{kong2007} are extended to single index regression model, where the strategy is to use a nonparametric method to estimate $g$. Extending the penalized regression or penalized likelihood idea to single index model is straightforward by approximating the unknown link function $g$ using the basis function expansion; see for example, \citet{wang2008, wang2009, peng2011, zeng2012, radchenko2015, li2017, bindele2019}, or  by employing the sufficient dimension reduction methods; see for example, \citet{zhu2009, zhu2011, wang2015,lin2019}.

Similar to the generalized linear model, it is natural to consider the generalized single index model by assuming the response $Y$ follows a distribution in the exponential family with probability density function $p(y|\theta, \phi) = \exp\{\frac{\theta y - b(\theta)}{a(\phi)}+c(y, \phi)\}$, where $a(\cdot)$, $b(\cdot)$ and $c(\cdot)$  are  measurable functions. Typically, we have $\mathbb{E}(Y)=\mu=b^\prime(\theta)$, and $\text{Var}(Y)=b^{\prime\prime}(\theta)a(\phi)$. Relating the covariates with the mean of $Y$ through the natural parameter $\theta$, i.e., $\theta=g(\bm{X}^T\bm{\beta})$, we get the generalized single index model.  There are some parameter estimation and variable selection methods proposed for this model; see, for example, \citet{cui2011, li2017}. Although these methods are proved to asymptotically have some nice properties, it is difficult to understand the impact of approximation error on the accuracy of variable selection in finite-sample settings.

Compared with single index regression model, less attention is paid to the generalized single index model. Hence,  in this paper, we focus on the Single Index Logistic Model (SILM) defined as
\begin{equation}
\label{equ:logit}
Y\sim\text{Bernoulli}\left(\frac{\exp\{g(\bm{X}^{T}\bm{\beta})\}}{1+\exp\{g(\bm{X}^{T}\bm{\beta})\}}\right),
\end{equation}
and consider the parameter estimation and variable selection under this model. Specifically, we propose a Bayesian method with a Gaussian process prior for unknown link function $g$ and a spike-slab prior for single index $\bm{\beta}$ to achieve variable selection. Because the single index vector $\bm{\beta}$ is assumed to be on the unit sphere for the identifiability, a posterior projection method with data augmentation is applied to rapidly sample from the posterior distribution.

The rest of the paper is organized as follows. Section \ref{sec:meth} presents the Bayesian inference for SILM with details on priors for parameters, posterior distributions and posterior sampling algorithm. In Section \ref{sec:num}, we present numerical results from simulation studies and real-data analysis. Section \ref{sec:conc} concludes this paper with further discussions.

\section{Bayesian Inference for SILM}
\label{sec:meth}

\subsection{Prior Distribution}
\label{subsec:prior}
In Bayesian inference framework, the prior distribution summarizes the prior understanding of the statistical model.
In this section, we present prior distributions for unknown parameters in the single index logistic model.  Before that, we introduce some notations for convenience. Let $\left\{(y_i,\bm{x}_i): i=1,2,\cdots, n\right\}$ be $n$ independent and identically distributed (i.i.d) samples from SILM (\ref{equ:logit}), where $y_i \in \{-1, 1\}$ is the binary response variable, and $\bm{x}_i \in R^p$ is the $p$-dimensional predictor vector. Let $\bm{\beta}_0 \in R^p$ be the true value of the single index with $s$ non-zero elements, and $g_0$ be the true link function.
\subsubsection{Prior for Link Function}
Estimating the unknown link function $g$ is key to estimate the single index and variable selection in both single index regression model and logistic model. Approximation methods such as splines and B-splines are commonly employed in both frequentist and Bayesian literature, see for example, \citet{antoniadis2004,wang2009spline,wang2009,radchenko2015,lu2016,li2017,feng2020}. Although these methods enjoy computational advantage, one should be careful on selecting the order and knots of spline basis functions and an additional penalty on the number of knots is necessary to avoid over-fitting, which introduces some additional computational inefficiency. Gaussian process is popular  in Bayesian nonparametric modelling \citep{williams2006}, and has been successfully applied to single index regression model \citep{choi2011} and single index quantile regression model \citep{hu2013}, which motivates us put a Gaussian process prior on the unknown link function $g$.

Gaussian process is a stochastic process such
that any finite subcollection of random variables follows a multivariate Gaussian distribution. The function $f:\mathcal{X}\rightarrow\mathbb{R}$ follows Gaussian process prior, or the collection of random variables $\{f(x): x\in\mathcal{X}\}$ is said to be drawn from Gaussian process with mean function $m(\cdot)$ and covariance function $k(\cdot,\cdot)$, denoted by $$f(\cdot)\sim\text{GP}\left(m(\cdot),k(\cdot,\cdot)\right),$$
if for any set with finite elements $x=\{x_1,\cdots,x_n\}\subset \mathcal{X}$, the corresponding random vector $\{f(x_1), \cdots, f(x_n)\}$ follows a multivariate Gaussian distribution with mean vector $m(x)=\left(m(x_1),\cdots,m(x_n)\right)^T$ and covariance matrix $K(x)$, where $K_{ij}= k(x_{i}, x_{j})$.

A Gaussian process is completely specified by its mean and covariance function. Usually, the mean function is chosen to be identically zero. But the covariance function $k(\cdot,\cdot)$, also called kernel function, must be chosen such that the resulting covariance matrix $K$ is always positive semidefinite. For the single index logistic model, we assume the unknown link function $g$ follows a Gaussian process prior given by
\begin{equation}
	\label{gpiror}
	g(\cdot) | \tau, l \sim \text{GP}(0,k(\cdot, \cdot)),
\end{equation}
with $k(\cdot, \cdot)$ defined by
\begin{equation}
\label{equ:ker}
k(\bm{x}_i, \bm{x}_j) = \tau\exp\{ -\frac{\left(\bm{x}_{i}^T\bm{\beta}-\bm{x}_{j}^T\bm{\beta}\right)^{2}}{l} \},
\end{equation}
where $\tau$ and $l$ are hyperparameters, and both assumed to have a inverse Gamma prior, that is, $$p(\tau)\propto\tau^{-(a_{\tau}+1)}\exp\{-\frac{b_{\tau}}{\tau}\}, \text{and}~ p(l)\propto l^{-(a_{l}+1)}\exp\{-\frac{b_{l}}{l}\}.$$

\subsubsection{Prior for Single Index}
In high-dimensional problems, it is often that only a small part of covariates truly contributes to the response, which requires that the estimated single index only has a small number of non-zero elements.  A classic framework to attain this requirement is to put a Spike-Slab prior (e.g., \citet{mitchell1988,george1993,ishwaran2005}) on the single index, that is, the prior of $\bm{\beta}=(\beta_1, \cdots, \beta_p)$ is given by
\begin{equation}
p(\beta_j|\delta_j) = (1-\delta_j)p_{\text{Spike}}(\beta_j) + \delta_j p_{\text{Slab}}(\beta_j),j=1,2,\cdots,p,
\end{equation}
where $\delta_j=1$ indicates the corresponding variable $X_j$ should be included in the model and $\delta_j=0$ indicates $X_j$ should not be included, $p_{\text{Slab}}(\beta_j)$ is a flat slab distribution (e.g. normal distribution) and $p_{\text{Spike}}(\beta_j)$ is a spike distribution (e.g. Dirac distribution) concentrating its mass to values close to zero.
To alleviate the sampling difficulty of posterior distribution,   \citet{george1993} proposed a hierarchical normal mixture spike-slab model, where the spike distribution is choose as a normal distribution with very small variance.
Inspired by this, we propose our spike-slab prior for $\bm{\beta}$ as follows:
\begin{equation}\label{priorb1}
\beta_j|\delta_j,\sigma_{\beta_j}\sim\text{Normal}\left(0,r(\delta_j)\sigma_{\beta_j}\right), \sigma_{\beta_j}\sim\text{InvGamma}(a_{\sigma_{\beta_j}},b_{\sigma_{\beta_j}}),j=1,2,\cdots,p,
\end{equation}
where $\delta_j\in\{0,1\}$, $r(0)=c\ll 1$, $r(1)=1$, and  $\{a_{\sigma_{\beta_j}}, b_{\sigma_{\beta_j}}\}_{j=1}^{p}$ are hyperparameters. Integrating out $\sigma_{\beta_j}$, $\beta_j$ marginally follows Student distribution,
\begin{equation}\label{priorb2} \beta_j|\delta_j\sim\text{t}_{2a_{\sigma_{\beta_j}}}\left(0,\sqrt{r(\delta_j)b_{\sigma_{\beta_j}}/a_{\sigma_{\beta_j}}}\right).
\end{equation}
Finally, we put independent Bernoulli prior on the binary variables $\delta_j$, i.e.,
\begin{equation}\label{priorb3}
 \delta_j|\pi_j \sim \text{Bernoulli}(\pi_j),j=1,2,\cdots,p.
\end{equation}
The prior means variable $X_j$ is assumed to be included in the model with probability $\pi_i$. After incorporating the information from data, we include variable $X_j$ if the posterior probability of $\delta_j=1$ is larger than 0.5.

An important thing to be mentioned is that the single index $\bm{\beta}$ is constrained on the unit sphere $S^{p-1}$ for model identifiability. However, $\bm{\beta}$ from models (\ref{priorb1})-(\ref{priorb3}) is not assured to satisfy this constraint. Usually, priors like Uniform distribution on the unit sphere or half of the unit sphere and the Fisher-von Mises distribution are considered in the literature \citep{antoniadis2004,wang2009,choi2011}. However, the sparsity is ignored in these settings.
\citet{patra2018} developed a general approach, called posterior projection, for constrained Bayesian inference, which samples firstly from an unconstrained or less constrained posterior and then projects the draws to the constrained space through minimal distance mapping. We adopt this strategy for sampling $\bm{\beta}$ with our spike-slab prior.

\subsection{Posterior Distribution}
\label{subsec:post}
In Bayesian framework, the posterior distribution summarizes the information from prior knowledge (prior distribution) and data (likelihood).
Before presenting the main results, we introduce some notations to simplify the presentation.
Let $\bm{y}=\left(y_1,\cdots,y_n\right)$, $\bm{x}=(\bm{x}_1, \cdots, \bm{x}_n)$, $\bm{g}=\left(g_1,\cdots,g_n\right)$ with $g_i = g(\bm{x}_i^T\bm{\beta})$, $\bm{\delta}=\left(\delta_1,\cdots,\delta_p\right)$, $\bm{\sigma_\beta} = \left(\sigma_{\beta_1},\cdots,\sigma_{\beta_p}\right)$, and $\bm{\pi}=\left(\pi_1,\cdots,\pi_p\right)$.
According to models (\ref{equ:logit})-(\ref{priorb3}), the posterior distribution of unknown parameters $\bm{g}$, $\bm{\beta}$, $\bm{\delta}$, and $\bm{\sigma_\beta}$, and hyperparameters $\bm{\pi},\tau$ and $l$, is given by
\begin{equation}\label{poster}
p(\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l|\bm{y}, \bm{x})\propto L(\bm{y} |\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l, \bm{x})p(\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l |\bm{x})
\end{equation}
where $L(\bm{y} |\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l, \bm{x})$ is the likelihood function,
$$p(\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l |\bm{x}) = p(\bm{g}|\bm{\beta},\tau,l ,\bm{x})p(\bm{\beta}|\bm{\sigma_\beta},\bm{\delta})p(\bm{\sigma_\beta})p(\bm{\delta}|\bm{\pi})p(\bm{\pi})p(\tau)p(l),$$ and $p(\bm{\pi})$, $p(\tau)$ and $p(l)$ is the prior of $\bm{\pi},\tau$ and $l$, respectively.

According to the single index logistic model \ref{equ:logit}, we have
\begin{equation} \label{likef}
L(\bm{y} |\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l, \bm{x})=p(\bm{y}|\bm{g})=\prod_{i=1}^{n} \frac{\exp\left\{y_i g_i \right\}}{1+\exp\left\{ y_i g_i \right\}}.
\end{equation}
Moreover, according to model (\ref{gpiror}), we have
\begin{equation*}
p(\bm{g}|\bm{\beta},\tau,l,\bm{x})\propto|K|^{-1/2}\exp\{ -\frac{1}{2}\bm{g}^{T}K^{-1}\bm{g}\}.
\end{equation*}
In addition, according to models (\ref{priorb1})-(\ref{priorb3}), we have
\begin{equation*}
p(\bm{\beta}|\bm{\sigma_\beta},\bm{\delta})\propto \prod_{j=1}^{p}\left(r(\delta_j)\sigma_{\beta_j}\right)^{-1/2}\exp\{-\frac{\beta_j^{2}}{2r(\delta_j)\sigma_{\beta_j}}\}\\
=|\Sigma_{\bm{\beta}}|^{-1/2}\exp\{ -\frac{1}{2}\bm{\beta}^{T}\Sigma_{\bm{\beta}}^{-1}\bm{\beta} \},
\end{equation*}
$$p(\bm{\sigma_\beta})\propto \prod_{j=1}^{p}\sigma_{\beta_j}^{-(a_{\sigma_{\beta_j}}+1)}\exp\{-\frac{b_{\sigma_{\beta_j}}}{\sigma_{\beta_j}}\},
\text{and}~ p(\bm{\delta}|\bm{\pi})=\prod_{j=1}^{p}\pi_j^{\delta_j}(1-\pi_j)^{1-\delta_j},$$
where $\Sigma_{\bm{\beta}} =\text{diag}\left(r(\delta_1)\sigma_{\beta_1},\cdots,r(\delta_p)\sigma_{\beta_p}\right)$.

\subsection{Posterior Sampling}
\subsubsection{Data Augmentation Scheme}
\label{subsec:aug}
Gibbs sampling algorithm is efficient for high-dimensional sampling problems by sampling from conditional marginal distributions. Usually, the Metropolis-Hasting (MH) algorithm \citep{hastings1970monte} is used to sample from the posterior distribution when the corresponding marginal distributions do not have a analytic form.  However, this MH within Gibbs algorithm is not so efficient for many problems due to a low accept rate.  Data augmentation is an efficient alternative for some problems to MH algorithm when the target distribution is not analytically available. For example, \citet{albert1993} proposed a data augmentation (DA)  approach for the probit model.  \citet{holmes2006} and \citet{polson2013} proposed DA methods for the logistic model. Following \citet{polson2013}, we have
\begin{equation}
	\label{equ:pg}
	\begin{aligned}
		\frac{\left(\exp\{\psi\}\right)^a}{(1+\exp\{\psi\})^b}&=2^{-b}\exp\{\kappa\psi\}\mathbb{E}\left(\exp\{-\omega\psi^{2}/2\}\right) \\ &=2^{-b}\exp\{\kappa\psi\}\int_{0}^{\infty}\exp\{-\omega\psi^2/2\}p(\omega|b,0)d\omega,
	\end{aligned}
\end{equation}
where $b>0$, $a\in\mathbb{R}$, $\kappa=a-b/2$ and $p(\omega|b,0)$ is the density of $\text{P{\'o}lya-Gamma}(b,0)$ distribution, denoted as $\text{PG}(b,0)$. Generally, a $\text{PG}(b,c)$ random variable has the density
\begin{equation*}

	p(\omega|b,c) = \cosh^b\left(\frac{c}{2}\right)\exp\{-\frac{c^2}{2}\omega\}p(\omega|b,0).
\end{equation*}
Thus, we can rewrite the likelihood function (\ref{likef}) as
\begin{equation}
	\label{equ:pglogit}
		p(\bm{y}|\bm{g})=\prod_{i=1}^{n}\frac{\exp\{y_i g_i\}}{1+\exp\{y_i g_i\}}
		=\prod_{i=1}^{n}2^{-1}\exp\{y_i g_i/2\}\int_{0}^{\infty}\exp\{-\omega_i g_i^2/2\}p(\omega_i|1,0)d\omega_i,
\end{equation}
which suggests that the likelihood function is proportional to the marginal density function of $p(\bm{y},\bm{w}|\bm{g})= 2^{-n}\exp\{\frac{1}{2}\bm{y}^{T}\bm{g} - \frac{1}{2}\bm{g}^{T}\Omega \bm{g}\}\prod_{i=1}^{n}p(\omega_i|1,0)$,
where $\bm{\omega}=\left(\omega_1,\cdots,\omega_n\right)$, and $\Omega=\text{diag}(\omega_1,\cdots,\omega_n)$.

By adding the auxiliary variables $\bm{\omega}$ that follow the P{\'o}lya-Gamma distribution, we have,
\begin{equation}\label{equ:post}
\begin{aligned}
p(\bm{g},\bm{\beta}| \bm{\omega},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l, \bm{y}, \bm{x})\propto& p(\bm{y},\bm{\omega}|\bm{g})p(\bm{g},\bm{\beta}| \bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l,\bm{x})\\
\propto& \exp\{\frac{1}{2}\bm{y}^{T}\bm{g} - \frac{1}{2}\bm{g}^{T}\Omega \bm{g}\}\prod_{i=1}^{n}p(\omega_i|1,0)\\
&\times p(\bm{g}, \bm{\beta} | \bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l,\bm{x}).
\end{aligned}
\end{equation}
This simple data augmentation provides an efficient sampling algorithm. Details are given in next section.
\subsubsection{Sampling Algorithm}
In this section, we present the sampling algorithm that samples from the posterior distribution (\ref{poster}).  According to results presented in previous sections, we have following marginal distributions with detailed derivation given in Appendix \ref{app:cond}.
\begin{equation}
\label{equ:post:g}
\bm{g}|\bm{y},\bm{\omega},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}}\sim\text{Normal}\left(\bm{\mu_{g}},\Sigma_{g}\right),
\end{equation}
$$\omega_i|\bm{y},\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}}\sim\text{PG}(1,g_i),i=1,2,\cdots,n,$$
\begin{equation}
\label{equ:post:beta}
p(\bm{\beta}|\bm{y},\bm{\omega},\bm{g},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}})\propto |K|^{-\frac{1}{2}}\exp\{ -\frac{1}{2}\bm{g}^TK^{-1}\bm{g} \}\times\exp\{ -\frac{1}{2}\bm{\beta}^{T}\Sigma_{\beta}^{-1}\bm{\beta} \},
\end{equation}
\begin{equation}
	\label{equ:post:delta}
	\delta_j|\bm{y},\bm{\omega},\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\pi},\tau,l{\color{black},\bm{x}} \sim\text{Bernoulli}\left( \left(1+\frac{p_{\text{Spike}}(\beta_j)(1-\pi_j)}{p_{\text{Slab}}(\beta_j)\pi_j}\right)^{-1}\right),j=1,\cdots,p,
\end{equation}
\begin{equation}
	\label{equ:post:tau}
	\tau|\bm{y},\bm{\omega},\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},l{\color{black},\bm{x}}\sim \text{InvGamma}\left(a_\tau+\frac{n}{2},\frac{1}{2}\bm{g}^TK_0^{-1}\bm{g}+b_\tau\right),~K_0 = K/\tau,
\end{equation}
\begin{equation}
	\label{equ:post:l}
	p(l|\bm{y},\bm{\omega},\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau{\color{black},\bm{x}})\propto |K|^{-\frac{1}{2}}\exp\{ -\frac{1}{2}\bm{g}^TK^{-1}\bm{g} \}\times\frac{1}{l^{a_l+1}}\exp\{-\frac{b_l}{l}\}.
\end{equation}
\begin{equation}
\sigma_{\beta_j}|\bm{y},\bm{\omega},\bm{g},\bm{\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}}\sim\text{InvGamma}\left(a_{\sigma_{\beta_j}}+\frac{1}{2},\frac{\beta_j^2}{2r(\delta_j)}+b_{\sigma_{\beta_j}}\right),j=1,\cdots,p,
\end{equation}
\begin{equation}
\pi_j|\bm{y},\bm{\omega},\bm{g},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\tau,l{\color{black},\bm{x}}\sim\text{Beta}\left(a_\pi+\delta_j,b_\pi+1-\delta_j\right),j=1,\cdots,p,
\end{equation}
where $\bm{\mu_g}=\frac{1}{2}\Sigma_g\bm{y}$, $\Sigma_g = K - K(K + \Omega^{-1})^{-1}K$,
$\Sigma_\beta = \text{diag}\left(r(\delta_1)\sigma_{\beta_1},\cdots,r(\delta_p)\sigma_{\beta_p}\right)$, and $p_{\text{Spike}}(\cdot)$ and $p_{\text{Slab}}(\cdot)$ represent the density functions for zero-mean Gaussian distributions with variance $c\cdot\sigma_{\beta_j}$ and $\sigma_{\beta_j}$ respectively.

It can be seen from equation (\ref{equ:post:beta}) and equation (\ref{equ:post:l}) that sampling  $\bm{\beta}$ and $l$ directly from their conditional posteriors is difficult, therefore a MH sampler is used. Note that the matrix $K$ is always ill-conditioned in practice, and sometimes $K$ may have a few small but negative eigenvalues due to the finite precision and round-off error, which causes problems to compute the inverse and determinant of matrix $K$ in equations (\ref{equ:post:beta}), (\ref{equ:post:tau}) and (\ref{equ:post:l}).  Partially collapsed Gibbs sampler \citep{van2008} is used to avoid this problem.  Specifically, by integrating out $\bm{g}$ from joint distribution of $(\bm{\beta},\bm{g})$, $(\tau,\bm{g})$ and $(l,\bm{g})$, we have following reduced conditional distributions for $\bm{\beta}, \tau$ and $l$.
\begin{equation}\label{equ:post:beta2}
\begin{aligned}
p(\bm{\beta}|\bm{y},\bm{\omega},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}})
=&\int p(\bm{\beta},\bm{g}|\bm{y},\bm{\omega},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau,l{\color{black},\bm{x}})d\bm{g}\\
\propto& |K+\Omega^{-1}|^{-\frac{1}{2}}\exp\{-\frac{1}{8}\bm{y}^{T}\Omega^{-1}(K+\Omega^{-1})^{-1}\Omega^{-1}\bm{y}\}\\
& \times \exp\{ -\frac{1}{2}\bm{\beta}^{T}\Sigma_{\beta}^{-1}\bm{\beta} \}.
\end{aligned}
\end{equation}
\begin{equation}\label{equ:post:tau2}
\begin{aligned}
p(\tau|\bm{y},\bm{\omega},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},l{\color{black},\bm{x}})\propto&|K+\Omega^{-1}|^{-\frac{1}{2}}\exp\{-\frac{1}{8}\bm{y}^{T}\Omega^{-1}(K+\Omega^{-1})^{-1}\Omega^{-1}\bm{y}\}\\
& \times \frac{1}{\tau^{a_{\tau}+1}}\exp\{-\frac{b_{\tau}}{\tau}\},
\end{aligned}
\end{equation}
\begin{equation}\label{equ:post:l2}
\begin{aligned}
p(l|\bm{y},\bm{\omega},\bm{\beta},\bm{\sigma_\beta},\bm{\delta},\bm{\pi},\tau{\color{black},\bm{x}})\propto& |K+\Omega^{-1}|^{-\frac{1}{2}}\exp\{-\frac{1}{8}\bm{y}^{T}\Omega^{-1}(K+\Omega^{-1})^{-1}\Omega^{-1}\bm{y}\}\\
& \times  \frac{1}{l^{a_{l}+1}}\exp\{-\frac{b_{l}}{l}\}.
\end{aligned}
\end{equation}
Now, we need the inverse and determinant of $K+\Omega^{-1}$ instead of $K$, which is much more stable in practice. There is one thing to be mentioned, a improper sampling order may destroy the desired stationary distribution due to the partially collapse of $\bm{g}$ \citep{van2008,van2015}. According to suggestions in \citet{van2015}, we draw in order  $\bm{\beta},\tau,l,\bm{g},\bm{\omega},\bm{\sigma_\beta},\bm{\delta}$ and $\bm{\pi}$ iteratively, and the full sampling algorithm is presented in Algorithm \ref{algo:mhpcg}.

\begin{breakablealgorithm}
	\caption{Sampling Algorithm for SILM}
	\label{algo:mhpcg}
 	\begin{algorithmic}
	\REQUIRE Initial values $\{\bm{\beta}^{0},\tau^{0},l^{0},\bm{g}^{0},\bm{\omega}^{0},\bm{\sigma_\beta}^{0},\bm{\delta}^{0},\bm{\pi}^{0}\}$; Tuning parameters for proposal distributions $\{\sigma_\beta, \sigma_\tau,\sigma_l\}$; Spike parameter $r(0)=c$; Parameters for prior: $\{a_{\sigma_{\beta_j}},b_{\sigma_{\beta_j}}\}_{j=1}^{p}$, $\{a_\tau,b_\tau\}$, $\{a_l,b_l\}$, $\{a_\pi,b_\pi\}$; Data $\{\bm{x_i},y_i\}_{i=1}^{n}$ where $y_i\in\{1,-1\}$.

	\ENSURE Given $\bm{\beta}^{(t)},\tau^{(t)},l^{(t)},\bm{g}^{(t)},\bm{\omega}^{(t)},\bm{\sigma_\beta}^{(t)},\bm{\delta}^{(t)},\bm{\pi}^{(t)}$, generate $\bm{\beta}^{(t+1)},\tau^{(t+1)},l^{(t+1)},\bm{g}^{(t+1)},\bm{\omega}^{(t+1)}$, $\bm{\sigma_\beta}^{(t+1)},\bm{\delta}^{(t+1)},\bm{\pi}^{(t+1)}$ as follows:
	\begin{enumerate}
		\item Sample $\bm{\beta}^{(t+1)}$ from reduced conditional posterior $p(\bm{\beta}|\bm{y},\bm{\omega}^{(t)},\bm{\sigma_\beta}^{(t)},\bm{\delta}^{(t)},\bm{\pi}^{(t)},\tau^{(t)},l^{(t)})$ by Metropolis-Hastings (MH) algorithm with a normal proposal distribution $N(\bm{\beta}^{(t)},\sigma_\beta I_p)$, and $\bm{\beta}^{(t+1)}\leftarrow \bm{\beta}^{(t+1)}/||\bm{\beta}^{(t+1)}||_2$.

		\item Sample $\tau^{(t+1)}$ from reduced conditional posterior $p(\tau|\bm{y},\bm{\omega}^{(t)},\bm{\beta}^{(t+1)},\bm{\sigma_\beta}^{(t)},\bm{\delta}^{(t)},\bm{\pi}^{(t)},l^{(t)})$ by MH algorithm with proposal distribution $\ln \tilde{\tau}^{(t+1)}\sim\text{Normal}(\ln \tau^{(t)}, \sigma_\tau)$.

		\item Sample $l^{(t+1)}$ from reduced conditional posterior $p(l|\bm{y},\bm{\omega}^{(t)},\bm{\beta}^{(t+1)},\bm{\sigma_\beta}^{(t)},\bm{\delta}^{(t)},\bm{\pi}^{(t)},\tau^{(t+1)})$ by MH algorithm with proposal distribution $\ln \tilde{l}^{(t+1)}\sim\text{Normal}(\ln l^{(t)}, \sigma_l)$.

		\item Sample $\bm{g}^{(t+1)}$ from $\text{Normal}\left(\bm{\mu_{g}},\Sigma_{g}\right)$ with $\Sigma_g=K - K(K + \Omega^{-1})^{-1}K$, $\bm{\mu_g}=\frac{1}{2}\Sigma_g\bm{y}$, $\Omega =\text{diag}(\bm{\omega}^{(t)})$ and $K_{ij} = \tau^{(t+1)}\exp\{ -\frac{\left(\bm{x_{i}}^T\bm{\beta}^{(t+1)}-\bm{x_{j}}^T\bm{\beta}^{(t+1)}\right)^{2}}{l^{(t+1)}} \}$.

		\item Sample $\omega_i^{(t+1)}$ from PG$(1,g_i^{(t+1)})$ for $i = 1,2,\cdots,n$.

		\item Sample $\sigma_{\beta_j}^{(t+1)}$ from InvGamma$\left(a_{\sigma_{\beta_j}}+\frac{1}{2},\frac{\left(\beta_j^{(t+1)}\right)^2}{2r(\delta_j^{(t)})}+b_{\sigma_{\beta_j}}\right)$ for $j=1,2,\cdots,p$.

		\item Sample $\delta_j^{(t+1)}$ from Bernoulli$\left(\left(1+\frac{p_{\text{Spike}}(\beta_j^{(t+1)})(1-\pi_j^{(t)})}{p_{\text{Slab}}(\beta_j^{(t+1)})\pi_j^{(t)}}\right)^{-1}\right)$ for $j=1,2,\cdots,p$ that $p_{\text{Spike}}$ and $p_{\text{Slab}}$ represent the density functions for zero-mean Gaussian distributions with variance $c\cdot\sigma_{\beta_j}^{(t+1)}$ and $\sigma_{\beta_j}^{(t+1)}$ respectively.

		\item Sample $\pi_j^{(t+1)}$ from Beta$\left(a_\pi+\delta_j^{(t+1)},b_\pi+1-\delta_j^{(t+1)}\right)$ for $j=1,2,\cdots,p$.
	\end{enumerate}
\end{algorithmic}
\end{breakablealgorithm}

To make sure that the Markov chain is well mixed,  the first half of the $N$ MCMC samples are taken as burn-in period, and the second half of the MCMC samples are used for statistical inference. We take $N=10000$ in our analysis and the posterior mean are reported as the point estimation for each parameter of interest. In addition, Markov chains with independent random initial values are run to make sure the convergence of the Markov chain, and the Potential Scale Reduction Factors (PSRF) \citep{brooks1998} is used to diagnose the convergence. In practice, a PSRF less than 1.2 indicates the convergence of Markov chains \citep{brooks1998}.
\subsection{Computational  Issues}
\label{subsec:comput}

The major computational cost for our sampling algorithm comes from computing the determinant and inverse, or the linear equations of positive definite (PD) matrix $K+\Omega^{-1}\in\mathbb{R}^{n\times n}$ in equations (\ref{equ:post:g}), (\ref{equ:post:beta2}), (\ref{equ:post:tau2}) and (\ref{equ:post:l2}). Typically, Cholesky factorization is commonly used to compute the determinant and inverse of PD matrix, which takes $O(n^3)$ time for the computation. When the sample size goes to large, the computation burden is large. Thus, iterative methods, such as conjugate gradient method  with proper preconditioner, are suggested for this case.

In addition, approximation methods can be applied to speed up the computation of the inverse matrix. For example, Nystr\"{o}m approximation method is an important low-rank matrix approximation technique for the kernel matrix and has been widely developed \citep{williams2001,drineas2005,zhang2008,kumar2009,li2010,wang2013,gittens2016}. The Nystr\"{o}m method approximates $K$ by $\tilde{K}=CW^{\dag}C^{T}$, where $C\in\mathbb{R}^{n\times m}$ is consist of $m$ randomly selected columns from $K$, $W^{\dag}$ denotes the (pseudo) inverse of $W\in\mathbb{R}^{m\times m}$, which is the intersection of the selected columns and rows. By Woodbury-Sherman-Morrison identity, we have
\begin{equation}
\label{equ:nys}
\left(\tilde{K} + \Omega^{-1} \right)^{-1}= \Omega-\Omega C\left(W+C^T\Omega C\right)^{-1}C^T\Omega,
\end{equation}
and thus, we only need the inverse of matrix $W+C^T\Omega C\in\mathbb{R}^{m\times m}, m<n$.
Consequently, the total computational time for computing the inverse matrix can be reduced by Nystr\"{o}m method, for example, to $O(m^2n)$ time complexity as shown in \citet{williams2001}.

There is one thing needed to be mentioned that the approximation for posterior covariance $\Sigma_g$ in equation (\ref{equ:post:g}) may lead to indefinite matrix $\tilde{\Sigma}_g =K - K(\tilde{K} + \Omega^{-1})^{-1}K$. To ensure
the positive definiteness of covariance, we take the following positive semidefinite (PSD) approximation procedure:
$$\tilde{\Sigma}_g^+ = \arg\min_{A \geq 0} \|A - \tilde{\Sigma}_g\|_{F}^{2},$$
where $\|\cdot\|_F$ is Frobenius norm and $A \geq 0$ is symmetric PSD matrix with order $n$. Let $\tilde{\Sigma}_g = U\Lambda U^T$ be the eigen decomposition of $\tilde{\Sigma}_g$, and the optimal solution $\tilde{\Sigma}_g^+$ has analytical form $\tilde{\Sigma}_g^+ = U\Lambda^+ U^T$ where $\Lambda=\text{diag}\left(\lambda_1,\cdots,\lambda_n\right)$ and $\Lambda^+=\text{diag}\left(\max\{\lambda_1,0\},\cdots,\max\{\lambda_n,0\}\right)$ \citep{higham1988}. Even though the time complexity for eigen decomposition is still $O(n^3)$, but such decomposition is inevitable to sample from $n$-dimensional multivariate Gaussian distribution.

Finally, several methods are proposed for approximating the determinant of a larger matrix, see for example, \citep{zhang2007,boutsidis2017}, \citep{han2015}, \citep{dong2017}, and \citet{gardner2018}, and can be used to speed up our MCMC algorithm.

\section{Numerical Results}
\label{sec:num}
In this section, we present results from simulation studies and real data analysis to evaluate the performance of the propose method.  For our sampling algorithm, parameters for proposal distributions are tuned such that the acceptance rates are around 20\% $\sim$ 30\%.  In addition, we take the spike parameter $c=r(0)=1/1000$, $a_{\sigma_{\beta_j}} =b_{\sigma_{\beta_j}}=0.5$ for $j=1, 2, \cdots, p$ and $a_\pi=b_\pi=a_l=b_l=a_{\tau}=b_{\tau}=0.5$. To show the advantages of the proposed method, we compare it with two state-of-art methods, the sparse sliced inverse regression method via Lasso (LassoSIR) \citep{lin2019} and distribution-based LASSO (DLASSO) method \citep{wang2015}. DLASSO has a tuning parameter  selected by the modified BIC criterion \citep{wang2015} with an extra parameter $\gamma$. Thus, we consider two different versions of DLASSO with $\gamma=0$ or $\gamma=0.5$, denoted by DLASSO$_0$ and DLASSO$_{0.5}$ respectively. Other settings of these two methods follow that given in the original paper.

\subsection{Simulations}
\label{subsec:simu}
Let $s$ be the number of variables truly contribute to the response, we consider the following three different models for the unknown like function $g$, and take the number of variables $p=10$. The link function $g$ in Model 1 is linear, which is the simplest one. In Model 2, $g$ is nonlinear and continuously differentiable. In Model 3, it is nonlinear and not differentiable at $t=0$, which adds some difficulty to the problem.

\textbf{Model 1}: $g(t) = 5t$ with $\bm{\beta}^T = (3,2,2,0,0,0,0,0,0,0)/\sqrt{17} $ and $s=3$;

\textbf{Model 2}: $g(t) = 5\left(t + \sin(t^3)\right)$ with $\bm{\beta}^T=(2,2,1,1,0,0,0,0,0,0)/\sqrt{10}$ and $s=4$;

\textbf{Model 3}: $g(t) = 10|t|\times\sin(t)$ with $\bm{\beta}^T = (1,1,1,1,1,0,0,0,0,0)/\sqrt{5}$ and $s=5$;

In our simulations, the predictors $\{\bm{x}_i\}_{i=1}^{n}$ are generated from multivariate Gaussian distribution N$(0,\Sigma_X)$. We firstly take $\Sigma_X=\Sigma_1 = I_p$ to show the performance of different methods on the case where predictors are independent, and then take $\Sigma_X=\Sigma_2=\left(0.5^{|i-j|}\right)_{1\leq i,j\leq p}$ to see their performance on the case where predictors are dependent. The sample size $n$ is taken as $\{60, 100, 140\}$.  For parameter estimation, the estimation bias with the corresponding standard deviation is reported to evaluate the performance of each method. For variable selection, the number of true positives and false positives are reported to evaluate their performance. All results are reported based on 200 independent repetitions.

\begin{table}
	\centering
	\caption{TP/FP (standard deviation) from each method under different models.}
	\label{table:simtpfp}
	\resizebox{140mm}{78mm}
	{
		\begin{tabular}{cccccc}
		\toprule
			Method & $n$ & $\Sigma_X$ & Model 1 ($s=3$) & Model 2 ($s=4$) & Model 3 ($s=5$)\\
			\midrule
			Proposed&60&$\Sigma_1$&2.85(0.38)/0.12(0.35)&3.29(0.64)/0.09(0.28)&4.29(0.84)/0.07(0.25)\\
			&&$\Sigma_2$&2.61(0.56)/0.20(0.44)&2.95(0.68)/0.07(0.25)&3.67(0.77)/0.16(0.39)\\
			&100&$\Sigma_1$&3.00(0.00)/0.04(0.19)&3.76(0.47)/0.02(0.14)&4.95(0.21)/0.06(0.27)\\
			&&$\Sigma_2$&2.92(0.27)/0.06(0.23)&3.29(0.53)/0.12(0.38)&4.54(0.62)/0.03(0.17)\\
			&140&$\Sigma_1$&3.00(0.00)/0.01(0.10)&3.93(0.25)/0.00(0.00)&5.00(0.00)/0.00(0.00)\\
			&&$\Sigma_2$&2.95(0.21)/0.03(0.17)&3.57(0.53)/0.01(0.10)&4.90(0.30)/0.00(0.00)\\
			\hline
			 LassoSIR&60&$\Sigma_1$&2.99(0.10)/2.79(2.01)&3.90(0.30)/2.84(1.69)&4.97(0.17)/2.86(1.37)\\
			&&$\Sigma_2$&2.96(0.19)/1.79(1.62)&3.76(0.42)/1.87(1.64)&3.92(0.96)/1.52(1.43)\\
			&100&$\Sigma_1$&3.00(0.00)/3.11(1.87)&3.96(0.24)/2.72(1.83)&5.00(0.00)/3.06(1.31)\\
			&&$\Sigma_2$&3.00(0.00)/2.25(1.79)&3.92(0.27)/2.28(1.65)&4.30(0.83)/1.27(1.42)\\
			&140&$\Sigma_1$&3.00(0.00)/3.17(1.98)&4.00(0.00)/3.10(1.57)&5.00(0.00)/2.78(1.48)\\
			&&$\Sigma_2$&3.00(0.00)/1.93(2.04)&3.99(0.10)/2.18(1.71)&4.75(0.53)/1.64(1.28)\\
			\hline
			DLASSO$_0$&60&$\Sigma_1$&2.24(1.07)/0.12(0.35)&2.05(1.30)/0.06(0.23)&0.77(1.60)/0.01(0.10)\\
			&&$\Sigma_2$&2.33(0.82)/0.06(0.23)&2.27(1.15)/0.02(0.14)&0.77(1.25)/0.01(0.10)\\
			&100&$\Sigma_1$&2.81(0.56)/0.16(0.36)&3.35(0.88)/0.11(0.31)&3.02(2.32)/0.10(0.30)\\
			&&$\Sigma_2$&2.90(0.30)/0.06(0.31)&3.36(0.81)/0.03(0.17)&1.46(1.61)/0.01(0.10)\\
			&140&$\Sigma_1$&3.00(0.00)/0.11(0.31)&3.61(0.73)/0.08(0.27)&4.68(1.19)/0.23(0.44)\\
			&&$\Sigma_2$&2.98(0.14)/0.08(0.30)&3.64(0.59)/0.03(0.22)&2.73(1.88)/0.08(0.27)\\
			\hline

			DLASSO$_{0.5}$&60&$\Sigma_1$&1.07(1.30)/0.01(0.10)&0.86(1.18)/0.00(0.00)&0.01(0.50)/0.00(0.00)\\
			&&$\Sigma_2$&1.76(1.13)/0.02(0.14)&1.50(1.25)/0.00(0.00)&0.15(0.65)/0.00(0.00)\\
			&100&$\Sigma_1$&2.87(0.44)/0.09(0.32)&2.86(1.30)/0.03(0.17)&1.53(2.27)/0.04(0.19)\\
			&&$\Sigma_2$&2.77(0.58)/0.03(0.17)&2.96(1.10)/0.01(0.10)&0.73(1.21)/0.02(0.20)\\
			&140&$\Sigma_1$&3.00(0.00)/0.05(0.21)&3.56(0.85)/0.09(0.28)&4.64(1.28)/0.31(0.61)\\
			&&$\Sigma_2$&2.97(0.17)/0.08(0.44)&3.48(0.65)/0.03(0.17)&1.38(1.71)/0.05(0.26)\\
			\bottomrule
		\end{tabular}
	}
\end{table}

Table \ref{table:simtpfp} shows the simulation results on variable selection from each method under different models. When sample size is small ($n=60$), the performance of {\color{black} LassoSIR} is slightly better than the proposed method in terms of a little higher TP, but the corresponding FP is relatively more higher than our method. When sample size is larger, say $n>100$, the proposed method almost has the best performance among these methods in terms of both TP and FP. The performance of DLASSO is the worst among them. Importantly,  the dependence between predictors has a strong impact on its performance under Model 3. This suggests that DLASSO may provide unsatisfied results in some cases.  For the proposed method, the impact of dependence between predictors is minor.

\begin{table}
	\centering
	\caption{Biases and standard error (S.E.) of estimators from the proposed method under different models.} \label{table:simbeta}
	\resizebox{130mm}{105mm}
	{
		\begin{tabular}{ccccccccccccc}
			\toprule
			Model & $n$ & & $\beta_1$ & $\beta_2$& $\beta_3$& $\beta_4$& $\beta_5$& $\beta_6$& $\beta_7$& $\beta_8$& $\beta_9$& $\beta_{10}$\\ \midrule
			\textbf{\textit{Model 1}}& \\
			\multirow{6}{*}{$\Sigma_X=\Sigma_1$}
			& \multirow{2}{*}{60} & Bias &-.030&-.052&-.048&-.001&.005&-.005&-.006&-.004&-.006&-.001 \\
			& & S.E. &.086&.135&.118&.046&.050&.067&.066&.057&.077&.053 \\
			& \multirow{2}{*}{100} & Bias
			&-.030&-.002&-.009&.001&.001&.001&-.001&.002&-.001 &-.008 \\
			& & S.E. &.052&.060&.065&.046&.038&.041&.045&.045&.043&.041\\
			& \multirow{2}{*}{140} & Bias &.052&.060&.065&.046&.038&.041&.045&.045&.043&.041\\
			& & S.E. &.038&.050&.057&.033&.036&.032&.038&.037&.032&.041\\
			\midrule
			\multirow{6}{*}{$\Sigma_X=\Sigma_2$}
			& \multirow{2}{*}{60} & Bias
			&-.077&-.087&-.066&.028&.006&-.003&-.005&.010&-.002&-.001 \\
			& & S.E. &.157&.199&.159&.116&.071&.058&.078&.068&.069&.085\\
			& \multirow{2}{*}{100} & Bias
			&-.046&-.033&-.020&.005&.007&.008&-.003&.001&.001&.002 \\
			& & S.E. &.079&.130&.123&.063&.047&.056&.044&.039&.041&.049\\
			& \multirow{2}{*}{140} & Bias &-.024&-.029&-.025&-.001&-.001&.005&-.005&-.001&.001&-.004\\
			& & S.E. &.066&.121&.103&.059&.035&.045&.042&.039&.037&.039\\
			\midrule
			\textbf{\textit{Model 2}}& \\
			\multirow{6}{*}{$\Sigma_X=\Sigma_1$}
			& \multirow{2}{*}{60} & Bias
			&-.025&-.031&-.062&-.066&-.006&.005&-.001&.006&.005&-.011\\
			& & S.E. &.108&.081&.127&.120&.054&.058&.065&.062&.058&.049\\
			& \multirow{2}{*}{100} & Bias
			&-.025&-.013&-.012&-.015&.002&-.002&.003&.003&-.006&.002\\
			& & S.E. &.061&.057&.081&.086&.036&.041&.035&.035&.050&.041\\
			& \multirow{2}{*}{140} & Bias
			&-.014&-.008&-.010&-.012&-.009&.001&.000&-.001&-.004&.001\\
			& & S.E. &.046&.046&.061&.073&.032&.037&.029&.034&.034&.029\\
			\midrule
			\multirow{6}{*}{$\Sigma_X=\Sigma_2$}
			& \multirow{2}{*}{60} & Bias
			&-.096&-.042&-.061&-.075&.004&.003&.006&.007&.002&.001\\
			& & S.E. &.153&.158&.188&.170&.056&.053&.057&.058&.061&.066\\
			& \multirow{2}{*}{100} & Bias
			&-.039&-.028&-.046&-.050&.029&.008&.004&-.001&-.005&.002\\
			& & S.E. &.092&.099&.139&.128&.082&.055&.049&.056&.050&.049\\
			& \multirow{2}{*}{140} & Bias
			&-.018&-.033&-.033&-.016&.009&-.001&-.003&-.003&-.001&-.002\\
			& & S.E. &.078&.098&.115&.100&.033&.041&.040&.040&.034&.036\\
			\midrule
			\textbf{\textit{Model 3}}& \\
			\multirow{6}{*}{$\Sigma_X=\Sigma_1$}
			& \multirow{2}{*}{60} & Bias
			&-.063&-.072&-.063&-.052&-.057&.004&.005&.000&.002&.001\\
			& & S.E. &.148&.144&.137&.162&.152&.052&.065&.064&.055&.064\\
			& \multirow{2}{*}{100} & Bias
			&-.025&-.014&-.024&-.018&-.022&-.000&.002&.003&-.004&.001\\
			& & S.E. &.082&.068&.069&.088&.091&.039&.051&.040&.046&.048\\
			& \multirow{2}{*}{140} & Bias
			&-.019&-.013&.002&-.018&-.016&-.001&.002&-.001&-.001&-.002\\
			& & S.E. &.065&.058&.062&.068&.061&.032&.042&.036&.031&.036\\
			\midrule
			\multirow{6}{*}{$\Sigma_X=\Sigma_2$}
			& \multirow{2}{*}{60} & Bias
			&-.122&-.092&-.025&-.062&-.131&.033&.001&.013&.006&-0.002\\
			& & S.E. &.167&.194&.189&.184&.183&.084&.090&.073&.065&0.061\\
			& \multirow{2}{*}{100} & Bias
			&-.038&-.038&-.041&-.037&-.056&.009&.000&.002&.002&-.004\\
			& & S.E. &.107&.139&.133&.150&.124&.047&.039&.043&.047&.044\\
			& \multirow{2}{*}{140} & Bias
			&-.030&-.017&-.023&-.029&-.020&.003&-.003&.002&-.001&.003\\
			& & S.E. &.086&.104&.103&.092&.086&.036&.038&.040&.030&.033\\
			\bottomrule
		\end{tabular}
	}
\end{table}

Furthermore, we show in Table \ref{table:simbeta} the estimation bias of parameters from the proposed method, and the corresponding results for LassoSIR and DLASSO are shown in Appendix~\ref{app:result}. Results in Table \ref{table:simbeta}  indicate that our proposed method estimates the single index $\beta$ well in all cases, and the biases and standard errors decline in general as the sample sizes increase. The bias from LassoSIR and DLASSO are larger than that from our method, which may partially explain the better performance of our method on variable selection.

\begin{table}
	\centering
	\caption{TP/FP (standard deviation) from the proposed method under Model 1 when Nystr\"{o}m approximation is used to speed up calculation.}\label{table:simtpfp2}
	\begin{tabular}{ccc}
		\toprule
		$m$ & $\Sigma_X=\Sigma_1$ & $\Sigma_X=\Sigma_2$ \\
		\midrule
		$60$&2.96(0.19)/0.58(0.92)&2.80(0.42)/1.47(1.31)\\
		$80$&2.99(0.10)/0.09(0.37)&2.93(0.29)/0.37(0.78)\\
		$100$&3.00(0.00)/0.00(0.00)&2.99(0.10)/0.07(0.35)\\ \bottomrule
	\end{tabular}
\end{table}

\subsection{Accelerating MCMC for Large Sample Problems}
\label{subsec:acce}

In this section, we evaluate the impact of the approximation in computing the inverse and determinant of positive definite matrix $K+\Omega^{-1}\in\mathbb{R}^{n\times n}$ in equations (\ref{equ:post:g}), (\ref{equ:post:beta2}), (\ref{equ:post:tau2}) and (\ref{equ:post:l2}), on variable selection performance of the proposed method.  Since the proposed method has similar performance on different link functions, we now focus on Model 1, and take sample size $n=500$.

We focus on the performance of the Nystr\"{o}m approximation method as described in Section~\ref{subsec:comput}. Specifically, we follow the method in \citet{williams2001} to approximate the inverse of matrix $K+\Omega^{-1}$, where $m \in \{60, 80, 100\}$ columns are chosen uniformly at random without replacement, and the determinant is approximated using GPytorch \citep{gardner2018}. Theoretically the matrix $W+C^T\Omega C$ in equation (\ref{equ:nys}) is at least positive semidefinite, but numerically it has a few small but negative eigenvalues like $K$. To tackle with the problem, jilters are added to the diagonal of $K$, i.e. $K$ is replaced by $K+\epsilon I_{n}$. In practice, $\epsilon= 1/100\cdot\tau$ and the inverse of $W+C^T\Omega C$ is computed by Cholesky factorization. Results given in Table~\ref{table:simtpfp2} show that the proposed method with Nystr\"{o}m approximation has a comparable performance with that with exact calculation as shown in Table~\ref{table:simtpfp}. This shows the hope of speeding up the MCMC algorithm for problems with a large sample size. There is one thing to be mentioned that the convergence of MCMC algorithm is fast and one thousand iterations are enough for obtaining converged MCMC samples.

\subsection{Real Data Examples}
\label{subsec:real}
In this section, we show the performance of the proposed method on three real examples. To eliminate the impact of units, predictor variables in all examples are scaled to have mean zero and standard deviation one.  We report the estimated $\beta$, $\bm{\hat{\beta}}$, from different methods. For the proposed method, we also report the standard deviation of $\bm{\hat{\beta}}$ (denoted as $s_{\bm{\hat{\beta}}}$), the posterior probability of $\delta_j=1$ (denoted as $\hat{\pi_j}$),  standard deviation of $\hat{\pi_j}$ (denoted as $s_{\hat{\pi_j}}$), and PRSF for convergence diagnosis of Markov chains.

\textbf{Example 1}: Swiss Banknote Data \citep{flury1988}.  This dataset, available in the R package uskewFactors, contains measurements on 200 Swiss banknotes including 100 genuine and 100 counterfeit. There are 6 predictor variables that are measurement results on the banknotes including length of bill, width of left edge, width of right edge, bottom margin width, top margin width and diagonal length. For convenience, these variables are denoted by $X_1,\cdots,X_6$ respectively. The response $Y$ is coded as -1 or 1, indicating whether the banknote is genuine or not.

\begin{table}
	\centering
	\caption{ Results for Swiss Banknote Data (\textbf{Example 1})}
	\label{table:banknote}
		\begin{tabular}{lcccccc}
			\toprule
			Method&$X_1$&$X_2$&$X_3$&$X_4$&$X_5$&$X_6$\\
			\midrule
			LassoSIR&0&-0.282&0.306&0.451&0.474&-0.630\\
			DLASSO$_0$&0&0&0&0.638&0.376&-0.671\\
			DLASSO$_{0.5}$&0&0&0&0.638&0.376&-0.671\\
			\midrule
			\multicolumn{7}{l}{\bf{\textit{Proposed}}}\\
			$\bm{\hat{\beta}}$&-0.001&0.031&0.029&\textbf{0.667}&\textbf{0.299}&\textbf{-0.603}\\
			$s_{\bm{\hat{\beta}}}$ &0.057&0.101&0.098&0.103&0.197&0.162\\
			$\hat{\pi_j}$&0.107&0.151&0.158&\textbf{0.888}&\textbf{0.592}&\textbf{0.861}\\
			$s_{\hat{\pi_j}}$&0.310&0.358&0.365&0.315&0.491&0.345\\
			PRSF&1.03&1.05&1.05&1.05&1.11&1.10\\
			\bottomrule
		\end{tabular}
\end{table}
 Table \ref{table:banknote} shows the estimation and variable selection results from different methods. For the propose method, 6 Markov chains are run from different initials and PSRFs for each $\beta_j$ are also shown in Table \ref{table:banknote}, which indicates Markov chain is converged.  The proposed method and DLASSO select $\{X_4, X_5, X_6\}$ as the true variables contribute to the response in this example, however LassoSIR selects two more variables, $X_2$ and $X_3$.

\textbf{Example 2}: Body Fat Data \citep{penrose1985}. This dataset, available in the R package mfp, contains 252 observations and 128 samples are left after deleting samples with errors in the R package mplot. The response in this dataset is the percentage of body fat determined by underwater weighing and the predictor variables include age, weight, height and ten body circumference measurements (neck, chest, abdomen, hip, thigh, knee, ankle, biceps, forearm and wrist). For convenience, these variables are denoted by $X_1,\cdots,X_{13}$, respectively.

This dataset has been widely studied based on single index regression model, see for example, \citet{li2017,peng2011} and \citet{li2017penalized}. In our study, to fit to our settings, the response variable is transformed to be binary depending on whether its value is greater than the sample median, indicating whether the man is overweight or not. Table \ref{table:fat} shows the estimation and variable selection results from different methods. It can be seen that the predictor $X_6$ (abdomen) is selected by the proposed method and DLASSO, which coincides with the empirical fact that the measurement of abdomen indeed generally reflect the body fat. However, LassoSIR selects much more predictors among which the measurements on chest $(X_5)$ and abdomen $(X_6)$ is the most influential.

\begin{table}
	\centering
	\caption{Results for Body Fat Data  (\textbf{Example 2})}
	\label{table:fat}
	\resizebox{160mm}{29mm}
	{
		\begin{tabular}{cccccccccccccc}
			\toprule
			Method&$X_1$&$X_2$&$X_3$&$X_4$&$X_5$&$X_6$&$X_7$&$X_8$&$X_9$&$X_{10}$&$X_{11}$&$X_{12}$&$X_{13}$\\
			\midrule
			LassoSIR &0.028&0.124&0&0.011&0.661&0.512&0.331&0&0.182&0&0&0.374&0\\
			DLASSO$_0$&0&0&0&0&0&1&0&0&0&0&0&0&0\\
			DLASSO$_{0.5}$&0&0&0&0&0&1&0&0&0&0&0&0&0\\
			\midrule
			\multicolumn{7}{l}{\bf{\textit{Proposed}}}\\ 	$\bm{\hat{\beta}}$&0.014&-0.009&0.015&-0.023&0.019&\textbf{0.965}&-0.025&-0.022&-0.074&-0.007&-0.026&-0.016&-0.048\\
			$s_{\bm{\hat{\beta}}}$&0.042&0.058&0.045&0.047&0.115&0.037&0.067&0.058&0.106&0.054&0.059&0.052&0.064\\
			$\hat{\pi_j}$&0.070&0.100&0.085&0.091&0.179&\textbf{0.879}&0.128&0.124&0.220&0.099&0.111&0.098&0.150\\
			$s_{\hat{\pi_j}}$&0.255&0.301&0.280&0.288&0.384&0.325&0.334&0.330&0.414&0.299&0.314&0.298&0.357\\
			PRSF&1.01&1.05&1.03&1.05&1.09&1.17&1.08&1.10&1.05&1.07&1.02&1.11&1.02\\
			\bottomrule
		\end{tabular}
	}
\end{table}

\textbf{Example 3}: Breast Cancer Data \citep{patricio2018}. This dataset,  available in the UCI machine learning repository, is developed to screen potential biomarkers of breast cancer based on routine blood analysis. Clinical features are observed or measured for 64 patients with breast cancer and 52 healthy controls. There are 9 predictors including Age, BMI, Glucose, Insulin, HOMA, Leptin, Adiponectin, Resistin and MCP-1, denoted by $X_1, \cdots, X_9$ respectively. Based on logistic regression, random forests and support vector machine methods, Age, BMI, Glucose and Resistin ($X_1,X_2,X_3,X_8$) are selected as potential biomarkers in \citet{patricio2018}.

\begin{table}
	\centering
	\caption{Results for Breast Cancer Data  (\textbf{Example 3})}
	\label{table:breast}
		\begin{tabular}{cccccccccc}
			\toprule
			Method&$X_1$&$X_2$&$X_3$&$X_4$&$X_5$&$X_6$&$X_7$&$X_8$&$X_9$\\
			\midrule
			LassoSIR&0.045&0.822&-0.455&-0.240&0&0&0&-0.237&0\\
			DLASSO$_0$&0&0&1&0&0&0&0&0&0\\
			DLASSO$_{0.5}$&0&0&1&0&0&0&0&0&0\\
			\midrule
		\multicolumn{7}{l}{\bf{\textit{Proposed}}}\\  $\bm{\hat{\beta}}$&0.053&\textbf{0.216}&\textbf{-0.698}&-0.131&-0.041&0.023&0.017&\textbf{-0.569}&0.011\\
			$s_{\bm{\hat{\beta}}}$&0.084&0.138&0.140&0.149&0.094&0.085&0.078&0.159&0.052\\
			$\hat{\pi_j}$&0.171&\textbf{0.539}&\textbf{0.888}&0.320&0.167&0.164&0.131&\textbf{0.849}&0.085\\
			$s_{\hat{\pi_j}}$&0.376&0.498&0.314&0.466&0.373&0.371&0.337&0.358&0.279\\
			PRSF&1.03&1.07&1.15&1.07&1.10&1.08&1.02&1.14&1.06\\
			\bottomrule
		\end{tabular}
\end{table}

Table~\ref{table:breast} shows the estimation and variable selection results from different methods. The results show that DLASSO methods only select predictor $X_3$ (Glucose). Comparing with results in \citet{patricio2018}, $X_1$ (Age) is not included in our selected model and $X_4$ (Insulin) is extra variable selected by LassoSIR. Thus, results from our method and LassoSIR are more closer to the underlying truth in \citet{patricio2018}.

\section{Conclusion}
\label{sec:conc}

In this paper, we propose a Bayesian estimation and variable selection approach for single index Logistic model with a relatively efficient MCMC sampling algorithm. Specially, the unknown link function $g$ is assumed to have a Gaussian process prior, and a Spike-Slab prior is assumed for single index $\bm{\beta}$ to find variables truly contributing to the response. The advantage of Bayesian model for single index Logistic model over frequentist methods is its ability to simultaneously estimate the unknown link function and the single index.  Numerical results from both simulation studies and real data analysis show the advantage of the proposed method.

Although we focus on parameter estimation and variable selection approach for single index Logistic model, the framework can be extended to other single index general models, for example, single index Poisson model. The problem for Bayesian estimation for single index models is the efficiency of MCMC algorithm. As discussed before, when the sample size is large, it will be very time consuming for sampling from the posterior distribution. Although some approximation methods (Section \ref{subsec:comput}) can be used to accelerate the MCMC algorithm and are shown to work well through simulations, the impact of this approximation on the distribution of MCMC samples are not well studied yet. It is in a strong necessity to explore the theoretical properties on the MCMC samples from algorithms with approximations. That is, we should study the convergence rate of MCMC samples from an approximated MCMC algorithms to the target distributions.
 In addition, for highly imbalanced problems, current sampling algorithm will be inefficient \citep{johndrow2019}. Thus, it is also in need to further investigation on how to tackle with the problem when samples are highly imbalanced. These interesting topics are left as future works.

\newpage
\bibliographystyle{chicago}

redpajama_set_name: RedPajamaArXiv


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Alex Winters and Keanu Reeves confirm third Bill & Ted movie will hit cinemas next year
Most excellent dudes, Alex Winters and Keanu Reeves have confirmed a third Bill & Ted movie is on the way.
The pair released a video on Twitter at Los Angeles Hollywood Bowl where they revealed they will reprise their titular roles as Bill S. Preston, Esq. and Ted (Theodore) Logan in Bill & Ted Face the Music, which will film this summer in New Orleans.
The film is scheduled for release in summer 2020, 28 years after the second instalment of the saga, Bill & Teds Bogus Journey.
The actors announced the news in a brief video standing in front of Los Angeles Hollywood Bowl, where Alex noted they will never play.
The pair then thanked fans for repeatedly demanding a third film.
Be excellent, dudes
We owe you a huge debt of gratitude, and we wanted to say thank you. And be excellent, said Alex with Keanu repeating: Be excellent.
An official logline for the movie teased the appearance of a few music legends as the two friends set out to save life as we know it.
Yet to fulfil their rock and roll destiny, the now middle-aged best friends set out on a new adventure, when a visitor from the future warns them that only their song can save life as we know it and bring harmony to the universe, it read.
Along the way, they will be helped by their families, old friends and a few music legends.
More: Film
Galaxy Quest filmmaker Dean Parisot will direct from a script written by previous Bill & Ted writers Chris Matheson and Ed Solomon.
The first movie, Bill & Teds Excellent Adventure was released in 1989 and followed the teenage pals time-travelling adventures while they tried to pass a high school history class.
The follow-up was released two years later and saw the pair battle their evil robotic duplicates.
Got a showbiz story?
If you've got a story, video or pictures get in touch with the Metro.co.uk Entertainment team by emailing us [email protected], calling 020 3615 2145 or by visiting our Submit Stuff page – we'd love to hear from you.
MORE: Grimes killing off stage name for new album: Itll be a public execution
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No matter what, refiners want to run crude, the more the better. It doesn't matter what comes out of the crude, and if you can get a refiner to process a secondary unit feedstock, then the stars have aligned, or you've died and gone to heaven! The trick is getting everybody on board and that takes detailed communication. This story is about how the stars did align, but the detailed communication was a little lacking.
Long ago, in a far away place…we had access to 100,000 barrels of waxy (high pour) vacuum gas oil to take advantage of available capacity on the refinery's "cat" unit.1 Of course, in deals like these there are always elements to work out. This situation was no different. To get it to the refinery we had to receive it on the water and then pump it several miles under the harbor. Not an easy feat, but we were willing to move mountains to make this happen.
Our Product Supply Operations Manager coordinated with the Pipeline Operations Manager in Dallas to make the movement. The Pipeline Operations Manager promptly refused. His concern was that if there were any problems with the movement, for instance if the movement was stopped, the waxy feedstock would solidify in the pipe and we'd have a two mile long candle to deal with. When my guy relayed this refusal to me, I asked him if he had communicated to the pipeline guy how much this was worth to us. His eyes lit up and he turned and walked out.
Sometime later he returned looking unhappy. He relayed that he contacted the pipeline guy again, although he used a more colorful word than "guy", and that he still refused the movement. I asked what he told the pipeline guy about the value of the deal, to which he replied, "It's worth a dime to us!" I reached into my pocket, pulled out a dime and threw it to him. He caught it, looked at it, looked at me, and walked out.
1 A fluid catalytic cracking unit (FCC) is a refinery unit used to convert the high-boiling, high-molecular weight hydrocarbon fractions of petroleum crude oils to more valuable gasoline, olefinic gases, and other products.


The objective of this thesis was to design an automatic big data analysis system for CPU and memory usage, which will be used to test performance critical devices in the radio wave communication network.
The objective of this thesis was to develop machine learning based models for CPU-load monitoring and reporting of anomalous test results.
The objective of this diploma thesis was to create a device for moisture percentage measurement during the grain drying. The meter enables the process automation and control over the internet.


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Ena from Science Fell In Love reinvents the classic Chunibyo archetype of anime
The charming anime of educational entertainment Science fell in love, so I tried to prove it elegantly blends the hard science of a modern lab with the adorable tricks of a life anime, mostly to determine if co-stars Himuro Ayame and Yukimura Shinya are really in love. But there is also room to explore the other characters and the archetypes they embody.
Although Shinya is a fairly conventional kuudere, little Ibarada Ena is a self-proclaimed chunibyo, and is not the only recent anime character to adapt to this mold. Ena has a talent for the dramatic like the other chunibyos, but she also has the ingenious mind of a scientist and a heart of gold. It is perhaps the best chunibyo of the anime yet.
RELATED: Science Fell in Love: Why Ayame Originally Fell in Love with Shinya
How Science Ena Fell Into Love adapts to the Chunibyo archetype
Episode 3 de Science fell in love sees Ena continue her chunibyo streak, but even though she's always acted that way in the anime, this is the first time she's commented on it. In fact, the The anime has rarely shown Ena's internal dialogue so far, giving fans the impression that she's a bit of a playful tsundere who hasn't taken anything seriously outside of her college studies.
The opening scenes of episode 3 are shown from Ena's point of view, giving a rare insight into her own worldview and personality outside of her external dialogue. She is depicted dressing for class in a dark bedroom while it rains outside, and is called chunibyo by her name to explain her gloomy worldview and her gothic dresses with ruffles under her lab coat.
For a time, Ena ominously portrays her life and the world around her in dark, dramatic tones, a classic sign of a chunibyo with supposedly hidden powers and ideas. Chunibyos as Ena i Komi cannot communicateOmoharu Nakanaka believes that they are tragic and unique beings that no one else understands, but in the end, Ena is not so tragic after all. In fact, he has a great life and is slowly realizing it.
RELATED: Science has fallen in love: the best lessons of Science Bear, Rikekuma
The undeniable joy of Ibarada Ena's university life
Ena really embodies the chunibyo archetype in some way, especially when episode 3 begins, but there's so much more for this little science student. Obviously he's just trying to have fun with his chunibyo lifestyle, but deep down, he doesn't really subscribe to it. A chunibyo can act out of boredom, self-pity, or a desire for attention, but based on Science fell in loveSo far, Ena doesn't need all this in the lab and she really has a lot to be thankful for. No one asks her to become a full-fledged genki girl, but again, she has no real reason to be a chunibyo, no matter how old she is.
Unlike an anxious, attention-seeking eighth-grader, Ena is a name-only chunibyo and is quick to dismiss that person when she sees the happy faces of her classmates. Chunibyos feels lonely and misunderstood, but Ena, as a member of the gang, feels welcome and happy in this science lab.
She tried to see the world in terms of darkness and despair, but she ate those words later when she proclaimed to herself that her friends, the light of the sun, and her purpose here are far stronger than her ways of chunibyo. With that in mind, Ena feels ready to continue her doctoral studies in Saitama with her friends by her side, all the self-pitying darkness forgotten. It doesn't hurt that she also develops a mutual love affair with Kosuke.
Science Fell in Love's Himuro Ayame softly echoes my Senpai is boring
Louis Kemner
(4007 articles published)
Louis Kemner has been a fan of Japanese animation since 1997, when he discovered Pokémon and Dragon Ball Z in elementary school. Now, he's a bigger fan of anime / manga than ever and is willing to share what he knows with readers around the world. He graduated from high school in 2009 and earned a bachelor's degree in creative writing from UMKC in 2013, and then put his skills to work in 2019 with CBR.com. He's always looking for a wonderful new anime to watch or a manga series to read.
More Louis Kemner
15 adorable little dog breeds
Historical donation of $ 159 million for scientific research at UdeM


Bradford S. Bell is an Associate Professor of Human Resource Studies and Director of the Center for Advanced Human Resource Studies. He received his B.A. in Psychology from the University of Maryland at College Park and his M.A. and Ph.D. in Industrial and Organizational Psychology from Michigan State University. Dr. Bell teaches courses in Human Resource Management, Training and Development, and Work Groups and Teams to graduate and undergraduate students. Dr. Bell worked in the management and organization development department of First USA Bank/Banc One and has worked as an HR consultant for multiple private and public firms. Dr. Bell is a member of the Academy of Management, Society for Industrial and Organizational Psychology, and American Psychological Association.
Dr. Bell's research interests include training and development, team development and effectiveness, and virtual work. His research has appeared in a number of journals, including the Journal of Applied Psychology, Personnel Psychology, Human Resource Management, Academy of Management Learning & Education, International Journal of Human Resource Management, Group and Organization Management, and the International Journal of Selection and Assessment. In addition, Dr. Bell has published numerous chapters that have appeared in edited research volumes and presents regularly at conferences, including the Academy of Management and the Society for Industrial and Organizational Psychology. Dr. Bell was awarded the Early Career Achievement Award by the HR Division of the Academy of Management in 2008 and he is a former Editor of Personnel Psychology.
Traci Sitzmann, Bradford Bell. 2017. The Dynamic Effects of Subconscious Goal Pursuit on Resource Allocation, Task Performance, and Goal Abandonment, Organizational Behavior & Human Decision Processes . 138:1-14.
Bradford Bell, Scott I. Tannenbaum, Raymond A. Noe, J. Kevin Ford, Kurt Kraiger. 2017. 100 years of training and development research: What we know and where we should go, Journal of Applied Psychology . 102(3):305-323.
Adam M Kanar, Christopher Collins, Bradford Bell. 2015. Changing an unfavorable employer reputation: the roles of recruitment message type and familiarity with employer, Journal of Applied Social Psychology .
Bradford Bell. 2014. Rigor and relevance, Personnel Psychology . 67(1):1-4.
Adam M Kanar, Bradford Bell. 2013. Guiding learners through technology-based instruction: The effects of adaptive guidance design and individual differences on learning over time, Journal of Educational Psychology . 105(4):1067-1081.
Bradford Bell, Jessica E Federman. 2013. E-learning in Postsecondary Education, Future of Children . 23(1):165-185.
Christopher Collins, Bradford Bell. 2012. The state of the art in performance management: Learnings from discussions with leading organizations, People & Strategy . 36(2):50-52.
Bradford Bell. 2012. Remote work: Examining current trends and organisational practices, International HR Adviser . 49:4-6.
Bradford Bell, Steve WJ Kozlowski. 2012. Three conceptual themes for future research on teams, Industrial and Organizational Psychology: Perspectives on Science and Practice . 5(1):45-48.
Adam A Kanar, Christopher Collins, Bradford Bell. 2010. A comparison of the effects of positive and negative information on job seekers' organizational attraction and attribute recall, Human Performance . 23(3):193-212.
Traci Sitzmann, Katherine Ely, Bradford Bell, Kristina Bauer. 2010. The effects of technical difficulties on learning and attrition during online training, Journal of Experimental Psychology: Applied . 16(3):281-292.
Bradford Bell, Jessica Federman. 2010. Self-assessments of knowledge: Where do we go from here?, Academy of Mgmt Learning & Education . 9(2):342-347.
Matthew J Pearsall, Aleksander PJ Ellis, Bradford Bell. 2010. Building the infrastructure: The effects of role identification behaviors on team cognition development and performance, Journal of Applied Psychology . 95(1):192-200.
Traci Sitzmann, Bradford Bell, Kurt Kraiger, Adam M Kanar. 2009. A multilevel analysis of the effects of prompting self-regulation in technology-based instruction, Personnel Psychology . 62(4):697-734.
Bradford Bell, Adam M Kanar, Steve WJ Kozlowski. 2008. Current issues and future directions in simulation-based training, International Journal of Human Resources Management . 19(8):1416-1434.
Bradford Bell, Steve WJ Kozlowski. 2008. Active learning: Effects of core training design elements on self-regulatory processes, learning, and adaptability, Journal of Applied Psychology . 93(2):296-316.
Bradford Bell, J Kevin Ford. 2007. Reactions to skill assessment: The forgotten factor in explaining motivation to learn, Human Resource Development Quarterly . 18(1):33-62.
Bradford Bell, Lee Sae-Won, Sarah K Yeung. 2006. The impact of eHR on professional competence in HRM: Implications for the development of HR professionals, Human Resource Management Journal (US) . 45(3):295-308.
Steve W J Kozlowski, Bradford Bell. 2006. Disentangling achievement orientation and goal setting: Effects on self-regulatory processes, Journal of Applied Psychology . 91(4):900-916.
Bradford Bell, Darin Wiechmann, Ann Marie Ryan. 2006. Consequences of organizational justice expectations in a selection system, Journal of Applied Psychology . 91(2):455-466.
Aleksander P J Ellis, Bradford Bell, Robert E Ployhart, John R Hollenbeck, Daniel R Ilgen. 2005. An evaluation of generic teamwork skills training with action teams: Effects on cognitive and skill-based outcomes, Personnel Psychology . 58(3):641-672.
Bradford Bell, Ann Marie Ryan, Darin Wiechmann. 2004. Justice expectations and applicant perceptions, International Journal of Selection and Assessment . 12(1-2):24-38.
Bradford Bell, Steve W J Kozlowski. 2002. Adaptive guidance: Enhancing self-regulation, knowledge, and performance in technology-based training, Personnel Psychology . 55(2):267-306.
Bradford Bell, Steve W J Kozlowski. 2002. Goal orientation and ability: Interactive effects on self-efficacy, performance, and knowledge, Journal of Applied Psychology . 87(3):497-505.
Bradford Bell, Steve W J Kozlowski. 2002. A typology of virtual teams: Implications for effective leadership, Group and Organization Management . 27(1):14-49.
Bradford Bell, Daniel R Ilgen. 2001. Informed consent and dual purpose research, American Psychologist . 56(12):1177.
Bradford Bell, Steve W J Kozlowski, R J Toney, M E Mullins, D A Weissbein, K G Brown. 2001. Developing adaptability: A theory for the design of integrated-embedded training systems, Advances in Human Performance and Cognitive Engineering Research . 1:59-124.
Bradford Bell, K J Klein. 2001. Effects of disability, gender, and job level on ratings of job applicants, Rehabilitation Psychology . 46(3):229-246.
Bradford Bell, Steve W J Kozlowski. 2007. A theory-based approach for designing distributed learning systems. in Toward a science of distributed learning. Washington, DC, United States: APA, 2007.
Bradford Bell, Steve W J Kozlowski. 2007. Advances in technology-based training. in Managing human resources in North America : current issues and perspectives. Oxford, United Kingdom: Routledge, 2007.
Aleksander P J Ellis, Bradford Bell. 2006. Capacity, collaboration, and commonality: A framework for understanding team learning. in Understanding teams: A volume in research in management. Charlotte NC, United States: Information Age, 2006. Linda L Neider, Chester A Schriesheim.
Bradford Bell, Steve W J Kozlowski. 2003. Work groups and teams in organizations. in Comprehensive Handbook of Psychology (vol. 12): Industrial and Organizational Psychology. New York NY, United States: Wiley, 2003.
Matthew Pearsall, Aleksander PJ Ellis, Bradford Bell. 2008. Slippage in the system: The Effects of errors in transactive memory behavior on team performance. Briarcliff Manor, NY, United States: Academy of Management, 2008.
An Examination of Barriers to Employee Participation in Health and Wellness Programs. Presented to Academy of Management. Chicago, IL. 2018.
When and How Dynamic Team Membership Affects Team Effectiveness. Presented to Academy of Management. Chicago, IL. 2018.
Dynamic Membership and Team Effectiveness: The Role of Team Emergent States. Presented to INGRoup. Bethesda, MD. 2018.
The Longitudinal Effects of Telecommuting on Individual and Team Outcomes. Presented to McGill University. Montreal, QC. 2018.
The Longitudinal Effects of Telecommuting on Individual and Team Outcomes. Presented to Work and Family Researchers Network. Washington, DC. 2018.
Building Inclusive Climates Top-Down and Bottom-Up. Presented to Society for Industrial and Organizational Psychology. Chicago, IL. 2018.
Not All Telework is Created Equal: A Closer Look at the Nature of Telework. Presented to Society for Industrial and Organizational Psychology. Chicago, IL. 2018.
An examination of the mediating effects of team emergent states on the relationship between multiple team membership and project team satisfaction. Presented to Academy of Management. Atlanta, GA. 2017.
Dynamic membership and team effectiveness. Presented to Academy of Management. Atlanta, GA. 2017.
Innovative and experiential approaches to teaching HRM. Presented to Academy of Management. Atlanta, GA. 2017.
Virtual work research: Crossing boundaries. Presented to Academy of Management. Atlanta, GA. 2017.
Theorizing about enhancing the effectiveness of project teams. Presented to INGRoup. St. Louis, MO. 2017.
Editorial landscape: Where we've been and where we're going. Presented to Society for Industrial and Organizational Psychology. Orlando, FL. 2017.
Teaching HR to different audiences. Presented to Academy of Management. Anaheim, CA. 2016.
Toward a unified approach to conceptualizing and measuring virtuality. Presented to Academy of Management. Anaheim, CA. 2016.
Talk and let talk: The effects of language proficiency on speaking up and expertise perceptions in multinational teams.. Presented to INGRoup. Helsinki, Finland. 2016.
Unpacking the context of telework: The role of team teleworking density. Presented to Work and Family Researchers Network. Washington, DC. 2016.
A quasi-experimental study of leader training for cultivating inclusive climates. . Anaheim, CA. 2016.
Dynamic effects of subconscious goals on resource allocation, performance, and goal abandonment. Presented to Academy of Management. Vancouver, BC. 2015.
The consequences of telework: An examination of individual and contextual moderators. Presented to LERA. Pittsburgh, PA. 2015.
Talk and let talk: The effects of language proficiency on speaking up and expertise perceptions in multinational teams.. Presented to Academy of Management. Vancouver, Canada. 2015.
The consequences of telework: An examination of individual and contextual moderators. Presented to Academy of Management. Orlando, FL. 2013.
A study of the influence of virtuality dimensions on leadership effectiveness. Presented to Academy of Management. Orlando, FL. 2013.
Developing and managing performance. Presented to Academy of Management. Orlando, FL. 2013.
The importance of employee learning. Presented to Academy of Management. Boston, MA. 2012.
Errors in Organizations. Presented to Society for Industrial and Organizational Psychology. San Diego, CA. 2012.
Individual and contextual influences on the transfer of diversity training. Presented to Society for Industrial and Organizational Psychology. San Diego, CA. 2012.
Modelisation de la distance subjective au sein des equipes virtuelles. Presented to AGRH. Marrakech, Morocco. 2011.
The effects of interruptions on self-regulated learning during web-based instruction. Presented to Academy of Management. San Antonio, TX. 2011.
Recent advances and future directions in research on self-regulation. Presented to Academy of Management. San Antonio, TX. 2011.
An examination of the effects of interruptions and learner characteristics on self-regulated learning during web-based instruction. Presented to Cognitive Science Society. Boston, MA. 2011.
The effects of interruptions and learner characteristics on self-regulated learning. Presented to American Psychological Society. Washington, DC. 2011.
Interpersonal dysfuntion: Understanding mistreatment and dysfunction in the workplace. Presented to Academy of Management. Montreal, QC. 2010.
Self-Assessment of Knowledge: Reflections and Reactions to Sitzmann et al.'s (2010) article. Presented to Academy of Management. Montreal, QC. 2010.
The effects of interruptions on self-regulation and learning in technology-delivered instruction. Presented to Academy of Management. Montreal, QC. 2010.
Toward a better understanding of subjective distance. Presented to Academy of Management. Montreal, QC. 2010.
The effects of technical interruptions on learning and attrition from web-based instruction. Presented to Academy of Management. Chicago, IL. 2009.
Emotion control training strategies in technology-delivered instruction. Presented to Society for Industrial and Organizational Psychology. New Orleans, LA. 2009.
Slippage in the system: The effects of errors in transactive memory behavior on team performance. Presented to Academy of Management. Anaheim, CA. 2008.
Training and Socialization. Presented to Academy of Management. Anaheim, CA. 2008.
Adaptive guidance in technology-based training: An aptitude-treatment perspective.. Presented to Society for Industrial and Organizational Psychology. San Francisco, CA. 2008.
Frontier Series Learning, Training, and Development in Organizations. Presented to Society for Industrial and Organizational Psychology. San Francisco, CA. 2008.


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Recent Engagements
SCORDATURA COLLECTIVE
Judith Weir, Unlocked
As we prepare our Unlocked Festival from 28th-30th August, we thought we would revisit a blogpost about the piece of music that gave it it's name. This has long been a favourite in our programmes, and you can hear 'Make Me a Garment' on 28th August, and there will be two chances to hear 'The Wind Blow East' on 29th August. If you would be interested in supporting the festival, you can also do so here or here.
This post originally appeared on The Hensel Project on May 29th 2016.
The position classical music has in our society does make it rather removed from current affairs. It's presented as this refuge from modern life, where you go to have a comfortable sleep, often in a concert. I don't think my music is removed from everyday life. Judith Weir (2005 Guardian interview)
In 2014, Judith Weir made history by becoming the first ever female Master of the Queen's Music. Born to a Scottish family in 1954, she had composition lessons with John Tavener while still at school before studying at King's College, Cambridge. Tom Service has described her compositions as 'addictive, scintillating music', and she has used a wide range of sources for her work from Scottish traditional music to Chinese opera and Taoist texts. 'Unlocked', written for Ulrich Heinen and first performed in Birmingham in 1999, is no exception. Weir describes it as a set of five cello 'fantasias' based on American folksongs collected by John and Alan Lomax. Many of these songs were collected from black prisoners in Southern jails, and 'Unlocked' explores ideas of captivity and the desire for freedom. 'Unlocked' is for solo cello, and uses a range of extended techniques. All song texts below are taken from John A. Lomax Our Singing Country.
1. Make Me a Garment
Mama, Mama, make me a garment And make it long, white and narrow. Mama, Mama, look on my pillow And you will find some money Get along boys, gather 'round me, Come pay my fine, come and get me. My true love died the other day I Believe I'll die tomorrow.
'Make Me a Garment' was collected from a prisoner named Roscoe McLean in 1936. John Lomax writes that McLean contributed many songs to the Folk Song Archive, but had been on the tuberculosis ward of the Florida Penitentiary. Lomax writes that in his most recent talk with Roscoe "he could only whisper as he peered through the woven wire netting, sobbing his despair. He will never sing again." This song is an example of a 'holler', a distinct type of African American folk singing. The singer vocalises with an open throat, improvising variations on a simple melody.
Weir uses the ornaments and freedom of melody of the holler, and follows Lomax's notation of the rhythms sung getting gradually more complex as the piece progresses. The song melody is disturbed by interjections of a menacing low E which is also used to frame the movement.
2. No Justice
Oh, we don't get no justice here in Atlanta, Oh, we don't get no justice here in Atlanta, For if you say the law ain't right, In the jail you'll spend the night, We don't get no justice here in Atlanta. Oh, if you say the judge ain't right, In the jail you'll spend the night, We don't get no justice here in Atlanta. Oh, we don't get no justice here in Atlanta, If you say the judge ain't right, In the gang you'll stay all night, You don't get no justice here in Atlanta.
John Lomax records the singer of this song only as a 'Negro man' in the Milledgeville State Penitentiary in Georgia. It is one of a number of protest songs about the treatment of black prisoners, and Lomax describes the man complaining of the cold and the weight of his chains.
Weir employs a number of unusual techniques to convey this angry, forceful melody. At the very opening the cellist is required to play using only 'finger resonance' i.e. bringing the left hand fingers down on the string hard enough to hear the pitch of the notes. She later asks the cellist to slam the left hand flat on the fingerboard to create a percussive effect, to press the bow right down into the string to create a 'scratch tone', to play legato with the wood of the bow, to stamp and to use the body of the cello as a drum.
3. The Wind Blow East
Oh the wind blow east, the wind blow west, The wind blow the Sunshine right down in town. Oh the wind blow east, the wind blow west, The wind blow the China right down in town. Oh the wind blow east, the wind blow west, The wind blow the Setting Star right down in town.
This is a song from the Bahamas, describing the effects of a hurricane. Three ships, the Sunshine, China and Setting Sun have been blown 'right down in town.' This is the only song of the piece with no direct link to prison or prisoners, with the Lomaxes collecting it in 1935 from a 'group of men and women with a drum'. Weir writes that the movement represents the prisoner's dream of a better life. She asks for the music to be 'dreamy and faint', and for a mute to be used throughout. Rather than a hurricane, the wind in this still and beautiful movement seems to be gentle and warm.
4. The Keys to the Prison
Mama, they're gonna give me the keys, To this jailhouse, yes, the keys to this old jailhouse. What do you mean, give you the keys to this old jail, When the turnkeys have them hung around their necks Yes, right around their necks. Mama, I mean they're coming to get me about nine this evening, Yes, and they're going to hang me at about ten tonight, Yes, about ten tonight.
Les Clefs de la Prison is an original composition by a fifteen year old Cajun girl names Elide Hofpauir. Lomax describes it as "a swift and acid dialogue between a condemned man and his father and mother [which] stands alone of its kind among American folksongs." The young man sings of how terrible it is to know in advance that he is going to die, and that his mother should be the one to retrieve his body.
Once again, Weir adapts the song to alter its meaning. She takes the initial excitement of the opening lines, and writes that the piece is "the prisoner's fantasy that the prison doors are suddenly wide open and the guards have all gone." The piece is fast and agile, and ends with a simple version of the melody in artificial harmonics.
5. Trouble, Trouble
Trouble, trouble, I had them all my day, Trouble, trouble, trouble, had them all my day. Well, it seem like trouble go'n' let me to my grave. Well, I'm gwine back South, Mamma, where de weather suit my clothes, Well, I'm go'n back South, babe, where de weather suit my clothes, Well, I'm gonna lay out on dat green grass, an' look up at de sky. Well, so many a day, Mamma, laid in my cell an' moan, Well, so many a day, Lawd, laid in my cell an' moan, Well, I'm thinkin' about my baby, Lawd, an' yo' happy home. Well, Mamma, Mamma, here an' listen to my second mind, Hey, hey, Mamma, listen to my second mind, Well, I don't b'lieve I'd 'a' been here, wringin' my hand an' cryin'.
This movement is a transcription of a blues song sung by a man names James Hale from Alabama. Weir treats the song simply, alternating the melody with tremolo played over the fingerboard.
Supporting Unlocked Festival
We are excited to be performing again, and will be bringing you all performances for free. However, there are costs involved in rehearsing and recording safely and as self-employed musicians many of our players have seen a significant drop of income in recent months. We would be incredibly grateful if you could give the price of a coffee to help bring this festival to life! https://ko-fi.com/scordaturacollective
Composer Focus: Antonia Bembo
Composer Focus: Barbara Strozzi
Composer Focus: Dorothy Rudd Moore


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HomeNews & Social MediaHaikou NewsGoldfish with tattoos of various symbols on display in Haikou
Goldfish with tattoos of various symbols on display in Haikou
Inked up: Goldfish with tattoos of various symbols are displayed at a tropical trade fair in Hainan, China
As a symbol of wealth and harmony, the goldfish is a popular choice of tattoo.
But in a craze sweeping China, goldfish are themselves being inked with patterns and characters intended to being their owners good fortune.
These ones were being displayed during a tropical products trade fair in Haikou in the Hainan Province of China on Wednesday.
The tattoos are said to be the result of an injection with special medicine which takes half a year to form. Other methods use lasers.
Tattooed fish first appeared on the market in 2005 and have become very popular in the last year or two.
The technique is not without its critics, however, who claim it is not only cruel, but against the laws of nature.
Earlier this year, the public reacted angrily when a magic trick with goldfish watched by hundreds of millions of Chinese on TV during the Lunar New Year celebrations was believed to involve abuse including implanted magnets.
Animal rights groups called for the stunt, in which six fish were seen swimming in perfect synchronisation under magician Fu Yandong's direction, to be banned.
Previous news: North Wales Katie shuns water even tears over rare allergy
Next news: Kungfu fighter Shangguan Pengfei dies after match in Haikou, Hainan


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They're back… Bird scooters to hit the streets again, at least temporarily
Bird scooters are making a comeback and will return to the city Thursday, Aug. 9. Several Bird scooter "nests"were spotted in Downtown Louisville July 19 but were removed by July 23. | Photo by Rebekah Alvey
Louisville Metro Government and scooter company Bird have reached a 30-day agreement, allowing access to no more than 100 scooters starting Thursday, Aug. 9.
On the day of the release, Bird will host a riding demonstration and helmet giveaway for the city at Fourth Street Live from 11 a.m. to 1 p.m.
According to the agreement, the Scooters are only accessible in an area defined by Louisville Metro. The area is bound by the Ohio River to the north and west and the Watterson Expressway to the east and south. Any Bird Scooter found outside this area will be considered a violation of the agreement, and Bird will pay $100 per Scooter found outside of the zone, the contract states.
Designated zone of operation, defined in the agreement. | Courtesy of LMG
When Bird scooters showed up in other cities, concerns were raised about parking issues and the potential obstruction of public walkways and sidewalks. To help alleviate those possible problems, the agreement states that Bird and Louisville Metro will continue to coordinate throughout the 30 days to identify designated parking areas for the scooters.
Bird will propose a weekly list of the daily locations of possible "nests," which is where scooters are initially stored and recharged. The city will review and approve the list.
The agreement states the scooters must be deployed before 6:00 a.m. daily and retrieved and placed in Nests by "sunset" each day. Failure to do so would be a breach of the agreement.
The agreement also outlines specific parking regulations for the scooters. If there is a breach in the agreement, Bird could face consequences, including not being able to request an operating permit from the city for up to two years. The punishment will depend on the severity of the breach, according to the agreement.
Louisville Metro holds the right to renew or terminate the agreement.
Bird scooters first made a Louisville debut July 19 but were quickly removed at the request of the city on July 23. Birds are standing, motorized scooters that can be found and rented through a mobile app.
Bird Agreement by Insider Louisville on Scribd
By Rebekah Alvey|2019-12-30T15:34:30-05:00August 8th, 2018|


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Oilers History
Oilers History: The Reijo Ruotsalainen Experiment
By Rob Soria July 28th, 2018
There has never been a player in the Edmonton Oilers storied history quite like Reijo Ruotsalainen. The talented blueliner suited up in just 26 regular season games for the organization but still managed to leave a mark on the 1987 and 1990 Stanley Cup winning clubs. Not too shabby for a player who was here for a brief cup of coffee and one that Glen Sather ultimately stole from the New York Rangers.
Reijo Ruotsalainen (photo courtesy Finnish Hockey Hall of Fame)
After being selected with the 119th selection in 1980 NHL Entry Draft, the young Finn put together five impressive campaigns to start his career. Though he was predominately used on defence during his time with the Rangers, the club decided to give Ruotsalainen a "look-see" at forward during the 1984-85 season. Despite being thrust into a different role, he managed to post career-highs of 28 goals and 73 points playing upfront.
With that said, he would finish the year with a career-worst plus/minus rating of -27 and though the stat is no longer viewed with the importance it once was, that number looks just as ugly today as it did 30-odd years ago. After playing a starring role under former bench boss Herb Brooks on Broadway, the native of Oulu, Finland fell out of favour with new head coach Ted Sator and made his back to Europe following the 1985-86 campaign.
An odd decision considering during his five-year stay in the Big Apple, Ruotsalainen never registered less than 51 points in a season and was a plus player every year except the aforementioned 1984-85 campaign. Losing him was a major blow to the Rangers offence and one that would go on to hamper the club for years to come. He possessed a skill-set only a handful of players in the league had and was among the very best skaters in the game.
Ruotsalainen A Rare Breed
Outside of Paul Coffey, no one came close to matching his smooth and seemingly effortless stride and like the Oilers No. 7, Ruotsalainen routinely left fans in awe with his breathtaking end to end rushes. Whereas Coffey would typically take the puck right up the middle of the ice, the man teammates affectionately referred to as "Rexi" would glide up ice and routinely fly by opposing players along the side boards as if they were stuck in the mud.
With Ruotsalainen wasting away over in Europe and seemingly nothing more than an afterthought to the Rangers organization, Sather recognized an opportunity to swoop in and grab a high-end talent for pennies on the dollar. To his credit, he would go on do exactly that and the Oilers already loaded lineup became that much better.
Glen Sather (Perry Nelson-USA TODAY Sports)
In acquiring Ruotsalainen's NHL rights, along with Clark Donatelli, Vile Kentala, and Jim Weimer, Edmonton agreed to part ways with veteran defenceman/tough guy Don Jackson, Mike Golden, Miloslav Horava and future considerations, which turned out to be Stu Kulak. In other words, it was highway robbery at its finest and viewed by many to be one of the very best trades Sather constructed during his time as general manager.
A Glen Sather Special
Jackson was a fan favorite and loved by teammates for his willingness to do whatever was required of him during his four-plus years with the organization. However, he would play just 22 games for the Rangers in 1986-87 before calling it quits and of the other three players Edmonton gave up in the deal, only Horava went on to play any significant time in the NHL and he scored all of five goals in 80 career games.
After completing his season in the Swiss League with SC Bern, Ruotsalainen joined the Oilers and played in 16 games during the 1986-87 season and was his usual prolific self. The 27-year old scored five times and put up thirteen points to close out the regular season and chipped in another seven points during the playoffs, dressing for all 21 of Edmonton's games en route to earning Lord Stanley's mug for the third time in four years.
On this date in 1987, #Oilers Reijo Ruotsalainen scored his 100th career #NHL goal.
— Jen (@NHLhistorygirl) March 6, 2017
After winning his first cup, Ruotsalainen decided to return to Europe and played the next two years over in Sweden and Switzerland. Prior to the 1989-90 season, the itch to return to North America had come back to surface but his NHL rights belonged to the New Jersey Devils after he was claimed from the Oilers during the 1987 Waiver Draft. His stay with the Devils lasted just 31 games before being shipped back to the Alberta capital.
Ruotsalainen Returns
Sather had to give up Jeff Sharples in order to get the trade done, who ended up never playing another game in the NHL following the 1988-89 season. After trading away Coffey during the 1987-88 season in a blockbuster deal for a Craig Simpson, the Oilers ability to transition the puck up ice was never quite the same. While they still won another championship in 1988, with Wayne Gretzky no longer part of the equation, Edmonton was in dire need of a presence on the point on the man advantage and Sather looked to an old friend for an answer.
Though he would be used sparingly after coming over from New Jersey, dressing for only ten games, he still picked up eight points in limited duty. Having just turned 30-years of age a day after the team's season finale, the hope was the veteran defender could kick it up another notch for one final playoff run, just as he did during his previous tenure in Oilers silks.
Ruotsalainen would once again go on to play in all of Edmonton's playoff games, finished up with 13 points in 22 games and tied Jari Kurri for the second-best plus/minus on the club at +13 during the post-season. Unlike the team that won in 1987, no one expected the Oilers to win it all in 1990, which made it that much sweeter for everyone involved and the Ruotsalaninen angle was icing on the proverbial cake.
Glen Sather managed to pull a cat out of the hat for a second time, in what was another brilliant move and it paid off in spades. It was a win-win for both the organization and player and one that will forever link the Edmonton Oilers to one of the most talented and underappreciated European players of his era in Reijo Ruotsalainen.
Reijo Ruotsalainen


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New Pacifica High gymnasium makes modern an old tradition
By OC Tribune Staff on May 4, 2021 • ( Leave a comment )
MIKE BUENAFE, boys' freshman basketball coach at Pacifica High School in Garden Grove, was selected to take the first shot in the new gymnasium. PHS is part of the Garden Grove Unified School District (Orange County Tribune photos by Christopher Abdelmalek),
Every journey starts with a single step. For the new gymnasium at Pacifica High School, it begins with a single basket.
Repeating a ceremony first held in 1965, the new Mariner gym was initiated on Saturday by having freshman basketball coach Mike Buenafe take the first hoops shot in the facility.
That's an echo of his father, Jeff Buenafe, christening the original PHS gym in 1965.
"We got the call to come down to the gym and I was told I was going to take the first shot," recalls Jeff, who was 11 at the time. His father, Al Buenafe, was the first principal at PHS. "Luckily, I made it."
Keeping the tradition alive, Mike was tapped to repeat the act at the new facility. "I don't like the attention to be on me," said Mike, "but this was something special. It's great for the players, great for the school, and great for the community."
The new gymnasium is a state-of-the art building that is an 18,250-square foot multi-function facility that houses three full-size basketball courts and can convert to nine volleyball courts. The gym allows for seating for up to 1,584 people and includes a lobby, locker rooms, coach's offices, equipment storage and more.
Divider curtains will allow for multiple events to be held at once.
But that means saying goodbye to the old place, which will be demolished this summer. "It's a bittersweet moment," said Mike, who played basketball for the Mariners and was graduated in 2004. "The [old] gym was full of great memories, but the roof was also falling in, and the floorboards were all messed up. This one is much nicer and allows for so much more activity for the students."
The "old" gym does have quite a history. In 1994, the boys' varsity basketball team, coached by Bob Becker, won a CIF Southern Section championship and advanced all the way to the state title game before losing.
More recently, guided by coach David Friedman, the M's have made the CIF-SS playoffs in 18 of 20 seasons, including 15 in a row at one point. Under Friedman, Mariner teams have won four league crowns and advanced to CIF-SS quarterfinals twice.
This year's team is off to a 6-5 start and won its inaugural game in the new palace, defeating Whitney High of Cerritos by the score of 43-31 on Saturday.
Oh, and Mike made his shot, too.
THE CURRENT MARINER boys' varsity basketball team with a balloon anchor.
Tagged as: Al Buenafe, Garden Grove, Garden Grove Unified School District, Jeff Buenafe, Mike Buenafe, Pacifica High School, Skyboxes
LA is "yellow," OC stays "orange"
Seniors view college-bound future


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Paid mods pulled from Steam Workshop, purchases to be refunded
By Brenna Hillier, Tuesday, 28 April 2015 00:41 GMT
Paid mods are no more.
"We thought this would result in better mods for everyone, both free and paid. But we underestimated the differences between our previously successful revenue sharing models, and the addition of paid mods to Skyrim's workshop."
Steam Workshop's controversial new paid mods feature has been axed in response to community feedback, Valve has announced.
"We're going to remove the payment feature from the Skyrim workshop. For anyone who spent money on a mod, we'll be refunding you the complete amount," Valve wrote in a Steam Community post.
"We talked to the team at Bethesda and they agree.
"We've done this because it's clear we didn't understand exactly what we were doing. We've been shipping many features over the years aimed at allowing community creators to receive a share of the rewards, and in the past, they've been received well. It's obvious now that this case is different."
Valve said its main goal with the feature was to allow mod makers to work on their creations full time if that was their wish, and to encourage better support for mod users.
"We thought this would result in better mods for everyone, both free and paid. We wanted more great mods becoming great products, like Dota, Counter-strike, DayZ, and Killing Floor, and we wanted that to happen organically for any mod maker who wanted to take a shot at it," Valve said.
"But we underestimated the differences between our previously successful revenue sharing models, and the addition of paid mods to Skyrim's workshop. We understand our own game's communities pretty well, but stepping into an established, years old modding community in Skyrim was probably not the right place to start iterating."
Valve said it missed the mark "pretty badly", but that there's a "useful feature" in there somewhere. It'll be looking at further feedback if you want to keep talking about it.
Bethesda also issued a short statement on the matter.
"After discussion with Valve, and listening to our community, paid mods are being removed from Steam Workshop. Even though we had the best intentions, the feedback has been clear – this is not a feature you want," the developer said.
"Your support means everything to us, and we hear you."
This is probably pretty disappointing to the mdoders involved; Bethesda said some Skyrim modders were earning more than its own staff.