text:
\section{Introduction}
Quantile regression, formally introduced by \citet{koenker1978regression}, has become a powerful tool for data analysis in a wide range of applications, from economics \citep{fitzenberger2013economic} to public health \citep{wei2019applications}. Quantile regression enables researchers to go beyond the modeling of conditional means; by modeling the effect of covariates at different conditional quantile levels of a response variable, we obtain more comprehensive information on the relationships between the response and the covariates. In particular, quantile regression enables us to reveal the differential effects of a covariate on the low and high end of the response distribution.

Because the sampling distributions of the quantile regression estimators involve the conditional density functions as nonparametric nuisance parameters, inferential methods have to approximate those quantities directly or indirectly. Existing methods include the use of plugged-in density estimates \citep{powell1991estimation,hendricks1992hierarchical}, rank-score tests \citep{gutenbrunner1993tests,koenker1999goodness}, re-sampling methods \citep{feng2011wild,pan2020multiplier}, and Bayesian computational approaches \citep{chernozhukov2003mcmc, yang2016posterior}.

The present paper follows the Bayesian computational framework to deliver frequentist inference for quantile regression. We investigate the asymptotic properties of posterior inference in a possibly sparse model and demonstrates desirable efficiency and stability of the approach.
We show that the pseudo-Bayesian approach based on a working likelihood and a shrinkage prior achieves automatic adaptation to sparsity, and it can provide asymptotically valid inference for quantile regression under heterogeneity. In the rest of our paper, we shall refer to our proposed pseudo-Bayesian approach as a ``Bayesian approach'', even though we pursue inference in the frequentist sense.

More specifically, we consider the asymmetric Laplace working likelihood  \citep{yu2001bayesian, yang2016posterior} with appropriate shrinkage priors in the spirit of common frequentist penalty functions \citep{wu2009variable}. With a random sample of size $n$ from a linear quantile regression model with $p \leq n$ covariates but only $s \le p$ active (non-zero) coefficients, our work offers the following insights into the posterior inference.

\vspace{3mm}

 1. The posterior distribution concentrates around the true quantile regression parameters at an adaptive rate: it achieves the $n^{-1/2}$ rate for the active coefficients and a super-efficient rate of $o(n^{-1/2})$ for the inactive (zero-valued) coefficients.

 2. The posterior mean for the active coefficients is asymptotically normal and oracle efficient: it achieves the same asymptotic variance as the quantile regression estimator as if we knew which coefficients are active/inactive.

 3. With an appropriate adjustment of the posterior variance, we can construct automatically adaptive confidence intervals in the frequentist sense: they are asymptotically oracle for the active coefficients, while they are super-efficient for the inactive coefficients with coverage probabilities tending to one.

 4. Even if we identify the active covariates correctly, optimally weighted quantile regression estimators cannot be obtained by focusing on only those active covariates. The Bayesian approach does not rely on any binary selection of active/inactive covariates; thus, it tends to offer performance advantages over variable selection approaches.

\vspace{3mm}

It is important to note that unadjusted Bayesian inference is not automatically valid since the posterior is constructed operationally from a mis-specified asymmetric Laplace working likelihood.
Even for finite-dimensional models without the use of shrinkage priors, the posterior distribution does not approximate the sampling distribution of the classical quantile regression estimator \citep{sriram2015sandwich,yang2016posterior}.
However, from the frequentist perspective, we find a relatively simple adjustment to the posterior variance that facilitates asymptotically valid and efficient inference in possibly sparse quantile regression models.
The Bayesian computational framework allows us to circumvent the nonparametric estimation of the conditional density functions as nuisance parameters \citep{chernozhukov2003mcmc}; therefore, it serves as a valuable tool to frequentist inference.

Bayesian modeling with shrinkage priors has been quite well studied in terms of estimation accuracy (error rates) of the parameters and variable selection in high-dimensional problems; see, e.g., \citet{song2017nearly,jiang2019bayesian} and \citet{gao2020general}.
The focus of the present paper is not estimation or variable selection consistency but the understanding of what can be accomplished in inference for possibly sparse quantile regression models, about which relatively little has been available in the literature even when the number of predictors $p$ is fixed.
The main challenge is adjusting for the mis-specification of the likelihood function under heterogeneity and model sparsity.
Our work also gives the first asymptotic analysis, as far as we know, for the posterior mean and variance in the Bayesian quantile regression framework with a shrinkage prior.
To simplify the technicalities and focus on the main points, we begin by working with the asymptotic framework where the sample size $n$ goes to infinity yet the covariate-dimension $p$ is kept fixed; We discuss an extension to the regime where $p$ grows with $n$ at a controlled rate later in the paper.

The key to understanding Bayesian approaches is the combination of likelihood and prior. Since the quantile regression model does not assume any parametric likelihood function, it is common to rely on a working likelihood to pursue posterior inference. Examples of other working likelihoods include the empirical likelihood \citep{yang2012bayesian,xi2016bayesian}, the score likelihood \citep{wu2021bayesian}, or the approximate likelihood \citep{feng2015bayesian}.
The use of different shrinkage priors is also prevalent in practice for more efficient estimation \citep{li2010bayesian,adlouni2018regularized,kohns2020horseshoe}. This paper adopts the asymmetric Laplace working likelihood and focuses on two easy-to-understand examples of shrinkage priors for their interpretability and computational attractiveness. Under this Bayesian framework, we provide new theoretical insight on the adaptation of posterior inference in possibly sparse quantile regression models. In particular, we propose new adaptive adjustment to the posterior variance that generalizes \citet{yang2016posterior} and provide thorough theoretical investigation of its properties.

The rest of the paper is organized as follows. In Section \ref{sec::setup}, we discuss the quantile regression problem and our Bayesian framework.
Then we give the corrected posterior inference approach in Section \ref{sec::main}, supported by the asymptotic properties of the posterior distribution.
In Section \ref{sec::increase-p}, we discuss the theoretical extension towards the asymptotic regime with moderately increasing covariate-dimension.
Section \ref{sec::simulation} shows some simulation results to demonstrate the effectiveness and stability of the proposed approach. We make some concluding remarks in Section \ref{sec::conclude}.

\section{Problem setup}
\label{sec::setup}
\subsection{The quantile regression model}

Let $Q_\tau(Y\mid X =\bm{x})$ be the $\tau$th conditional quantile of the response variable $Y$ given covariates $X = \bm{x}$, where $\bm{x}= (x_0,\ldots,x_p)^{{ \mathrm{\scriptscriptstyle T} }}$ includes an intercept term $x_0 = 1$ and $p$ covariates, and  $\tau\in(0,1)$ is a pre-specified quantile level of interest. We consider the linear quantile regression model
\begin{equation}
\label{eq::truemodel}
Q_\tau(Y\mid X = \bm{x}) = \bm{x}^{{ \mathrm{\scriptscriptstyle T} }}\bm{\beta}^0(\tau),
\end{equation}
where $\bm{\beta}^0(\tau) = (\beta_0^0(\tau),\ldots,\beta_p^0(\tau))^{{ \mathrm{\scriptscriptstyle T} }}$ is the true quantile regression coefficient vector.
The conditional median of $\tau =0.5$ is a special case, and high or low quantile levels of $\tau$ are often of interest in applications, ranging from financial risk quantification \citep{taylor2019forecasting} to public health assessment \citep{wei2019applications}. Since we focus on a fixed $\tau$ in the model, we often suppress the index $\tau$ in $\bm{\beta}^0(\tau)$ hereafter.

In this paper, we consider Model (\ref{eq::truemodel}) to be possibly sparse.
Let $\mathcal{S} = \{0\}\cup \{ j \in \{1,\ldots p\}:\; \beta_j^0 \neq 0 \}$ be the index set of the active (non-zero) coefficients, including the intercept term; whereas $\mathcal{S}^c = \{0,\ldots,p\}\backslash \mathcal{S}$ is for the inactive coefficients. Let $s = |\mathcal{S}| - 1$ be the number of active covariates. A possibly sparse model refers to $0 \leq s \leq p$ for some integer $s$; yet we do not know $\mathcal{S}$ in advance.
For now, we suppose the covariate-dimension $p$ is fixed, and discuss an extension towards the case when $p$ can increase in Section \ref{sec::increase-p}.

Here we briefly review the classical quantile regression analysis.
Let $\mathbb{D}_n = \{(\bm{x}_i,y_i):i=1,\ldots,n\}$ be a random sample of size $n$ that satisfies Model (\ref{eq::truemodel}).
The quantile regression estimator \citep{koenker1978regression} is
\begin{equation}
\label{eq::quantobj}
\widehat{\bm{\beta}} = \argmin_{\bm{u}\in\mathbb{R}^{(p+1)}}\;\sum_{i=1}^n \rho_\tau(y_i - \bm{x}_i^{{ \mathrm{\scriptscriptstyle T} }}\bm{u}),
\end{equation}
where $\rho_\tau(v) = v\{\tau - 1(v<0)\}$ and $1(\cdot)$ is the indicator function.
With $p \ll n$, statistical inference for Model (\ref{eq::truemodel}) can be carried out based on the asymptotic properties of the estimator $\widehat{\bm{\beta}}$; we refer to \citet{koenker_2005} and \citet{koenker2017handbook} for more details on quantile regression. Here we highlight two aspects for the estimator $\widehat{\bm{\beta}}$: (i) it does not account for the possible model sparsity, therefore it does not achieve the optimal efficiency when Model (\ref{eq::truemodel}) is sparse; (ii) its asymptotic variance-covariance matrix involves the conditional density function of $Y$ given $X$, which requires non-parametric estimation and can be unstable in practice.

\subsection{A Bayesian framework}
\label{sec::bayes-model}
In this Section, we give the Bayesian framework for modeling the quantile regression coefficient $\bm{\beta}$ in Model (\ref{eq::truemodel}).
We adopt the asymmetric Laplace working likelihood:
\begin{equation}
\label{eq::ALlikelihood}
\mathcal{L}(\mathbb{D}_n \mathrel{\mid} \bm{\beta}) \propto
\exp\left\{ -\sum_{i=1}^n\rho_\tau(y_i - \bm{x}_i^{{ \mathrm{\scriptscriptstyle T} }}\bm{\beta}) \right\},
\end{equation}
where $\propto$ means equality up to a multiplicative factor that does not depend on $\bm{\beta}$.
We call $\mathcal{L}(\mathbb{D}_n | \bm{\beta})$ a working likelihood because it does not correspond to the true data-generating mechanism of $\mathbb{D}_n$ under parameter value $\bm{\beta}$; in fact, there is no `true' likelihood function as Model (\ref{eq::truemodel}) itself does not fully specify a conditional distribution of $Y$ given $X$.
Choosing a working likelihood in the form of (\ref{eq::ALlikelihood}) enjoys two benefits: (i) it allows the maximum working likelihood estimator to coincide with the classical quantile regression estimator $\widehat{\bm{\beta}}$ in (\ref{eq::quantobj}); (ii)
its Fisher information matrix shares a critical component with the variance-covariance matrix of $\widehat{\bm{\beta}}$ \citep{yang2016posterior}.

To incorporate the possible model sparsity, in this paper we consider two examples of shrinkage priors in the spirit of common penalty functions:
\begin{eqnarray}
\pi_{AL}(\bm{\beta}) &\propto& \exp\left\{ - n^{1/2}\lambda_n\sum_{j=1}^p w_j\lvert \beta_j \rvert \right\}, \label{eq::priorAlasso}\\
\pi_{CA}(\bm{\beta}) &\propto& \exp\left\{ - n\sum_{j=1}^p  p_{\lambda_n}(\beta_j) \right\},\label{eq::priorSCAD}
\end{eqnarray}
where $w_j$ and the function $p_{\lambda_n}(\cdot)$ will be given below; the tuning parameter $\lambda_n$ depends on the sample size, but the subscript $n$ is often suppressed in the paper when there is no confusion.
The prior (\ref{eq::priorAlasso}) corresponds to the Adaptive Lasso (AL) penalty \citep{zou2006adaptive}, where $w_j = 1/\lvert \hat{\beta}_j\rvert$ for $j\in\{1,\ldots,p\}$ as in \citet{wu2009variable} and $\hat{\beta}_j$ is the $j$-th component of $\widehat{\bm{\beta}}$ defined in (\ref{eq::quantobj}).
The Clipped Abosolute (CA) prior (\ref{eq::priorSCAD}) is motivated from the Smoothly Clipped Absolute Deviation (SCAD) penalty of \citet{fan2001variable}, where we define the penalty function $p_{\lambda}(u) = \lambda(|u|\wedge \lambda)$. Our choice of $p_{\lambda}(u)$ in (\ref{eq::priorSCAD}) is in the same spirit of the SCAD penalty, but we remove the smoothing component to simplify the theoretical derivation; See Figure \ref{fig::SCADprior} for a visual comparison.
Note for either (\ref{eq::priorAlasso}) or (\ref{eq::priorSCAD}), we impose an improper flat prior for the intercept $\beta_0$, i.e, $\pi(\beta_0) \propto 1$; therefore $\beta_0$ is not penalized.

\begin{figure}[hbt]
\centering
\includegraphics[width = .85\textwidth]{SCADcompare86.eps
\caption{Comparison between the prior $\pi_{CA}(u)$ and the prior induced by the SCAD penalty in \citet{fan2001variable}; $a$ is a tuning parameter in the SCAD penalty and we set $a = 2$ in the plot. Both priors are flat when $|u| > a\lambda$.
}
\label{fig::SCADprior}
\end{figure}

Given the working likelihood (\ref{eq::ALlikelihood}) and any prior $\pi(\bm{\beta})$, we have the formal posterior density:
\begin{equation}
\label{eq::posterior_1}
p\left(\bm{\beta} \mathrel{\mid} \mathbb{D}_n\right) \propto \mathcal{L}(\mathbb{D}_n \mathrel{\mid} \bm{\beta})\times \pi(\bm{\beta}).
\end{equation}
Under either the AL (\ref{eq::priorAlasso}) or CA (\ref{eq::priorSCAD}) prior, existing Markov Chain Monte Carlo (MCMC) algorithms enable efficient sampling from the posterior; See \citet{li2010bayesian} and \citet{alhamzawi2012bayesian} for the prior (\ref{eq::priorAlasso}); \citet{li2011bayesianSCAD} and \citet{adlouni2018regularized} for priors similar to (\ref{eq::priorSCAD}). In the rest of the paper, we shall examine the asymptotic properties of the posterior distribution, from which we derive valid and adaptive confidence intervals in the frequentist sense.

\begin{remark}
Priors (\ref{eq::priorAlasso}) and (\ref{eq::priorSCAD}) are both examples from a wide range of shrinkage priors, which aims to shrink smaller coefficients towards zero while keep those larger coefficients unbiased; See e.g., \citet{song2017nearly} and \citet{jiang2019bayesian} for more discussions in the linear regression context.
In this paper we shall only focus only on two examples to simplify theory and computation for quantile regression, through which we hope to shed some light on the use of shrinakge priors for efficient statistical inference in quantile regression.
\end{remark}

\section{Adaptive posterior inference}

\label{sec::main}

While posterior inference seems straightforward with a Bayesian framework, its validity is not warranted since the working likelihood is mis-specified \citep{yang2016posterior}.
In this section, we begin by investigating the asymptotic properties of the posterior distribution from the frequentist perspective.
Next, we propose an adjustment to the posterior variance-covariance matrix and show that it can lead to valid confidence intervals that also adapts to model sparsity.
In the last subsection, we discuss an extension of using a weighted working likelihood to obtain the optimal efficiency.
Throughout this section, the covariate-dimension $p$ is fixed in Model (\ref{eq::truemodel}).

\subsection{Notation}
\label{sec::notations}
Recall that we have $\bm{\beta}^0 = (\beta_0^0,\ldots,\beta_p^0)^{ \mathrm{\scriptscriptstyle T} }$ as the true regression coefficient in Model (\ref{eq::truemodel}), and that $\mathcal{S}$ is the index set of the active (non-zero) coefficients, including the intercept term.
Without loss of generality, we assume $\mathcal{S} = \{0, 1,\ldots,s\}$.
Recall $\widehat{\bm{\beta}}$ is the classical quantile regression estimator in (\ref{eq::quantobj}); let $\tilde{\bm{\beta}}_{\mathcal{S}}\in\mathbb{R}^{s+1}$ be the oracle quantile regression estimator, which solves (\ref{eq::quantobj}) using only the active covariates.
For any vector $\bm{v} = (v_0,\ldots,v_p)^{ \mathrm{\scriptscriptstyle T} }$, let $\bm{v}_\mathcal{S}= \{v_j:\;j\in\mathcal{S}\}$ and $\bm{v}_{\mathcal{S}^c} = \{v_j:\;j\not\in\mathcal{S}\}$. For any matrix $A \in\mathbb{R}^{(p+1)\times (p+1)}$, we partition
\[
A =
\left(
\begin{array}{cc}
A_{\mathcal{S}}&A_{\mathcal{S},\mathcal{S}^c}\\
A_{\mathcal{S}^c,\mathcal{S}}&A_{\mathcal{S}^c}
\end{array}
\right),
\]
where $A_{\mathcal{S}} \in \mathbb{R}^{(s+1)\times (s+1)}$; for $i,j\in\{0,\ldots,p\}$, we shall write $A(i,j)$ as the $(i+1,j+1)$th entry of A.

Recall that $\mathbb{D}_n$ contains a random sample of size $n$ from the distribution $(X,Y)\sim\mathrm{pr}^*$ whose $\tau$th conditional quantile of $Y$ satisfies Model (\ref{eq::truemodel}). We will also use $\mathrm{E}^*(\cdot)$ as the expectation operator under $\mathrm{pr}^*$. Let $\epsilon = Y - X^{{ \mathrm{\scriptscriptstyle T} }}\bm{\beta}^0$, and  $f_{\epsilon\mid X}$ (or $f_{\epsilon\mid X_{\mathcal{S}}}$) be the conditional density function of $\epsilon$ given $X$ (or $X_{\mathcal{S}}$).
Furthermore, let $D = \mathrm{E}^*(XX^{{ \mathrm{\scriptscriptstyle T} }})$ and $G = \mathrm{E}^*\{XX^{{ \mathrm{\scriptscriptstyle T} }}f_{\epsilon\mid X_{\mathcal{S}}}(0)\}$.
Given the data $\mathbb{D}_n$ and the prior $\pi$, we consider the posterior probability measure as
\[
 \Pi\left(A\mathrel{\mid} \mathbb{D}_n\right)  = \int_A p(\bm{\beta} \mathrel{\mid} \mathbb{D}_n)\,\mathrm{d}\bm{\beta},
\]
for any measurable set $A\subset\mathbb{R}^{(p+1)}$, where  $p(\bm{\beta} \mid \mathbb{D}_n)$ is the posterior density in (\ref{eq::posterior_1}).

We also use the following set of notations in the paper.
For a vector $\bm{v}$, let $\lVert \bm{v} \rVert$ and $\lVert \bm{v}\rVert_\infty$  be its $\ell_2$ norm and its maximum norm, respectively.
For a matrix $A$, we denote its maximal/minimal eigenvalue by $\theta_{\max}(A)$ and $\theta_{\min}(A)$, respectively.
For probability density functions $h(x)$ and $g(x)$, we denote their total variation distance by $\lVert h-g \rVert_{TV}=\int |h-g|\mathrm{d}x$. For covariance matrices $A$ and $B$, we write $A\preceq B$ if $B-A$ is positive semi-definite.
For two deterministic sequences $a_n$ and $b_n$, we write $a_n \ll b_n$ if $a_n=o(b_n)$ and $a_n \lesssim b_n$ if there exists a universal constant $C_1 > 0$ such that $a_n \leq C_1b_n$.
For stochastic sequences $A_n$ and $B_n$, we use the notations $A_n \ll_{\mathrm{pr}^*} B_n$ and $A_n \lesssim_{\mathrm{pr}^*}B_n$ to denote $A_n = o_{\mathrm{pr}^*}(B_n)$ and $A_n  = O_{\mathrm{pr}^*}(B_n)$, respectively; we define $A_n \asymp_{\mathrm{pr}^*} B_n$ if both $A_n = O_{\mathrm{pr}^*}(B_n)$ and $B_n = O_{\mathrm{pr}^*}(A_n)$ hold.

\subsection{Posterior asymptotics}
\label{subsec::theory}
In this subsection, we present the large-sample properties of the posterior distribution defined in (\ref{eq::posterior_1}).  To this end, we need the following technical assumptions.

\begin{assumption}[Identification]
\label{aspt::id}
For any $\delta > 0 $, there exists $\varepsilon > 0$, such that
\[
\lim\limits_{n\to \infty}\;\mathrm{pr}^*\left[
\sup_{\bm{\beta}:\lVert \bm{\beta} - \bm{\beta}^0\rVert \geq \delta} \left\{\frac{L_n(\bm{\beta}^0) - L_n(\bm{\beta})}{n}\right\} \leq -\varepsilon
\right]
= 1,
\]
where $L_n(\bm{\beta}) = \sum_{i=1}^n \rho_\tau(y_i - \bm{x}_i{^{ \mathrm{\scriptscriptstyle T} }}\bm{\beta})$.
\end{assumption}

\begin{assumption}[Covariates]
\label{aspt::covar}
The covariate-vector $X$ has bounded support on $\mathcal{X}\subset \mathbb{R}^{p+1}$. Furthermore, the eigenvalues of $D = \mathrm{E}^*(XX{^{ \mathrm{\scriptscriptstyle T} }})$ are all bounded away from $0$ and $+\infty$.
\end{assumption}

\begin{assumption}[Conditional densities]
\label{aspt::density}
The conditional density function of $\epsilon = Y - X^{ \mathrm{\scriptscriptstyle T} }\bm{\beta}^0$ given $X=\bm{x}$ satisfies: (i)
there exists $L > 0$ such that for all $u,u'\in\mathbb{R}$,
\[
\sup_{\bm{x} \in\mathcal{X} }|f_{\epsilon\mid X = \bm{x}}(u) - f_{\epsilon\mid X = \bm{x}}(u')| \leq L|u - u'|;
\]
and (ii) there exist two constants $\underline{f} $ and $\overline{f}$, such that
\[
0< \underline{f} \leq \inf_{\bm{x} \in\mathcal{X}} \left\{f_{\epsilon\mid X=\bm{x}}(0) \right\} \leq
\sup_{\substack{u\in\mathbb{R}\\\bm{x}\in\mathcal{X}}} \left\{ f_{\epsilon\mid X=\bm{x}}(u)\right\} \leq \overline{f}.
\]

\end{assumption}

\begin{assumption}[Separation]
\label{aspt::sparsity}
For some constant $b_0 >0 $, we have
\begin{gather*}
\min_{j\in\mathcal{S}\setminus \{0\}}\;\lvert \beta_{j}^0\rvert  > b_0.
\end{gather*}
\end{assumption}

We briefly discuss the assumptions. Assumptions \ref{aspt::id}--\ref{aspt::density} are standard in Bayesian modeling with a working likelihood \citep{chernozhukov2003mcmc,yang2016posterior} and the quantile regression literature \citep{knight1998limiting,pan2020multiplier}; see also \citet[Section 4]{koenker_2005}.
In particular, the two assertions in Assumption \ref{aspt::density} hold for the conditional density $f_{\epsilon\mid X_{\mathcal{S}}}(u)$ as well; Furthermore, Assumption \ref{aspt::covar} implies the eigenvalues of $G = \mathrm{E}^*\{XX^{{ \mathrm{\scriptscriptstyle T} }}f_{\epsilon\mid X_{\mathcal{S}}}(0)\}$ are also bounded. Assumption \ref{aspt::sparsity}  holds automatically when we posit a fixed model as (\ref{eq::truemodel}) where $p$ is a constant;
Similar separation conditions are needed to achieve consistent model selection \citep{fan2001variable,wu2009variable,belloni2011}.

Now we present the main theoretical result regarding the posterior distribution defined in (\ref{eq::posterior_1}).

\begin{theorem}
\label{thm::Unified}
Consider the posterior distribution under either the AL prior (\ref{eq::priorAlasso}) or the CA prior (\ref{eq::priorSCAD}). Suppose Assumptions \ref{aspt::id}--\ref{aspt::sparsity} hold, and the tuning parameter $\lambda$ satisfies $n^{-1/2} \ll \lambda\ll 1$, then we have the following results.
\begin{enumerate}
\item Adaptive rate-of-contraction: for any sequence $M_n\to+\infty$,
\[
\Pi\left( \left\lVert\bm{\beta}_{\mathcal{S}} - \bm{\beta}^0_{\mathcal{S}}\right\rVert \leq \frac{M_n}{n^{1/2}} \,,\, \left\lVert\bm{\beta}_{\mathcal{S}^c} \right\rVert_{\infty} \leq \frac{M_n}{n\lambda} ~\middle\vert~ \mathbb{D}_n\right) \to 1,
\]
in $\mathrm{pr}^*$-probability.
\item Distributional approximation: for some density functions $\pi_j(u) = O_{\mathrm{pr}^*}(1)$ $(u\in\mathbb{R}, j\in\mathcal{S}^c)$,
\[
\left\lVert p\left(\bm{\beta} ~\middle\vert~  \mathbb{D}_n\right) -  \phi\left(\bm{\beta}_S;\tilde{\bm{\beta}}_S,\frac{1}{n}G_{\mathcal{S}}^{-1}\right)\times \prod_{j\not\in\mathcal{S}} \left\{ n\lambda  \pi_j(n\lambda \beta_j)\right\}\right\rVert_{TV} \to 0,
\]
in $\mathrm{pr}^*$-probability,
where $\phi(\cdot\,;\,\mu,\Sigma)$ is the density function of a multivariate-Gaussian distribution. In particular, $\pi_j(u) = (n^{-1/2}w_j/2)\exp\{-n^{-1/2}w_j|u|\}$ if we use the AL prior (\ref{eq::priorAlasso}), and  $\pi_j(u) = (1/2)\exp\{-|u|\}$  if we use the CA prior (\ref{eq::priorSCAD}).
\end{enumerate}
\end{theorem}

Theorem \ref{thm::Unified} shows that, despite the likelihood mis-specification, the posterior under either prior can separate the active and inactive coefficients with a wide range of choices of $\lambda$.
With $n\lambda \gg n^{1/2}$, part 1 of Theorem \ref{thm::Unified} shows the posterior for the inactive coefficients concentrates towards $0$ at a second-order rate, which is super-efficient.
Furthermore, part 2 of Theorem \ref{thm::Unified} shows the posterior for $\bm{\beta}_{\mathcal{S}}$ and $\bm{\beta}_{\mathcal{S}^c}$ are approximately independent. In particular, the posterior for $\bm{\beta}_{\mathcal{S}}$ is  `oracle', i.e., the Gaussian limiting posterior for $\bm{\beta}_{\mathcal{S}}$ is the same as if we knew the true model $X_\mathcal{S}$ in advance \citep{sriram2015sandwich} regardless of the prior we use.
Thus, with the two shrinkage priors in Section \ref{sec::bayes-model}, the posterior distribution can automatically adapt to the model sparsity.

Although slightly different in the limit, the posterior shares the same adaptation principle under both the AL and CA priors in Section \ref{sec::bayes-model}.
For an active coefficient, asymptotically the prior casts no effect on the posterior distribution; For an inactive coefficient $\beta_j (j\in\mathcal{S}^c)$, the shrinkage prior dominates over the working likelihood since the limiting posterior density $n\lambda\times\pi_j(n\lambda\beta_j)$ is proportional to the corresponding prior for $|n\lambda \beta_j| = O(1)$. Therefore, the shrinkage prior can separate the inactive coefficient from those active ones. This phenomena is in line with that in the Gaussian linear model setting under general shrinkage priors \citep[Theorem 2.4]{song2017nearly}.

\subsection{Confidence intervals from posterior moments}
\label{subsec::inference}
Since the working likelihood (\ref{eq::ALlikelihood}) is likely mis-specified, the posterior needs to be properly calibrated to deliver valid frequentist inference for quantile regression. However, the correction on the posterior variance proposed in \citet{yang2016posterior} is no longer valid with the use of shrinkage priors. In light of Theorem \ref{thm::Unified}, we give a modified adjustment that yields confidence intervals based on posterior moments that are automatically adaptive to model sparsity.

We construct the confidence intervals for $\bm{\beta}^0$ based on the posterior mean
$\check{\bm{\beta}} = (\check{\beta}_0,\ldots,\check{\beta}_p), $
and posterior variance-covariance matrix $\check{\Sigma}$  obtained from any posterior sampling algorithm.  We start from the adjustment used in \citet{yang2016posterior} by letting
$\widehat{D} = \sum_{i=1}^n\bm{x}_i\bm{x}_i^{{ \mathrm{\scriptscriptstyle T} }}/n$,
 and   $\check{\Sigma}_{adj} = {n\tau(1-\tau)}\check{\Sigma}\,\widehat{D}\,\check{\Sigma}$.
Then our proposed level $1-\alpha$  confidence interval for each $\beta_j^0$ takes the form
\begin{equation}
\label{eq::CI}
\check{\beta}_j \pm z_{\alpha/2}\eta_j \left\{\check{\Sigma}_{adj}(j,j)\right\}^{1/2},\quad j\in\{0, 1,\ldots,p\},
\end{equation}
where $\eta_j = \min\{n^{1/2}\lambda,\max\{1,\lambda/\lvert\hat{\beta}_j\rvert\}\}$ is the adjustment weight, and $z_{\alpha/2}$ is the upper $\alpha/2$ quantile of the standard normal distribution.
Theorem \ref{prop::moments} below reveals the property of the proposed interval (\ref{eq::CI}).

\begin{theorem}
\label{prop::moments}
Consider the posterior distribution under either the AL prior (\ref{eq::priorAlasso}) or the CA prior (\ref{eq::priorSCAD}). Under the conditions of Theorem \ref{thm::Unified}, we have the following results.
\begin{enumerate}
  \item Convergence of the posterior mean:
  \begin{eqnarray*}
     n^{1/2}(\check{\bm{\beta}}_{\mathcal{S}} -\bm{\beta}_{\mathcal{S}}^0) &\rightarrow&  \mathrm{N}\left\{0\,,\,\tau(1-\tau)G_{\mathcal{S}}^{-1}D_{\mathcal{S}}G_{\mathcal{S}}^{-1}\right\}, \\
     n\lambda(\check{\bm{\beta}}_{\mathcal{S}^c} - \bm{0})
&\rightarrow& 0,
\end{eqnarray*}
in distribution as $n \to \infty$.
\item Properties of the adjusted variance:
\begin{eqnarray*}
&n\,\check{\Sigma}_{adj,\mathcal{S}} & \; = \;  \tau(1-\tau)G_{\mathcal{S}}^{-1}D_{\mathcal{S}}G_{\mathcal{S}}^{-1} + o_{\mathrm{pr}^*}(1),\\
(n^{1/2}\lambda)^{-2}\;\lesssim_{\mathrm{pr}^*}\;&(n\lambda)^2\,\check{\Sigma}_{adj}(j,j)&\; \ll_{\mathrm{pr}^*} \; 1,\quad j\not\in \mathcal{S}.
\end{eqnarray*}
\end{enumerate}
\end{theorem}

Theorem \ref{prop::moments} informs us of several aspects of the posterior inference. First, the posterior mean for the active coefficient is first-order equivalent to the oracle quantile regression estimator as if we knew the set $\mathcal{S}$. Furthermore, the adjusted posterior variance-covariance matrix captures the sampling variance-covariance of the posterior mean. For those coefficients, the adjustment weight $\eta_j =1 + o_{\mathrm{pr}^*}(1)$ due to $n^{-1/2} \ll \lambda\ll 1$, so the confidence intervals in the form of (\ref{eq::CI}) can be viewed as standard Wald-type intervals in the oracle model.

Next we consider any inactive coefficient $\beta_j$ for $j\not\in\mathcal{S} $. In this case, $\hat{\beta}_j = O_{\mathrm{pr}^*}(n^{-1/2})$, so the adjustment weight $\eta_j \asymp_{\mathrm{pr}^*} n^{1/2}\lambda \to \infty$ in (\ref{eq::CI}) works to inflate the Wald-type interval. Theorem \ref{prop::moments} implies
\begin{eqnarray*}
\frac{\check{\beta}_j - 0}{\eta_j\left\{\check{\Sigma}_{adj}(j,j)\right\}^{1/2}} \to 0,\quad
n^{1/2}\;\eta_j\left\{\check{\Sigma}_{adj}(j,j)\right\}^{1/2}\to 0,\quad j \not\in \mathcal{S},
\end{eqnarray*}
in $\mathrm{pr}^*$-probability, and therefore, the confidence interval in (\ref{eq::CI}) will achieve a conservative 100\% asymptotic coverage probability but the interval length remains super-efficient at the order of $o_{\mathrm{pr}^*}(n^{-1/2})$.

In summary, the proposed intervals (\ref{eq::CI}) are automatically adaptive to the possible sparsity in the model without relying on a dichotomous variable selection step. In a sparse model  ($s<p$), such interval estimates are more efficient than the classical quantile regression inference using all the coefficients. Empirically we will see later that the proposed intervals are less sensitive to tuning than direct quantile regression inference following model selection.

\begin{remark}
We remark on the value of the Bayesian computational framework: Theorem \ref{thm::Unified} implies the posterior variance-covariance matrix approximates $G_{\mathcal{S}}^{-1}$, which is an essential quantity for inference in quantile regression and otherwise requires non-parametric estimation \citep{yang2016posterior}. We refer the readers to \citet{chernozhukov2003mcmc} for an in-depth discussion of frequentist inference via MCMC.
\end{remark}

\subsection{Optimally-weighted posterior inference}
\label{subsec::weighted}

In the presence of heteroscedasticity, it is well-known from \citet{newey1990semiparametric} that the following optimally-weighted quantile regression estimator is semi-parametric efficient for estimating $\bm{\beta}^0$ when no sparsity is at play:
\begin{equation*}
\hat{\bm{\beta}}^{(w)} = \argmin_{\bm{u}\in\mathbb{R}^{(p+1)}}\;\sum_{i=1}^n \zeta_i\rho_\tau(y_i - \bm{x}_i^{{ \mathrm{\scriptscriptstyle T} }}\bm{u}),
\end{equation*}
where $\zeta_i = f_{\epsilon\mid X = \bm{x}_i}(0)$. In a possibly sparse quantile regression model, a natural question is whether we can achieve the optimal semi-parametric efficiency by using only the data on $(X_\mathcal{S} , y)$, i.e., after `oracle' model selection is attained. The answer is, somewhat surprisingly, negative, because the "optimal" weights $f_{\epsilon\mid X_\mathcal{S}}(0)$ under the "oracle model" does not capture the full heteroscedasticity in data. Instead, we show that the statistical efficiency can be further improved by the posterior inference using the the optimally-weighted asymmetric Laplace working likelihood:
\begin{equation}
\label{eq::ALlikelihood-weight}
\mathcal{L}^{(w)}(\mathbb{D}_n \mathrel{\mid} \bm{\beta}) \propto
\exp\left\{ -\sum_{i=1}^n\zeta_i\rho_\tau(y_i - \bm{x}_i^{{ \mathrm{\scriptscriptstyle T} }}\bm{\beta}) \right\}.
\end{equation}
Coupling (\ref{eq::ALlikelihood-weight}) with the shrinkage priors in Section \ref{sec::bayes-model}, we obtain the posterior density  $p^{(w)}(\bm{\beta} \mathrel{\mid} \mathbb{D}_n)$, and we denote the posterior mean by $\check{\bm{\beta}}^{(w)}$. The following result gives the sampling distribution of the posterior mean for the active coefficients.
\begin{proposition}
\label{prop::SCAD-weighted}
Consider the weighted working likelihood (\ref{eq::ALlikelihood-weight}) and either of the prior (\ref{eq::priorAlasso}) or (\ref{eq::priorSCAD}). Under the same conditions in Theorem \ref{thm::Unified}, the posterior mean satisfies:
  \begin{eqnarray*}
     n^{1/2}(\check{\bm{\beta}}^{(w)}_{\mathcal{S}} -\bm{\beta}_{\mathcal{S}}^0) &\;\rightarrow\;&  \mathrm{N}\left\{0\,,\,\tau(1-\tau)Q_{\mathcal{S}}^{-1}\right\},

\end{eqnarray*}
in distribution,
where $Q_{\mathcal{S}} = \mathrm{E}^*\{X_{\mathcal{S}}X_{\mathcal{S}}^{{ \mathrm{\scriptscriptstyle T} }}f_{\epsilon\mid X}^2(0)\}$.
\end{proposition}

On the other hand, if classical quantile regression is applied to $(X_\mathcal{S} , Y)$ with the inactive covariates left out, the `optimally' weighted quantile regression has an asymptotic variance of $\tau(1-\tau){V}_{\mathcal{S}}^{-1}$ \citep{newey1990efficient}, where $V_\mathcal{S} = \mathrm{E}^*\{X_\mathcal{S} X^{{ \mathrm{\scriptscriptstyle T} }}_\mathcal{S} f^2_{\epsilon\mid X_{\mathcal{S}}}(0)\}$ relies only on the active covariates. We show in  the Supplementary Materials that
\begin{equation}
Q_{\mathcal{S}}^{-1} \preceq  V_{\mathcal{S}}^{-1},\label{eq::efficiency-compare}
\end{equation}
which reveals that focusing only on the oracle quantile regression model (even when it is available) does not lead to optimal efficiency for the active coefficients.

There is a simple reason why the inactive set of covariates should not be abandoned. Even though $X_{\mathcal{S}^c}$ does not affect the conditional $\tau$th quantile of $Y$ given $X$, it may still impact other aspects of the conditional distribution of $Y$ given $X$, in particular the density function $f_{\epsilon\mid X}(0)$ may depend on $X_{\mathcal{S}^c}$. Unless  $f_{\epsilon\mid X}(0) = f_{\epsilon\mid X_{\mathcal{S}}}(0)$, the optimal efficiency of quantile regression analysis cannot be achieved if we only focus on those active covariates. In general, a truly `oracle' model should also identify covariates that affect the conditional density function $f_{\epsilon\mid X}(0)$, in addition to $X_{\mathcal{S}}$.

\begin{remark}
To focus on the main idea, we suppose that the optimal weight $\zeta_i = f_{\epsilon \mid X = \bm{x}_i}(0)$ in (\ref{eq::ALlikelihood-weight}) is known. In practice, it is possible to use the estimated weights while still achieve the same asymptotic efficiency as if we knew $\zeta_i$; see, e.g., \citet{newey1990efficient,koenker1994estimatton} and \citet{zhao2001asymptotically} for some theoretical investigations.
 \end{remark}

\section{Posterior asymptotics with increasing dimensions}
\label{sec::increase-p}

In this Section, we present some
extensions of the results in Section \ref{subsec::theory}
when the dimension $p=p_n$ diverges with, while still of smaller order than, the sample size $n$; We also allow the size of the active covariates, $|\mathcal{S}| = s_n$, to depend on the sample size. For illustration purposes, we only focus on the CA prior (\ref{eq::priorSCAD}) in this Section, and we show that the posterior distribution still achieves adaptation to sparsity, even in the regime of moderately increasing dimensions.

The asymptotic regime with an increasing dimension is often of practical interest.
When modeling the conditional quantile function, it is common to consider Model (\ref{eq::truemodel}) where the complexity may depend on the available sample size. A leading example is when we approximate the unknown conditional quantile function by a linear combination of series/basis expansions, e.g., B-splines, polynomials, and wavelets \citep{chao2017quantile,belloni2019conditional}. To control the approximation error, the number of basis function typically increases with the sample size at a certain rate \citep{he1994convergence}.
The regime also covers the so-called `many regressors' model in econometrics, where a large number of variables are often necessary to model economic theories \citep{cattaneo2018inference}.

We first discuss some generalizations of the technical assumptions in Section \ref{subsec::theory} when the dimension $p_n \to \infty$. With $p_n = o(n)$, Assumptions \ref{aspt::id} and \ref{aspt::density} are standard in the quantile regression literature \citep{belloni2019conditional,pan2020multiplier}.
On the other hand, Assumptions \ref{aspt::covar} and \ref{aspt::sparsity} may not be suitable for the increasing dimensional regime, therefore we make the the following substitutions for them.

\begin{asptref}{aspt::covar}[Covariates]
There exists a constant $\sigma_0 > 0$, such that for all $\lVert u\rVert = 1$ and $t>0$:
\begin{equation}
\mathrm{pr}^*\left(|u^TD^{-1/2}X| \geq \sigma_0 t\right)\leq 2\mathrm{e}^{-t}.
\label{eq::sub-exponential}
\end{equation}
Furthermore, the eigenvalues of the matrix $D = \mathrm{E}^*[XX^T]$ satisfies
\begin{equation}
p_n^{-1}\lesssim\theta_{\min}(D) \leq\theta_{\max}(D)\lesssim p_n,\quad\text{and}\quad \theta_{\min}(D_{\mathcal{S}}) \geq \theta_{1} > 0,
\label{eq::modify-eigen}
\end{equation}
for some constant $\theta_1 > 0$.
\label{aspt::modify-covar}
\end{asptref}

\begin{asptref}{aspt::sparsity}[Sparsity]
There exists a sequence $\underline{b}_n > 0$ such that
\[
\min_{j\in\mathcal{S}\setminus \{0\}}\lvert \beta_{j}^0\rvert  > \underline{b}_n.
\]
\label{aspt::modify-sparsity}
\end{asptref}

Assumption \ref{aspt::modify-covar} consists of two parts:
First, (\ref{eq::sub-exponential})
states that the standardized covariate $D^{-1/2}X$ is sub-exponential, which strengthens the boundedness of $X$ in Assumption \ref{aspt::covar}; We refer to \citet[Section 3.3]{vershynin2018high} for examples of sub-exponential distributions in high-dimensions. Second, (\ref{eq::modify-eigen}) relaxes Assumption \ref{aspt::covar} by allowing some eigenvalues to vanish or diverge as $p_n\to\infty$, implying that there could be some degree of co-linearity among the $p = p_n$ covariates.
Finally, Assumption  \ref{aspt::modify-sparsity} requires all non-zero coefficients to be sufficiently separated from $0$, yet the threshold $\underline{b}_n$ is allowed to shrink towards zero as the sample size grows.

The result below generalizes Theorem \ref{thm::Unified} to an increasing dimensional regime, where we drop the subscript $n$ in $s$ and $p$ for simplicity.

\begin{theorem}
Consider the posterior distribution under the CA prior (\ref{eq::priorSCAD}) and $p\to\infty$. Suppose Assumptions \ref{aspt::id}, \ref{aspt::modify-covar}, \ref{aspt::density} and \ref{aspt::modify-sparsity} hold. If  $s^4p^2\log^2 n = o(n)$, and the tuning parameter $\lambda$ is chosen such that
\begin{equation}
\frac{s^{1/2}p\log^{3/2} p}{n^{1/2}} \ll \lambda \ll \min\left\{s^{-1/2},\;\underline{b}_n,\;\underline{b}_n[\theta_{\min}(D)]^{1/2}\right\},
\label{eq::lambda-in-p}
\end{equation}
then we have the following results:
\begin{enumerate}
\item Adaptive rate-of-contraction: for any sequence $M_n\to+\infty$.
\[
\Pi\left( \left\lVert\bm{\beta}_{\mathcal{S}} - \bm{\beta}^0_{\mathcal{S}}\right\rVert \leq M_n\sqrt{\frac{s}{n}} \,,\, \left\lVert\bm{\beta}_{\mathcal{S}^c} \right\rVert_{\infty} \leq M_n\frac{s\log p}{n\lambda} ~\middle\vert~ \mathbb{D}_n\right) \to 1,
\]
in $\mathrm{pr}^*$-probability
\item Distributional approximation: for $\pi_j(u) = (1/2)\exp\{-|u|\}$, $(\forall j\in\mathcal{S}^c)$,
\[
\left\lVert p\left(\bm{\beta} ~\middle\vert~  \mathbb{D}_n\right) -  \phi\left(\bm{\beta}_S;\tilde{\bm{\beta}}_S,\frac{1}{n}G_{\mathcal{S}}^{-1}\right)\times \prod_{j\not\in\mathcal{S}} \left\{ n\lambda  \pi_j(n\lambda \beta_j)\right\}\right\rVert_{TV} \to 0,
\]
in $\mathrm{pr}^*$-probability, where $\phi(\cdot\,;\,\mu,\Sigma)$ is the density function of a multivariate-Gaussian distribution; $\tilde{\bm{\beta}}_{\mathcal{S}}$ and $G$ are defined in Section \ref{sec::notations}.
\end{enumerate}
\label{thm::SCAD-in-p}
\end{theorem}

Theorem \ref{thm::SCAD-in-p} explicitly characterizes the effect of increasing model dimension on the posterior. Since $(n\lambda)/(s\log p) \gg (np)^{1/2}$, part 1 of Theorem \ref{thm::SCAD-in-p} shows the posterior distribution for all inactive coefficients concentrates simultaneously towards zero at a second-order rate, despite that there may be a diverging number of them.
For part 2 of Theorem \ref{thm::SCAD-in-p}, it is sometimes more informative to consider a one-dimensional linear combination of parameters $\bm{\alpha}^T\bm{\beta}$ for $\lVert \bm{\alpha} \rVert = 1$ in the regime of increasing dimension \citep{fan2004nonconcave}.
 If $\bm{\alpha}_{\mathcal{S}}\neq \bm{0}$, then the posterior for $\bm{\alpha}^T\bm{\beta}$ would be asymptotically `oracle'; otherwise the posterior would have a scale at the order of $p^{1/2}/(n\lambda) \ll n^{-1/2}$, which is super-efficient.

While Theorem \ref{thm::SCAD-in-p} covers a wide range of models under general design and spasity conditions, the range (\ref{eq::lambda-in-p}) may imply additional conditions on $\underline{b}_n$, $p_n$, or the eigenvalues of $D$ in a given setting. To better explain the conditions in Theorem \ref{thm::SCAD-in-p}, here we consider an example with a sparse model, i.e., $s_n = s_0$ stays fixed yet $p_n\to\infty$. In addition to Assumptions \ref{aspt::id}, \ref{aspt::modify-covar}, \ref{aspt::density} and \ref{aspt::modify-sparsity}, we suppose the design matrix satisfy $\theta_{\min}(D) \geq \theta_0 > 0$, which aligns with the setting in \citet{belloni2019conditional}.
Under this model setting, the conclusions in Theorem \ref{thm::SCAD-in-p} hold if
\begin{align*}
&p^2\log^2 n = o(n),\quad\underline{b}_n\gg \frac{p\log^{3/2} p}{n^{1/2}},\\
&\text{and}\qquad
\frac{p\log^{3/2} p}{n^{1/2}} \ll \lambda_n \ll \underline{b}_n,
\end{align*}
where $\underline{b}_n$ is defined in Assumption \ref{aspt::modify-sparsity}.
With a sparse model, the above conditions are more intuitive and are comparable with the literature on shrinkage estimation with moderately increasing dimensions \citep{fan2004nonconcave,huang2008adaptive,armagan2013generalized}, even though we work with a mis-specified likelihood.

\begin{remark}
For high-dimensional sparse problems with $s_n \ll n \ll p_n$, it is often more practical to employ a screening step first to reduce the dimension to a manageable scale \citep{fan2008sure}, and then pursue statistical inference via the Bayesian approach. For example, the Sure Independence Screening for quantile regression in \cite{he2013quantile} or \citet{wu2015conditional} can select a $d_n$-dimensional sub-model, such that $d_n = o(n)$ but all active covariates are retained with probability approaching one.
Our asymptotic regime in this Section then becomes relevant if we focus on the selected model, and Theorem \ref{thm::SCAD-in-p} applies to the $d_n$-dimensional posterior distribution post-screening.
\end{remark}

\section{Simulation}
\label{sec::simulation}

We use a Monte Carlo simulation to demonstrate that the asymptotic properties established in this paper are visible in finite-sample problems. A limited comparison with some other inferential methods in quantile regression is also included. We only highlight several key findings here, whereas more detailed results are relegated to the online Supplementary Materials.

We generate random samples of size $n$ from the following regression model
\[
Y = 1 + 3X_2 - 5X_5 +\left\{\frac{1+ (X_6 - 1)^2}{3}\right\}  e,
\]
where $e\sim \mathrm{N}(0,1)$ is independent of the covariate vector $X = (X_1,\ldots,X_6)^T \sim\mathrm{N}(0,\Sigma)$ with the $(i,j)$th entry of $\Sigma$ being $0.8^{|i-j|}$ for $i,j \in \{1, \cdots , 6\}$. The data generating process satisfies Model (\ref{eq::truemodel}) at $\tau = 0.5$, where $X_2$ and $X_5$ are active but $X_6$ is inactive for the conditional median of $Y$ given $X$.  We consider two different sample sizes $n = 200$ and $n = 500$, and use $2,000$ Monte Carlo data sets in each simulation.

We compare the proposed posterior inference method with four other approaches for constructing $90\%$ confidence intervals of the median regression coefficients. Three of competing approaches are the robust rank-score method of \citet{koenker1999goodness} applied to: (i) the full model with $(X_1,\ldots,X_6)$ included, (ii) the oracle model with $(X_2,X_5)$ included, and (iii) the selected model from adaptive lasso variable selection, respectively.
The fourth competing approach is the wild bootstrap for the adaptive lasso quantile regression proposed recently by \citet{wang2018wild}. Not all possible methods are included in this study, but those competitors are known to have generally good performance under heteroscedastic models.

We first compare the performances of those methods under a fixed tuning parameter in Table \ref{tab::qid}. To make a fair comparison, the tuning parameter $\lambda$ for both the shrinkage prior and the adaptive lasso model selection are kept the same across all Monte Carlo data sets at a given sample size. We relegate further implementation details, including tuning parameter specification, to the online Supplementary Materials.

\renewcommand{\arraystretch}{0.5}
\begin{table}[t!]
\begin{threeparttable}
\caption{Empirical coverage probabilities and average lengths ($\times 100$) for $90\%$ confidence intervals.}%
\begin{tabular}{lcccccc}
\toprule
 &\multicolumn{3}{c}{Empirical coverage}& \multicolumn{3}{c}{Average length (s.e.)}\\
 \midrule
 \multicolumn{7}{c}{$n = 200$}\\
 &$\beta_2$&$\beta_5$&$\beta_{zeros}$&$\beta_2$&$\beta_5$&$\beta_{zeros}$\\
 \midrule
Full &  92 &  91 & 90 & \hphantom{00}43.7 (0.25) & 43.9 (0.25) & 41.7 (0.10) \\
  Oracle &  89 &  93 & 100 & \hphantom{00}22.5 (0.13) & 32.6 (0.21) &  0.0 (0.00) \\
  Refit &  84 &  86 &  89 & \hphantom{00}28.1 (0.21) & 34.7 (0.22) & 11.4 (0.06) \\
  WildPen &  85 &  84 &  89 &\hphantom{00} 27.0 (0.13) & 30.2 (0.14) & 20.1 (0.08) \\
  BayesAdj &  93 &  93 &  96 & \hphantom{00}28.1 (0.11) & 32.7 (0.12) & 11.7 (0.04) \\
  \midrule
  \multicolumn{7}{c}{$n = 500$ }\\
  &$\beta_2$&$\beta_5$&$\beta_{zeros}$&$\beta_2$&$\beta_5$&$\beta_{zeros}$\\
\midrule
    Full &  91 &  91 &  90 &\hphantom{00} 26.7 (0.12) & 26.6 (0.12) & 25.6 (0.06) \\
  Oracle &  90 &  93 & 100 & \hphantom{00}14.0 (0.06) & 20.5 (0.11) &  0.0 (0.00) \\
  Refit &  81 &  86 &  89 & \hphantom{00}17.2 (0.11) & 21.5 (0.11) &  6.3 (0.05) \\
  WildPen &  84 &  85 &  91 &\hphantom{00} 16.7 (0.07) & 18.9 (0.07) & 11.3 (0.06) \\
  BayesAdj &  89 &  91 &  95 &\hphantom{00} 16.3 (0.06) & 19.3 (0.06) &  6.1 (0.03) \\
  \bottomrule
\end{tabular}
\label{tab::qid}
\begin{tablenotes}
\linespread{1}\footnotesize
\item 'Full' refers to the rank-score method applied to all the covariates, 'Oracle' uses only the active covariates for the conditional median, and 'Refit' is the rank-score method applied to a model selected by adaptive lasso.
'WildPen' is the wild bootstrap approach of \citet{wang2018wild}.
'BayesAdj' refers to the proposed adjusted posterior inference in Section \ref{subsec::inference}.
For the 'Refit' and 'Oracle' methods, if a covariate is not included in the model, we report its confidence interval as a singleton $\{0\}$.
The column $\beta_{zeros}$ averages over all inactive coefficients $\beta_1$, $\beta_3$, $\beta_4$ and $\beta_6$.
The numbers shown in the parentheses are the estimated standard errors. For the coverage estimates, their standard errors are all below $0.9$. For penalization/shrinkage, we used $\lambda = 0.066$ when $n=200$ and $\lambda = 0.051$ when $n = 500$.
\end{tablenotes}
\end{threeparttable}
\end{table}

Table~\ref{tab::qid} suggests that the adjusted posterior inference indeed achieves adaptive performance.
For the active coefficients, the adjusted posterior inference gives much shorter intervals than those from  the full model, and it is reasonably competitive with the results from the oracle model.
For the inactive coefficients, the adjusted posterior inference gives much shorter intervals than those under the full model with higher-than-nominal coverage probability.
On the other hand, the  wild bootstrap approach and the rank-score method applied to the selected model from adaptive lasso both fall short in coverage. Part of the reason for their under-performance is that the adaptive lasso does not achieve 'oracle' selection often enough in this case due to limited sample size, even when $n = 500$; See Figure S2 in the Supplementary Materials. Therefore, those approaches based on variable selection may fail to fully account for the uncertainty induced by selection.

In addition, the adjusted posterior inference gives more stable confidence intervals, as the standard errors for average interval lengths are among the smallest of all methods in Table \ref{tab::qid}. Such finite-sample stability of the adjusted posterior inference reflects its avoidance of pursuing dichotomous variable selection. We refer to Figure S1 and comments thereof in the Supplementary Materials for more details.

Next, we examine the impact of the tuning parameter in the comparisons of shrinkage-based methods. To this end, we vary $\lambda$ at a wide range of values and compare the performances in Figure \ref{fig_sensitivity} when sample size $n = 500$; see also Figure S3 in the Supplementary Materials for the results when $n = 200$.
We note that the coverage probabilities of the adjusted posterior inference method for the active coefficients are more stable around the nominal levels than other methods for a wide range of $\lambda$ values.
For the inactive coefficient $\beta_6$, the coverage probability for the proposed method remains high without any sacrifice in the lengths of the intervals relative to other non-oracle methods. Finally, we note from our empirical studies that the adjusted posterior inference tends to lose coverage if the shrinkage parameter $\lambda$ is too large. As a practical guide, we suggest choosing a $\lambda$ that is slightly smaller than what one would obtain from the cross-validation method for the adaptive lasso.

\begin{figure}[t!]
\centering
         \includegraphics[width = 0.85\textwidth]{test86n500.eps}\vspace{-0.5em}
\caption{Empirical coverage probabilities and average lengths for 90\% confidence intervals with different $\lambda$ when $n = 500$. The true regression coefficients are $\beta_2^0 = 3$, $\beta_5^0 = -5$ and $\beta_6^0 = 0$. The value of $\lambda$ marked by a vertical broken line is used to produce Table ~\ref{tab::qid}, and the abbreviated method names are the same as Table~\ref{tab::qid}. }
\label{fig_sensitivity}
\end{figure}

\section{Conclusion and discussion}
\label{sec::conclude}

In this paper, we show that Bayesian computational framework can be useful for constructing frequentist confidence intervals in possibly sparse quantile regression analysis.
By employing appropriate shrinkage priors, we show the posterior inference can adapt automatically to model sparsity. Asymptotically, the proposed confidence intervals are oracle efficient for the active coefficients, and are super-efficient for the inactive coefficients.
The posterior inference approach enjoys two distinct advantages:
(i) it avoids the need to pursue dichotomous variable selection by employing a continuous shrinkage prior;
(ii) it offers an alternative to classical frequentist computation by trading nuisance-parameter estimation for MCMC sampling. The proposed approach demonstrates desirable finite-sample stability in simulation studies.

We highlight a few possible extensions of our work.
In the paper, we demonstrate that the asymptotic adaptation phenomena of the posterior in problems of moderately increasing dimensions.
Nonetheless, it remains an interesting problem to understand the value of posterior inference in the high-dimensional setting where $p \gg n$.
In addition, exploiting the Bayesian computational methods would be even more appealing in more complex settings, e.g., censored quantile regression \citep{yang2016posterior,wu2021bayesian} where the objective function can be highly non-convex \citep{powell1984least,powell1986censored}. The Bayesian approach can avoid direct optimization of the objective function while incorporating possible model sparsity.

\section*{Supplementary Materials}

The online supplementary material contains some additional results from the simulation study, as well as the proofs of all the results in this paper.
\par
\section*{Acknowledgements}
The work is partially supported by the NSF Awards DMS-1914496 and DMS-1951980.
\par

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FeaturedJune 30, 2018
What's Is It Like Living in Sarasota: U.S. News World Report
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WCBTT- Black Nativity

redpajama_set_name: RedPajamaCommonCrawl


Frosty Loch Ard Loch Ard, Aberfoyle, Scotland.
Winter tends to drain most colours from the land combine it with a good hard frost and the thin white coat dilutes existing colours to mere pastel shades. I like this muted palette and find film responds very well to it subtlely bringing out any hint of tonality however faint. The air of serenity and stillness is nicely conveyed by the mirror reflection of a gentle sky in the surface of the loch.


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Have you ever wanted to do something perfectly without any defects? This is commonly known as perfectionism. Want work to be all perfect, generally someone who is too perfectionist will feel worried and feel guilty excessively when making mistakes in a job he does.
Well, do you feel familiar with this condition? Or even you are someone who is too perfectionist? Who would have thought that giving up perfectionism turned out to provide good benefits, you know. Here are 5 benefits that you will get when you dare to let go of perfectionism in life.
Who says reducing perfectionism means doing things carelessly? You can still do your best even though perfectionism has become your habit. When slowly overcoming it, you also have a new mindset that is to do something with the best effort, not with an obsession to pursue perfection.
When you want to always do something perfectly, then you will take more time because you are constantly unsure. Conversely, letting go of perfectionism makes your work faster to complete. Besides that, another benefit that you can get is more work that you can do. Very profitable, right?
There will be a feeling of anxiety approaching when you too want everything to be done perfectly. Whether for fear of making a mistake or for other reasons, the overly perfectionist nature is indeed a good idea to let go. With the choice of eliminating the tendency to always be perfect, your mind is guaranteed to be more relaxed and calm.
No longer feeling too worried if you make a mistake, the sign is that you have become a bolder figure. Rather than feeling excessive fear, you will be big-spirited to forgive mistakes rather than blaming yourself constantly. In addition, you are willing to improve and learn from the mistakes you have previously made.
When you no longer prioritize perfectionism, you can feel the difference yourself. Your life must be a lot happier. You feel that because you have stopped being haunted by fear and anxiety due to the tendency to do everything with perfection.
That was 5 good benefits that you can get when you no longer make perfection a benchmark for something you are doing. How about it, guys ? Are you interested in giving up too perfectionist attitude?


text:
A 9019 Eucommia (ideiglenes jelöléssel 1987 QF3) a Naprendszer kisbolygóövében található aszteroida. Eric Walter Elst fedezte fel 1987. augusztus 28-án.

Kapcsolódó szócikkek
 Kisbolygók listája (9001–9500)

Jegyzetek

A Naprendszer kisbolygói

redpajama_set_name: RedPajamaWikipedia


Shining a light on collaborative sustainability initiatives for Global Organisations.
Most food businesses are facing key challenges such as improving efficiency of water use, maintaining good on-farm soil health to ensure productivity, reducing GHG emissions and rooting out slavery and human trafficking in product supply chains. Faster progress can be made in addressing these major sustainability challenges by working collaboratively. Help has arrived in the form of the new Spotlight Tool, an online platform developed for the Sustainable Agriculture Initiative (SAI) Platform members who include major food and drinks manufacturers retailers, international agribusinesses and food service companies.
The Sustainable Agriculture Initiative (SAI Platform) is an industry group supporting food businesses sustainable sourcing agendas. ADAS's Sustainable Food and Farming Team (SFF), in partnership with RSK's Business Solutions (BSL) developed a new web-based tool for SAI Platform which has changed the way they operate by catalysing collaboration between members. The Spotlight tool, coined "the tinder of sustainable agriculture", was designed to generate cross industry partnerships by matching SAI Platform's business members based on shared interests in tackling sustainability challenges.
SAI Platform's food company members can enter information into the tool about their sustainability challenges, highlighting commodities, sustainability topics (e.g. deforestation, climate change), geographies and how they can support projects. This information can then be visualised over a mapping platform. The tool is changing the way food businesses engage with each other by shining light on their priorities and finding new collaborative opportunities.
What is SAI Platform's Spotlight?
The Spotlight Tool is an innovative pre-competitive space for food businesses to a) work together on shared sustainability challenges b) reduce barriers that exist between them as competitors and c) build on their combined resources and impact.
Food companies are asked to enter information about their key sustainability challenges, highlighting specific commodities, sustainability 'topics' (e.g. deforestation), geographies and what role they would like to play in any future partnership. As you can imagine this process creates a valuable data set, provided by the members for use by the members and by the SAI Platform Secretariat to inform the agendas of their working groups.
SAI Platform members can then search the online tool or interact with interactive mapping features developed using a software platform to see which companies may have similar interests, and to spark conversations.
The tool provides a pre-competitive space for businesses to identify ways they may work together to make faster progress on their sustainability agendas.
What are the key benefits of using Spotlight?
The Spotlight tool was launched in February and there has been a quick uptake from the SAI Platform membership in getting involved in Spotlight. So far members have entered data describing 200 opportunities across 50 countries covering 50 crops and 20 topic areas, and groups of businesses have begun to engage with each other in new ways. The tool is already shining light on the potential for collaboration based on members' interests with five country level opportunities having greater than seven companies expressing interest on similar topics. The Spotlight working group meets frequently to discuss how the tool might be developed and deployed further, with some great suggestions already floated for future developments. Watch this space!


text:
<?php

namespace ApplicationFeatureApi\Exception;

use Zend\Http\Response;

class ServiceUnavailableException extends AbstractException
{
    public function __construct(
        $message = "Service Unavailable",
        $code = Response::STATUS_CODE_503,
        \Exception $previous = null
    ) {
        parent::__construct($message, $code, $previous);
    }
}

redpajama_set_name: RedPajamaGithub


The San Francisco Caltrain terminus is located at 4th & King Streets. The current station opened on June 23, 1975 replacing the 1914 station at 3rd & Townsend one block away, giving today's station a modern appearance. The station at 3rd & Townsend was a historic Southern Pacific depot. It was demolished to make way for a freeway project. The current station was given some renovations between 1997 and 1999. Including extending the station building an extra 20 feet towards the platforms along with making the station ADA compliant although wheelchairs could board in San Francisco staring in July 1995 when CalTrain began accepting these passengers. The Caltrain ticket office (along with the one in San Jose Diridon) was permanently closed (saving $650,000 total) on October 11, 2010 leaving TVMs as the only venue for purchasing Caltrain tickets (although Clipper Cards can be refilled in person at various places). It's final hours were just 5:30am to 1:00pm weekdays for one shift although until 2009 it was still open full time (5:30am to 10:00pm weekdays with modified hours on weekends and holidays).
The station has 12 terminal tracks with 6 low-level island platforms that are the standard for Caltrain. Caltrain routinely stores trains in the station between runs so the station is usually full of trains. The platforms are left exposed to the elements with tall lampposts that have small signs that say San Francisco except at their northern ends. A modern canopy with light blue steel beams begins and covers about one car length (the canopy for tracks 1 and 2 is longer, about 3 car lengths). This canopy provides a covered walkway beyond the bumper blocks on each platform and through doors into the station. Trains always run in push mode northbound, pull mode southbound so the locomotives are always at the southern end of the platforms and never reach the bumper blocks.
To leave the platforms all passengers must pass through glass doors (that are basically platform gates) and into the simple glass station house. This station house is a single story building with a tall ceiling (held up by blue beams) and yellow framing around the windows out to the platforms. An old fashioned system of white signs manually changed inside the building near the doors out to each of the platforms are used to inform passengers of the track for each departure. These information panels are two fold: high up is a small white sign with the departure time, blank indicates a local (making all regularly scheduled station stops), a yellow or red dot a Limited Train or Baby Bullet Express. Closer to eye level outside each track is another changeable panel that has the train number, departure time and lists the scheduled stops with their scheduled arrival times. Since the station has so many tracks signs for departure tracks can be indicated many hours in advance (the track and signage for the 12:01am departure the next morning was already up at 5:51pm.) Inside the narrow building are a few concessions, the remains of the former ticket office (all POP ticket sales are now from Vending machines) and a few benches outside the track gates for waiting passengers (there isn't a formal waiting room). Leaving the building passengers are along 4th Street bounded by Townsend Street and King Street.
Since January 10, 1998 the station has had local rail connections provided by the Muni Metro: In the median of King Street across from the train platforms is the terminus of the Muni Metro's extension from the downtown tunnel to the Caltrain Station. This is where N Judah Trains terminate. Since 2007 this station has been supplemented by the 4th & King Street station across this intersection in the median of 4th Street. This station is an intermediate stop on the T Line from Sunnydale to Downtown. At the 4th & King Street corner of the Caltrain station is a sign telling passengers the boarding locations of the respective Muni Lines beneath a NextMuni Countdown clock that scrolls through information for both stations with the times beneath N-Center of King, T-Across King. This allows transferring commuters to know which platform to go to for the next MUNI train. Passengers relied on special MUNI Caltrain express shuttle buses to and from locations in downtown before the MUNI Metro extension opened.


I saw the movie Jem And The Holograms. This movie stars Aubrey Peeples (Austin & Ally), Stefanie Scott (A.N.T Farm), Hayley Kiyoko (Lemonade Mouth), Ryan Guzman (Step Up All In), Molly Ringwald (The Breakfast Club), and Juliette Lewis (Catch & Release).
This was a great movie. I enjoyed it quite a bit.


This past Thursday evening, some friends and I attended a Marc Jacobs event sponsored by GQ and Ciroc featuring the artist Bäst, a street artist who's style is a combination of collage and graffiti.
The scene was lively, drinks were flowing, GQ mags were being passed, and the art was abundant. Immediately upon entering was a disco-eqsue massive triangle globe made of shattered glass. All around the store were neon light images of Bäst's work and his collaboration with Marc Jacobs sneakers were displayed gallery style towards the back of the store. Ciroc, Brooklyn Brewery beer, Sophia Coppola champagne, and light bites kept the night flowing. It was a fun event and it's always dope experiencing the collaboration between two visionaries.


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text:
rc=0
start() {
  if [ ! -e /sys/block/bcache0/bcache/cache ]; then
    /sbin/wipefs -a $CACHE_DEVICE
    /usr/sbin/make-bcache -C $CACHE_DEVICE
    for DEV in $CACHE_DEVICE; do
      echo $DEV > /sys/fs/bcache/register
    done
    sleep 1
    name=`basename /sys/fs/bcache/*-*-*`
    echo $name > /sys/block/bcache0/bcache/attach

  fi

  touch /var/lock/subsys/bcache
  return 0
}

stop() {
  rm -f /var/lock/subsys/bcache
  return 0
}

tune() {
  # Some performance tuning, we assume our ebs volume is always
  # going to be slower that local SSDs
  echo 0 > /sys/block/bcache0/bcache/sequential_cutoff
  echo $BCACHE_MODE > /sys/block/bcache0/bcache/cache_mode
  sleep 2
  echo 0 > /sys/block/bcache0/bcache/cache/congested_read_threshold_us
  echo 0 > /sys/block/bcache0/bcache/cache/congested_write_threshold_us
}

case "$1" in
  start)
    start
    tune
    rc=$?
    ;;
  stop)
    stop
    rc=$?
    ;;
  tune)
    tune
    rc=$?
    ;;
esac

exit $rc


text:
Behind the piano: Thomas Hewitt Jones
I have previously written and forced you to listen to the fantastic music by British composer Thomas Hewitt Jones. And now it's time to dig a bit deeper into out knowledge about this man!
Where are you from? And where do you live?
I was born in Dulwich, London and my family came from Gloucestershire. Many of my relations are still there and it feels like home, in some kind of way. I currently live in Bickley, near Bromley.
How long have you been playing the piano, and do you play other instruments as well?
I have been playing the piano and cello from when I was about 4 years old. I also play the organ and keyboards in general – and odd instruments which lie around my studio.
Tell us about how you started playing music.
Many of my family are musicians, and so I grew up around it. There was also science in the family, but music won me over!
How long have you been making piano music?
I have been writing as long as I remember, and from a young age found myself interested in writing down and structuring my own music, as well as enjoying performing and communicating it to others. Without sounding overly pretentious, the ability of music to communicate that which words cannot is a lifetime's fascination for me.
Tell us something about that moment you realized you could make songs yourself!
I was given an old tape recorder with a microphone. I initially used it to record pretend 'radio programmes' as a kid, but quickly became interested in recording music which I had written. My school also had a printer and binding machine, so I started printing my own schoolboy music as well!
What are your favorite artists in this "piano genre"?
I love listening to all kinds of performers and pianists. The ones which really speak to me make whichever music they are playing incredibly fresh and appealing, as if it has just been written.
Is there one song which you play over and over again as soon as you sit down by a piano?
Playing Bach at the keyboard is something I do to relax, and I absolutely love it. His music is contrapuntal as well as just vertical, and so it is a feast for ears, eyes and brain.
What rules (in making music) needs to be broken?
Music would be boring if everyone wrote in diatonic harmony all the time. That said, our musical ancestors have evolved music to the point it is at, and so to compose music today without at least being aware of the basic rules of harmony and counterpoint is not only disrespectful but limiting. This is why I believe we need to deliberately break the rules in our own ways to achieve an individual style, rather than totally disregard them. In the 1960s and 1970s musical composition went as far as it can ever go away from any sense of order, and I personally feel that human emotion can now be expressed in all kinds of creative ways through music without totally throwing out what came before.
How do you record your music?
I usually record large orchestras in central London in studios designed for the task, and smaller projects and overdubs at my home studio. I do sometimes mix myself, but tend to get mastering done elsewhere as a separate pair of ears (of a mastering engineer you trust) can help a record.
Whats your take on sampled instruments?
I use them, but usually write away from the computer first and record as many live instruments as the budget allows, because writing music straight into a sequencer to make samples sound good can be extremely limiting.
Anything else you want to share?
Every human on this earth should have access to music as it helps us communicate – cavemen knew this! It has an untold number of benefits to people's lives, despite governments' unwillingness to allocate enough funding for a proper musical education which should be a birthright.
I would add that writing music is an obsessive, tiring, wonderful, addictive, intense, fulfilling occupation.
The last question is asked by my 5 year old son:
Where do all your songs come from?
God knows…I don't think any creative really knows!
Thank you very much for this interview Thomas!
For more information, please check out any of the following links:
Website / Spotify
Behind the piano: Philip G Anderson
Behind the piano: Barry Hudson-Taylor
Behind the piano: Christopher Colaço


FreeBets.ws is a directory of free bets for punters from UK and Ireland. You can find all free bets from various bookmakers here, and we have a unique search tool that allows you to find the perfect bookmaker offers based on your search criteria such as type of offer, deposit option, etc.
Click on one of the options and then click Filter. Only the offers related to that option will appear.
You can also search for multiple options! For example, you may search for Enhanced Odds offers at bookmakers that accept Western Union. And you'd get this.
You can also achieve the same result if you first choose your desired offer type in the top menu, and then further refine your search by other parameters and then click Filter.


You don't have to travel far to enjoy the best of Michigan's great outdoors, with Kensington Metropark, one of the most popular parks in Michigan. Kensington's 4,481 sprawling acres of wooded, hilly terrain surrounds beautiful Kent Lake, and is home to an abundance of wildlife and waterfowl. Kensington Metropark offers a multitude of recreational activities throughout the year, from biking and boating to cross-country skiing and tobogganing. In addition to striking sunrises and sunsets, 1,200-acre Kent Lake offers plenty of fun activities: swim at Martindale or Maple beaches, get soaked at the Splash 'n' Blast, or just spend the day fishing, boating or picnicking along the water. Take a tour of the lake aboard the Island Queen II in the summer and fall. Or, enjoy a winter day ice-fishing or skating on frozen lake waters.
This first-class recreational area also features an 18-hole regulation golf course, 27-hole disc course, nature center, farm center, beautiful picnic areas and scenic hiking and biking trails for hours of enjoyment. With two and a half million visitors every year Kensington Metropark is a favorite place to enjoy Michigan's natural treasures.
Michigan's $10 Recreation Passport, purchased for your Michigan license plate, allows entry into all Michigan State Parks. The Huron-Clinton Metroparks is a separate regional park system, and requires its own entry permit.
*The 2012 Joint Permit for Huron-Clinton Metroparks and Oakland County Parks offers entry to all parks within both park systems.


text:
Q: Preparing graphics in WPF for HiDPI ("retina") screens I'm developing an application with WPF, xaml etc.
I want to make sure that my icons are ready for HiDPI screens, so I will create them at 2x their original size.
But I don't know how to use these higher resolution versions the best in XAML then. Do I manually scale them down to 50%, and if the user has his DPI set to 200%, will it use the original version of the icon? Or will it blow up the resized version?
Or, is it just necessary to put the DPI of the image internally to 192 dpi? WPF seems sensitive for that kind of things.

redpajama_set_name: RedPajamaStackExchange


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Toriart Anri Wood Music Box, Plays "New York, New York"
Victorian Diecut EASTER BUNNY Le Petit Museum WOOD MUSIC BOX "Peter Cottontail"


text:
In the first of two articles, Digby Anderson looks at Christian friendship as a safeguard against loneliness
Practising Christians are an increasingly small minority of the population in England; fully practising Catholics a minority of that minority and Anglican Catholics an even tinier minority. Anglican-Catholicism is a lonely business. 'Lonely' means not only being alone but being aware of it. We are not just socially but intellectually and morally lonely. We know that few people believe and practice what we do.
Dangers of isolation
Intellectual loneliness is dangerous. Some people respond to it by giving up their beliefs, unable to persist with something which is so unusual. After all, what is so unusual is abnormal, eccentric, bizarre. Another reaction, the opposite, is to throw up the barricades and have nothing to do with non-believers. It is ironic that minority believers who desperately need more members often pursue a purity of belief that further isolates them. Look at the fissiparous tendencies of Trotskyists or some continuing churches.
More often lonely believers seek a rapprochement with the majority by translating their beliefs into the language of the majority. Thus Christians say they still believe in sin but call it maladjustment. Too late they discover that in translating their ideas they have lost them.
Social scientists would diagnose us as suffering from cognitive dissonance and being in a cognitive minority. But there's a reason to stick with the less pretentious loneliness! 'Cognitive minority' suggests the answer lies in altering the relative numbers of the minority and majority, whereas there is a much simpler answer to loneliness: friendship. If you want to persist in Anglican-Catholicism without going fanatical or barmy, cultivate a friendship with one or two similarly believing people.
Listen to Edmund Burke in his Letter to the Sheriffs of Bristol: 'The only method that has ever been found effectual to preserve any man against the corruption of nature and example, is an habit of life and communication of councils with the most virtuous and public-spirited men.' It is dangerous to attempt to live the moral and religious life alone. Such a person, Burke says, 'must be either an angel or a devil.' 'Good men [must] cultivate friendships.'
Sharing moral purpose
To be sure the friendship we are called to is not what many modern persons mean by that word. It is not based on Australian lager, shared gossip or flattery. It is classical, high friendship, two persons sharing moral purpose, pursuing the good together. Far from flattery, the friend is one of the few persons who will tell you the truth about yourself.
Friendship is based on honesty, sincerity, loyalty and permanence of character. Jonathan loved David as he loved his own soul! Coleridge wrote of his friendship with Southey, 'our pursuits were similar, our final aspirations similar ...he was a far better man.' Newman called friendship a special test of our virtue!
There is so much in our tradition on the importance of friendship. Was not St John Our Lord's 'bosom friend'? Were not Judas' betrayal and Peter's denial the more shocking because they were the betrayals of friendship? Aristotle thought friendship the best thing in the world.
Euripides thought 'one loyal friend is worth 10,000 relatives.' Augustine was devastated by the death of a friend, All that we had done together was now a grim ordeal without him.'
Most eloquent was St Gregory Nazianzan about his friend St Basil: 'We were all in all to each other, sharing the same roof, the same table, the same sentiments, our eyes fixed on one goal...we seemed to have a single soul animating two bodies.'
Writings of the saints
There's St Theresa of Avila, St Aelred of Rievaulx and St Ambrose too. St Jerome praised friendship through letter writing. C.S. Lewis wrote of friendship that friends think they have chosen each other - not so: A secret master of ceremonies has been at work. Christ who once said to his disciples, 'Ye have not chosen me, but I have chosen you^ can truly say to every group of Christian friends, 'You have not chosen one another but I have chosen you for one another.' At this feast let us not reckon without our Host.'
The point of so many quotations is not rhetorical. It is that friendship was once considered a great thing by great civilizations and by great saints in the Church. It is now all but forgotten. The Church says little about it and nothing of much worth. Indeed the last Christian writer to take it seriously was C.S. Lewis.
Christians, I'm sure, have friends still, and some of these friends are Christian. A few, perhaps, of the friendships are Christian friendships. But we need to recover the ideal of Christian friendship and have public acknowledgement of its importance in the church. It is an arena for the practice of central Christian virtues, an essential building block of any Christian community and the chief safeguard against apostasy, fanaticism and madness in this otherwise lonely Catholic life.
Digby Andersons book,
Losing Friends,
is published by
The Social Affairs Unit


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