text:
\section{\label{sec:intro}Introduction}
Methods for the integrative analysis of multiple datasets are becoming increasingly important. This is especially true in genetics and genomics, where petabytes of public data are readily available for integrative analysis \citep{richardson2016statistical, ritchie2015methods}. For example, \citet{pickrell2016detection} analyzed summary statistics from genome-wide association studies of 42 human traits and found that multiple traits were influenced by several hundred common genetic variants. In a cross-species example, \citet{shpigler2017deep} combined results from a honey bee gene expression study with a database of autism-associated genetic variants and found evidence for evolutionary conservation of genes associated with both honey bee sociality and human autism spectrum disorder. Comparing and contrasting existing data, or combining them with new data, can lead to novel insights that would have been difficult or impossible to uncover with a single dataset alone \citep{tseng2015integrating}.

Integrative analysis strategies can take many forms, and one particularly common implementation is to leverage side information from one or several auxiliary studies for the purpose of improving the analysis of some primary dataset of interest. Examples abound in the multiple testing literature, where methods such as $p$-value weighting and false discovery rate regression incorporate auxiliary information to improve the power to detect true signals in a primary dataset \citep{genovese2006false, ramdas2017unified}. In the genomic risk prediction literature, \citet{hu2017joint} and \citet{zhao2017integrative} showed that summary statistics from previously conducted genome-wide association studies can be used to improve the performance of polygenic risk scores.

Growing interest in these ideas gives rise to an important statistical question: what is the best way to leverage side information? This paper studies this question in a simple but nontrivial problem: the simultaneous estimation of a vector of normal means. The classical version of this problem considers a sequence of independent $X_{i1} \sim N(\theta_{i1}, \sigma_1^2)$ for $i = 1, \ldots, n$ with known $\sigma_1^2$, where the goal is to estimate the $\theta_{i1}$ \citep{johnstone2017gaussian}. The integrative version, studied here, investigates how a auxiliary sequence of Gaussian random variables can be used to improve estimation of the means $\theta_{i1}$ of the primary Gaussian sequence.

This classical Gaussian sequence model is simplistic, but studying data integration in this setting is nevertheless instructive. First, the model is still important for many applications \citep{cai2012minimax, johnstone2017gaussian}. Second, more accurate estimation of the mean vector has immediate implications for high-dimensional classification in genomics \citep{greenshtein2009application}, which will be demonstrated in Section~\ref{sec:data}. Finally, this simple problem can reveal general statistical phenomena that arise in integrative data analysis. More complicated variations of the Gaussian sequence model have been studied, for example involving unknown variances that differ across different indices $i$; see Section \ref{sec:previous}. Extensions of the present work to these more realistic settings are important directions for future work.

Section~\ref{sec:problem} formalizes this integrative estimation task as a compound decision problem and summarizes previous related work. The optimal way to leverage side information is derived in Section~\ref{sec:oracle}, which presents an oracle integrative decision rule that achieves the best risk within a certain class of estimators. This section also introduces a regularized version of the oracle rule that has the same asymptotic risk. A data-driven estimate of this regularized oracle rule is introduced in Section~\ref{sec:data-driven}, and is shown to asymptotically achieve the optimal risk. Its good performance is illustrated in simulations in Section~\ref{sec:sims} and in two genomic risk prediction problems in Section~\ref{sec:data}. A discussion is presented in Section~\ref{sec:disc} and additional simulations and proofs can be found in the Appendix.

\section{\label{sec:problem}Normal means problem with side information}
\subsection{\label{sec:statement}Problem statement}
As in the classical Gaussian sequence problem, consider a sequence of independent $X_{i1} \sim N(\theta_{i1}, \sigma_1^2)$ for $i = 1, \ldots, n$, with $\sigma_1^2$ known. The side information problem studied in this paper further supposes that a second sequence of independent $X_{i2} \sim N(\theta_{i2}, \sigma_2^2), i = 1, \ldots, n$ is available, with $\sigma_2^2$ known. The goal is to estimate the $\theta_{i1}$, just as in the classical problem, but here both the $X_{i1}$ and the $X_{i2}$ can be used for estimation. In this sense, the $X_{i1}$ play the role of a primary dataset, and the $X_{i2}$ act as side information from an auxiliary dataset. This paper assumes that the $X_{i1}$ are independent of the $X_{i2}$ for each $i$, though extensions to dependent $X_{i1}$ and $X_{i2}$ are discussed in Section \ref{sec:disc}.

This formulation is motivated by applications in integrative genomics. The indices $i$ represent different genomic features, such as different genes, and the $X_{i1}$ and $X_{i2}$ represent different measurements on feature $i$ from different studies. For example, in the genomics classification problem described in Section \ref{sec:data}, each $X_{i1}$ estimates a classifier parameter $\theta_{i1}$ corresponding to the $i$th gene from a primary study of interest, and each $X_{i2}$ is the $Z$-score for the $i$th gene reported by an auxiliary study of a related phenotype. The goal is to improve classification accuracy in the primary study by leveraging both $X_{i1}$ and $X_{i2}$ to better estimate the $\theta_{i1}$.

In the above example, the $X_{i1}$ and $X_{i2}$ are paired for each $i$, as both correspond to the same genomic feature. The informativeness of this pairing is crucial for the good performance of data integration. For example, because the phenotypes considered by the two studies in Section \ref{sec:data} are related, genes with significant $Z$-scores in the auxiliary study are also likely to be important features for classification in the primary study, so combining the studies is likely to be fruitful. In contrast, if the phenotypes were unrelated, $X_{i2}$ would likely not be informative about $\theta_{i1}$. The challenge is to develop an estimation procedure that can make optimal use of $X_{i2}$, incorporating them when appropriate and discarding them otherwise. This is addressed by the method proposed in this paper.

To more formally state the problem, define $\mathbf{X}_{\cdot d} = (X_{1d}, \ldots, X_{nd})$, $\boldsymbol{\theta}_{\cdot d} = (\theta_{1d}, \ldots, \theta_{nd})$ for $d = 1, 2$, and $\boldsymbol{\theta} = (\boldsymbol{\theta}_{\cdot 1}, \boldsymbol{\theta}_{\cdot 2})$. Then the normal means problem with side information is to find a decision rule $\boldsymbol{\delta}(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = \{\delta_1(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}), \ldots, \delta_n(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2})\} : \mathbb{R}^{2n} \rightarrow \mathbb{R}^n$ that minimizes the risk function
\begin{equation}
  \label{eq:risk}
  R_n(\boldsymbol{\theta}, \boldsymbol{\delta}) = \frac{1}{n} \sum_{i = 1}^n E [\{\theta_{i1} - \delta_i(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2})\}^2]
\end{equation}
over some class of decision rules. An important class, namely the class of separable estimators, will be considered in this paper and is discussed in Section~\ref{sec:oracle}. This paper adopts the frequentist framework where the $\boldsymbol{\theta}_{\cdot d}$ are fixed nonrandom constants. The auxiliary data are thus statistically independent of the primary data, and it is interesting that they can still provide useful information for estimating $\boldsymbol{\theta}_{\cdot 1}$.

To illustrate the complexities of this problem, first suppose that it were known that $\theta_{i2} = \theta_{i1}$ for all $i = 1, \ldots, n$ and that $\sigma_1 = \sigma_2$. The best way to integrate the auxiliary dataset would clearly be to apply existing optimal estimation methods for a single Gaussian sequence to the sequence of averaged observations $(X_{i1} + X_{i2}) / 2$. Next consider a slightly more complicated setting: $\theta_{i2} = \theta_{i1}$ for all but one $i$, but the $i$ for which $\theta_{i2} \ne \theta_{i1}$ is unknown. The auxiliary sequence is clearly still informative for estimating the $\boldsymbol{\theta}_{\cdot 1}$, but how it should be used is no longer obvious. Finally, consider an even more complicated scenario where $\theta_{i2} = h(\theta_{i1}) + e_i$ for some unknown function $h(t)$, where the $e_i$ are unknown perturbations that exhibit no patterns with respect to $\theta_{i1}$. If the magnitudes of the $e_i$ are small relative to the $\theta_{i1}$, $\mathbf{X}_{\cdot 2}$ should still be useful when estimating $\boldsymbol{\theta}_{\cdot 1}$, but it is even less clear how to optimally integrate it into the estimation procedure. This paper provides one approach.

\subsection{\label{sec:previous}Previous work}
The classical normal means estimation problem without side information, which aims to minimize the risk function \eqref{eq:risk} using decision rules that can depend only on $\mathbf{X}_{\cdot 1}$ and not $\mathbf{X}_{\cdot 2}$, has inspired an enormous literature \citep{johnstone2017gaussian}. \citet{stein1956inadmissibility} found that the maximum likelihood estimator $\delta_i(\mathbf{X}_{\cdot 1}) = X_{i1}$ is inadmissible, and since then research has focused on finding alternative estimators with better risk properties. Several different but intimately related perspectives on this problem have been developed.

The shrinkage perspective is exemplified by the James-Stein estimator \citep{james1961estimation, stigler19901988}, which estimates $\theta_{i1}$ by scaling $X_{i1}$ towards zero. The empirical Bayes perspective \citep{robbins1985empirical} treats the $\theta_{i1}$ as random draws from a prior distribution, uses the $X_{i1}$ to estimate any unknown parameters in the prior, then estimates each $\theta_{i1}$ by its posterior mean conditional on $X_{i1}$. \citet{efron1973stein} showed that the James-Stein estimator is an empirical Bayes estimator assuming a normal prior for the $\theta_{i1}$. The compound decision perspective \citep{robbins1951asymptotically, zhang1997empirical} treats the $\theta_{i1}$ as nonrandom constants and directly derives the decision rule that minimizes the risk. Under certain conditions, the optimal solution from this perspective is closely related to nonparametric empirical Bayes estimators \citep{brown2009nonparametric, jiang2009general, zhang2003compound}.

More complicated versions of the classical normal means problem have also been intensely studied. For example, specialized methods have been developed for estimating sparse normal means, where most of the $\theta_{i1}$ are assumed to equal zero \citep{castillo2012needles, donoho1994ideal, donoho1995adapting, martin2014asymptotically}. Heteroscedastic normal sequences, where the $X_{i1}$ can have different variances for different indices $i$, have also been considered, both when the variances are known \citep{fu2019nonparametric, tan2016steinized, weinstein2018group, xie2012sure, zhang2017empirical} and when they are unknown but estimates are available \citep{feng2018approximate, gu2017empirical, jing2016sure}.

So far, however, most work on the normal means problem and its variants has considered only a single sequence of observations $X_{i1}$, and it appears that the side information problem \eqref{eq:risk} has not yet been widely studied. \citet{jiang2010empirical}, \citet{cohen2013empirical}, \citet{tan2016steinized}, and \citet{kou2017optimal} proposed methods that can integrate $X_{i2}$, but these essentially require knowledge of the nature of the relationship between $\theta_{i1}$ and $X_{i2}$, and may not work well when this relationship is misspecified. \citet{banerjee2018adaptive} studied the side information problem, but only for sparse $\boldsymbol{\theta}_{\cdot 1}$. Very recently \citet{saha2017nonparametric} and \citet{koudstaal2018multiple} considered two or more Gaussian sequences, but minimized the risk of estimating the means of all of the sequences, rather than the means of just one of them as in \eqref{eq:risk}. In contrast to existing work, this paper studies the optimal use of $\mathbf{X}_{\cdot 1}$ and $\mathbf{X}_{\cdot 2}$ for estimating possibly non-sparse $\boldsymbol{\theta}_{\cdot 1}$.

\section{\label{sec:oracle}Oracle integrative separable rules}
Without any restrictions, the optimal decision rule is simply $\delta_i(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = \theta_{i1}$, which is not useful because the performance of this rule cannot realistically be achieved using the observed data alone. Instead, this paper only considers rules in the class
\begin{equation}
  \label{eq:S}
  \mathcal{S} = \{ \boldsymbol{\delta}(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) : \delta_i(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = f(X_{i1}, X_{i2})\},
\end{equation}
where $f$ is some fixed real-valued function that is applied to each pair $(X_{i1}, X_{i2})$ in order to estimate $\theta_{i1}$. In other words, the estimate of $\theta_{i1}$ is calculated by applying $f(x_1, x_2)$ to only the $i$th pair of observations $(X_{i1}, X_{i2})$, and $f(x_1, x_2)$ cannot vary with $i$.

Rules in $\mathcal{S}$, called ``separable'' rules, are appealing because of their simplicity and have been extensively studied \citep{brown2009nonparametric, cai2012minimax, robbins1951asymptotically, zhang2003compound}. The maximum likelihood estimator $\boldsymbol{\delta}(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = X_{i1}$ belongs to $\mathcal{S}$, and the James-Stein estimator approximates the optimal separable rule that is linear in $X_{i1}$ \citep{jiang2009general}. The minimum risk among all separable estimators has been shown to be asymptotically equivalent, in a certain sense, to the minimum achievable risk over the larger class of permutation invariant estimators \citep{greenshtein2009asymptotic}.

The following proposition describes the oracle optimal integrative rule in $\mathcal{S}$ for estimating $\boldsymbol{\theta}_{\cdot 1}$, which will perform no worse than any separable rule that relies only on $\mathbf{X}_{\cdot 1}$. It is a direct consequence of the fundamental theorem of compound decision problems \citep{robbins1951asymptotically, jiang2009general}. Let $\phi(x)$ denote the standard normal density and define
\begin{equation}
  \label{eq:density}
  \begin{aligned}
    p(x_1, x_2; t_1, t_2)
    =\,&
    \frac{1}{\sigma_1} \phi \left( \frac{x_1 - t_1}{\sigma_1} \right)
    \frac{1}{\sigma_2} \phi \left( \frac{x_2 - t_2}{\sigma_2} \right),\\
    p_i^0(x_1, x_2)
    =\,&
    p(x_1, x_2; \theta_{i1}, \theta_{i2}),
  \end{aligned}
\end{equation}
so that the density of $(X_{i1}, X_{i2})$ can be abbreviated by $p_i^0(x_1, x_2)$. As mentioned in the problem statement in Section \ref{sec:statement}, this paper assumes that the $X_{i1}$ and $X_{i2}$ are independent, but the following result is easily extended to settings where $X_{i1}$ and $X_{i2}$ are correlated; see Section \ref{sec:disc}.

\begin{proposition}
  \label{prop:oracle}
  Define the decision rule $\boldsymbol{\delta}^\star = (\delta^\star_1, \ldots, \delta^\star_n)$ where $\delta^\star_i(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = f^\star(X_{i1}, X_{i2})$ and
  \begin{equation}
    \label{eq:oracle}
    f^\star(x_1, x_2)
    =
    \frac{
      \sum_{j = 1}^n \theta_{j1} p_j^0(x_1, x_2)
    }{
      \sum_{j = 1}^n p_j^0(x_1, x_2)
    }.
  \end{equation}
  Then $R_n(\boldsymbol{\theta}, \boldsymbol{\delta}) \geq R_n(\boldsymbol{\theta}, \boldsymbol{\delta}^\star)$ for any $\boldsymbol{\delta} \in \mathcal{S}$ \eqref{eq:S} for $R_n(\boldsymbol{\theta}, \boldsymbol{\delta})$ in \eqref{eq:risk}.
\end{proposition}

The oracle rule $\boldsymbol{\delta}^\star$ also has a useful interpretation as a Bayes rule. If the $\theta_{i1}$ are viewed as independent draws from the discrete prior distribution
\begin{equation}
  \label{eq:G}
  G_n(t_1, t_2) = \frac{1}{n} \sum_{i = 1}^n I(\theta_{i1} \leq t_1, \theta_{i2} \leq t_2),
\end{equation}
then the posterior expectation $E(\theta_{i1} \mid X_{i1}, X_{i2})$ of $\theta_{i1}$ is exactly equal to \eqref{eq:oracle}. This is an example of the close connection between compound decision problems and nonparametric empirical Bayes procedures. The dependence between $\theta_{i1}$ and $\theta_{i2}$ under $G_n$ quantifies the amount of information that can be borrowed from $X_{i2}$.

While appealing, this Bayesian interpretation is not necessary for Proposition \ref{prop:oracle}, which holds for fixed and constant $\boldsymbol{\theta}_{\cdot 1}$ and $\boldsymbol{\theta}_{\cdot 2}$. Interestingly, under this frequentist setting Proposition \ref{prop:oracle} shows that $\mathbf{X}_{\cdot 2}$ can improve the estimation of $\boldsymbol{\theta}_{\cdot 1}$ even though $\mathbf{X}_{\cdot 1}$ and $\mathbf{X}_{\cdot 2}$ are statistically independent, as long the sequences $\boldsymbol{\theta}_{\cdot 1}$ and $\boldsymbol{\theta}_{\cdot 2}$ are related in some sense. There need not be an obvious functional relationship between the two mean vectors.

The above view of side information is slightly different from that of existing frameworks. Previous methods \citep{jiang2010empirical, kou2017optimal, tan2016steinized} posit some functional relationship, typically linear, between $\theta_{i1}$ and the observed $X_{i2}$, rather than between $\theta_{i1}$ and the true mean $\theta_{i2}$. For example, \citet{kou2017optimal} assume that $\theta_{i1} = h(X_{i2}) + e_i$ for some error term $e_i$, where $h(x)$ must be known up to a finite-dimensional parameter. These methods treat the $X_{i2}$ as fixed, while the proposed framework acknowledges that the $X_{i2}$ are random variables. The difference between existing work and the present setting is akin to the difference between classical regression methods and those that take into account covariate measurement error.

\begin{figure}[h!]
  \centering
  \includegraphics[width = \textwidth]{estimators.pdf}\par
  \caption{\label{fig:estimators}Oracle estimators with and without side information for $n = 20$ pairs $(X_{i1}, X_{i2})$. Each curve plots the estimate of $\theta_{i1}$ as a function of $X_{i1}$. Each dot corresponds to an observed $X_{i1}$ along with its true mean $\theta_{i1}$.}
\end{figure}

Figure \ref{fig:estimators} illustrates the oracle rule $\boldsymbol{\delta}^\star$ \eqref{eq:oracle} and compares it to the best separable estimator that does not use $\mathbf{X}_{\cdot 2}$, which is the posterior expectation of $\theta_{i1}$ under the prior $G_n$ conditional only on $X_{i1}$ \citep{zhang2003compound}. In both panels, $\theta_{i1}$ was generated by drawing $n = 20$ values from a standard normal distribution. In the left panel, $\boldsymbol{\theta}_{\cdot 2} = \boldsymbol{\theta}_{\cdot 1}$, so $\mathbf{X}_{\cdot 2}$ was highly informative for $\boldsymbol{\theta}_{\cdot 1}$. Thus $f^\star(x_1, 3)$ gave the best estimates of $\theta_{i1}$ for large $X_{i1}$ and $f^\star(x_1, -3)$ was most accurate for small $X_{i1}$. In the right panel of Figure \ref{fig:estimators}, the $\theta_{i2}$ were generated from an independent standard normal so that $\mathbf{X}_{\cdot 2}$ was completely uninformative. In this non-informative setting, $\boldsymbol{\delta}^\star$ may not have the same performance as the optimal non-integrative separable rule for any given set of $\mathbf{X}_{\cdot 1}$ and $\mathbf{X}_{\cdot 2}$, but in expectation Proposition \ref{prop:oracle} guarantees that it will have equal or lower risk.

The oracle separable integrative rule $\boldsymbol{\delta}^\star$ described in \eqref{eq:oracle} cannot be implemented in practice because it requires knowing the true $(\theta_{i1}, \theta_{i2})$ up to permutation of the indices. Section~\ref{sec:data-driven} will introduce a data-driven rule that targets the performance of $\boldsymbol{\delta}^\star$, though for technical reasons it will be more convenient to target a regularized version of the oracle rule. This will be denoted by $\boldsymbol{\delta}^\star_\rho = (\delta^\star_{\rho 1}, \ldots, \delta^\star_{\rho n})$, with $\delta^\star_{\rho i}(\mathbf{X}_{\cdot 1}, \mathbf{X}_{\cdot 2}) = f^\star_\rho(X_{i1}, X_{i2})$ for
\begin{equation}
  \label{eq:regularized}
  f^\star_\rho(x_1, x_2)
  =
  x_1
  +
  \frac{
    \sum_{j = 1}^n (\theta_{j1} - x_1) p_j^0(x_1, x_2)
  }{
    \rho + \sum_{j = 1}^n p_j^0(x_1, x_2)
  }
\end{equation}
and $\rho$ a small positive constant that prevents the denominator from being too close to zero. Under some assumptions, $\boldsymbol{\delta}^\star_\rho$ will have the same asymptotic risk as the oracle $\boldsymbol{\delta}^\star$.

\begin{assumption}
  \label{a:theta}
  There exist positive constants $C$ and $\eta$ such that $\vert \theta_{id} \vert \leq C n^{1/4 - \eta}$ for $i = 1, \ldots, n$ and $d = 1, 2$.
\end{assumption}

\begin{theorem}
  \label{thm:regularized}
  Under Assumption \ref{a:theta}, $\lim_{n \rightarrow \infty} \{ R_n(\boldsymbol{\theta}, \boldsymbol{\delta}^\star_\rho) - R_n(\boldsymbol{\theta}, \boldsymbol{\delta}^\star) \} = 0$.
\end{theorem}

Assumption \ref{a:theta} determines how quickly the magnitudes of $\theta_{id}$ can grow. To put this rate into perspective, if the $\theta_{id}$ were random draws from a normal distribution, then $\max_i \vert \theta_{id} \vert$ would be $O(\log^{1/2} n)$ almost surely. Related assumptions, which essentially restrict how variable the $\theta_{id}$ can be, have been made in previous work on normal means estimation without side information. For example, \citet{xie2012sure} require $\lim n^{-1} \sum_i \theta_{i1}^2 < \infty$, and \citet{jiang2009general} and \citet{zhang2009generalized} control the rate of the $p$-th weak moment of the distribution function $n^{-1} \sum_i I(\theta_{i1} \leq t_1)$.

\section{\label{sec:data-driven}Data-driven separable estimator}
\subsection{\label{sec:npeb}Existing nonparametric empirical Bayes approach}
By Proposition \ref{prop:oracle} and Theorem \ref{thm:regularized}, the regularized oracle $\boldsymbol{\delta}^\star_\rho$ \eqref{eq:regularized} is asymptotically optimal within the class of separable estimators \eqref{eq:S}, but it cannot be implemented in practice. It therefore remains to develop a fully data-driven estimator for the $\theta_{i1}$. Two classes of approaches already exist. They have been termed $f$- and $g$-modeling \citep{efron2014two, efron2019bayes} and are based on nonparametric empirical Bayes principles that pretend that the $(\theta_{i1}, \theta_{i2})$ are random variables with prior distribution $G_n(t_1, t_2)$ \eqref{eq:G}.

In $f$-modeling, the oracle estimator \eqref{eq:oracle} would be re-expressed as
\[
f^\star(x_1, x_2)
=
x_1 + \frac{\tilde{p}'(x_1, x_2)}{\tilde{p}(x_1, x_2)},
\]
where $\tilde{p}'(x_1, x_2) = \partial \tilde{p} / \partial x_1$ and
\[
\tilde{p}(x_1, x_2) = \int p(x_1, x_2; t_1, t_2) dG_n(t_1, t_2)
\]
with $p(x_1, x_2; t_1, t_2)$ from \eqref{eq:density}. If the $(\theta_{i1}, \theta_{i2})$ were truly random, $\tilde{p}(x_1, x_2)$ could be interpreted as the marginal density of $(X_{i1}, X_{i2})$, and $\tilde{p}(x_1, x_2)$ and $\tilde{p}'(x_1, x_2)$ could be estimated nonparametrically using kernel density estimators. In $g$-modeling, the oracle estimator would be re-expressed as
\[
f^\star(x_1, x_2) = x_1 + \frac{\int (t_1 - x_1) p(x_1, x_2; t_1, t_2) dG_n(t_1, t_2)}{\int p(x_1, x_2; t_1, t_2) dG_n(t_1, t_2)},
\]
and if the $(\theta_{i1}, \theta_{i2})$ were truly random, a nonparametric estimate of $G_n(t_1, t_2)$ could be obtained by maximizing the marginal log-likelihood \citep{kiefer1956consistency}
\[
\argmax_G \prod_{i = 1}^n \int p(X_{i1}, X_{i2}; t_1, t_2) dG(t_1, t_2).
\]

Both $f$- and $g$-modeling have been used in normal means problems without side information, where they are asymptotically optimal even in the frequentist framework where the $\theta_{i1}$ and $\theta_{i2}$ are nonrandom \citep{brown2009nonparametric, feng2018approximate, fu2019nonparametric, jiang2009general, koenker2014gaussian, koenker2014convex, saha2017nonparametric, zhang2009generalized}. However, neither approach directly estimates the oracle decision rule, with $f$-modeling proceeding through the intermediate quantity $\tilde{p}(x_1, x_2)$ and $g$-modeling proceeding through $G_n(t_1, t_2)$.

\subsection{\label{sec:proposed}Proposed direct risk minimization approach}
This paper explores a more direct approach to estimating the oracle integrative separable classifier. Motivated by the regularized oracle \eqref{eq:regularized}, consider separable rules of the form $\boldsymbol{\delta}^t_\rho = (\delta^t_{\rho 1}, \ldots, \delta^t_{\rho n})$ with
\begin{equation}
  \label{eq:delta^t}
  \delta^{\mathbf{t}}_{\rho i}(x_1, x_2)
  =
  x_1
  +
  \frac{
    \sum_{j = 1}^n (t_{j1} - x_1) p(x_1, x_2; t_{j1}, t_{j2})
  }{
    \rho + \sum_{j = 1}^n p(x_1, x_2; t_{j1}, t_{j2})
  },
\end{equation}
for a given $\mathbf{t} = (t_{11}, \ldots, t_{n1}, t_{12}, \ldots, t_{n2})$. By Theorem \ref{thm:regularized}, the optimal $\mathbf{t}$ equals $(\theta_{11}, \ldots, \theta_{n1}, \theta_{12}, \ldots, \theta_{n2})$, but the $\theta_{id}$ are not known. The challenge is to choose a $\mathbf{t}$ in a data-driven fashion that can still asymptotically achieve the optimal risk.

Choosing $\mathbf{t}$ to minimize the risk \eqref{eq:risk} of $\boldsymbol{\delta}^{\mathbf{t}}_\rho$ \eqref{eq:delta^t} should give an estimator with good performance. However, calculating the risk requires knowing the true $\theta_{id}$. On the other hand, Stein's Lemma \citep{stein1981estimation} can be used to give an unbiased estimate of the true risk as a function only of $\mathbf{t}$:
\begin{equation}
  \label{eq:sure}
  \begin{aligned}
    & \textsc{sure}(\mathbf{t})\\
    =\,&
    \frac{2}{n} \sum_{i = 1}^n
    \frac{
      \sum_j (t_{j1} - X_{i1})^2 p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }{
      \rho + \sum_j p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }
    -
    \frac{2}{n} \sigma_1^2 \sum_{i = 1}^n
    \frac{
      \sum_j p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }{
      \rho + \sum_j p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }
    \,- \\
    &
    \frac{1}{n} \sum_{i = 1}^n
    \left\{
    \frac{
      \sum_j (t_{1j} - X_{i1}) p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }{
      \rho + \sum_j p(X_{i1}, X_{i2}; t_{j1}, t_{j2})
    }
    \right\}^2
    +
    \sigma_1^2.
  \end{aligned}
\end{equation}
The following theorem shows that $\textsc{sure}(\mathbf{t})$ is also a good approximation to the actual loss
\begin{equation}
  \label{eq:loss}
  \ell_n(\mathbf{t}) = \frac{1}{n} \sum_{i = 1}^n \{\theta_{i1} - \delta^{\mathbf{t}}_{\rho i}(X_{i1}, X_{i2})\}^2
\end{equation}
uniformly over the set
\begin{equation}
  \label{eq:T}
  \mathcal{T} = \{ \mathbf{t} : \vert t_{jd} \vert \leq C n^{1/4 - \eta}, j = 1, \ldots, n, d = 1, 2 \}.
\end{equation}
\begin{theorem}
  \label{thm:uniform}
  Under Assumption \ref{a:theta}, if $0 < \rho \leq 1$, then
  \[
  \lim_{n \rightarrow \infty} E \sup_{\mathbf{t} \in \mathcal{T}} \vert \textsc{sure}(\mathbf{t}) - \ell_n(\mathbf{t}) \vert = 0.
  \]
\end{theorem}

The tuning parameter $\mathbf{t}$ can now be chosen by minimizing this estimated risk, as a proxy for minimizing the unknown true risk. The proposed estimator is therefore defined to be
\begin{equation}
  \label{eq:proposed}
  \boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho \mbox{ as in \eqref{eq:delta^t} with } \hat{\mathbf{t}} = \argmin_{\mathbf{t} \in \mathcal{T}} \textsc{sure}(\mathbf{t}).
\end{equation}
This strategy of direct risk minimization is common in the compound decision literature \citep{jing2016sure, kou2017optimal, tan2016steinized, weinstein2018group, xie2012sure, xie2016optimal, zhang2017empirical}, but has not yet been used to approximate an optimal separable rule like \eqref{eq:regularized}. The following theorem shows that \eqref{eq:proposed} can asymptotically achieve the same the performance as the optimal separable decision rule.

\begin{theorem}
  \label{thm:opt_risk}
  Under the same conditions as Theorem \ref{thm:uniform}, $\lim_{n \rightarrow \infty} \{ E \ell_n(\hat{\mathbf{t}}) - R_n(\boldsymbol{\theta}, \boldsymbol{\delta}^\star) \} \leq 0$, where $E \ell_n(\hat{\mathbf{t}})$ is the risk of the proposed estimator $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ \eqref{eq:proposed}.
\end{theorem}

\subsection{\label{sec:implementation}Implementation}
The proposed estimator has been implemented in the R package \verb|cole|, available at \verb|github.com/sdzhao/cole|. In practice, the exact value of $\rho$ appears to make little difference, and $\rho = 0$ works well in most cases. When the range of the $X_{id}$ is very large or the variances $\sigma_d^2$ are very small, problem may arise when calculating $\textsc{sure}(\mathbf{t})$ due to numerical precision, in which case setting $\rho = 10^{-12}$ seems to be sufficient. Throughout this paper, the proposed method was implemented with $\rho = 0$.

Because the value of $C n^{1/4 - \eta}$ that defines the feasible set $\mathcal{T}$ \eqref{eq:T} is not known, in practice the minimization in \eqref{eq:proposed} can be performed over
\[
\hat{\mathcal{T}}
=
\prod_{i = 1}^n
[X_{i1} - M \sigma_1, X_{i1} + M \sigma_1] \times [X_{i2} - M \sigma_2, X_{i2} + M \sigma_2]
\]
for some sufficiently large positive constant $M$, so that $\hat{\mathcal{T}}$ contains $(\boldsymbol{\theta}_{\cdot 1}, \boldsymbol{\theta}_{\cdot 2})$ with probability $\Phi(-M)^n$, where $\Phi$ is the cumulative distribution function of a standard normal. By default, \verb|cole| uses $M = 5$, so that $\hat{\mathcal{T}}$ contains $(\boldsymbol{\theta}_{\cdot 1}, \boldsymbol{\theta}_{\cdot 2})$ with probability 0.99 when $n = 10,000$. Optimizing $\textsc{sure}(\mathbf{t})$ over $\hat{\mathcal{T}}$ is sensible because it is known from Theorem \ref{thm:regularized} that $E\{\textsc{sure}(\mathbf{t})\}$ achieves a global minimum at $t_{jd} = \theta_{jd}$. This method works well, but bridging the gap between the theoretical procedure and its practical implementation is an important direction for future work.

Minimizing $\textsc{sure}(\mathbf{t})$ is difficult because it is a nonconvex function. The implementation in \verb|cole| performs a simple coordinate descent. At initialization, $t_{id}$ is set to $X_{id}$, and at each iteration one $t_{id}$ is updated by optimizing over $K$ equally spaced candidates in $[X_{id} - M \sigma_d, X_{id} + M \sigma_d]$. By default, \verb|cole| uses $K = 10$, and all analyses in this paper use $K = 10$ unless otherwise stated. The coordinates of $\mathbf{t}$ are updated in the order $t_{11}, t_{21}, \ldots, t_{n1}, t_{12}, \ldots, t_{n2}$, and convergence is reached when all of the coordinates have been cycled through once without changing the value of $\textsc{sure}(\mathbf{t})$ by more than a small  $\epsilon$, which \verb|cole| sets to $10^{-5}$ by default.

\section{\label{sec:sims}Simulations}
\subsection{\label{sec:sims_classical}Normal means problem without side information}
The direct risk minimization approach proposed in this paper for estimating optimal separable decision rules appears to be novel in the compound decision literature. This section thus first illustrates how this idea performs in the classical normal means problem without side information. The optimal separable estimator and its corresponding unbiased risk estimate will look like \eqref{eq:oracle} and \eqref{eq:sure}, respectively, but with the density $p(x_1, x_2; t_1, t_2)$ replaced by $\phi\{(x_1 - t_1) / \sigma_1\} / \sigma_1$, where $\phi(x)$ is the standard normal density. Similar to \eqref{eq:proposed}, a data-driven estimator of the oracle rule can be obtained by minimizing the risk estimate over $\mathbf{t}_1$ using the coordinate descent algorithm described in Section \ref{sec:implementation}; this is available in the \verb|cole| package. Analogs of Theorems \ref{thm:regularized}--\ref{thm:opt_risk} can also be proved.

The direct estimator was compared to the $g$-modeling procedure of \citet{jiang2009general}, which is also asymptotically risk-optimal. One independent sequence $X_{i1}, i = 1, \ldots, 1,000$ was generated from N$(\theta_{i1}, 1)$, with the goal of estimating $\boldsymbol{\theta}_{\cdot 1}$ using only $\mathbf{X}_{\cdot 1}$. The $\theta_{i1}$ equaled either 0 or $\mu$ and the number of nonzero $\theta_{i1}$ equaled either 5, 50, or 500. Table \ref{tab:nonintegrative} displays the average total squared errors over 100 replications. Results for the estimator of \citet{jiang2009general} were taken directly from their Table 1, while the proposed estimator was implemented using a coordinate descent algorithm that optimized over $K = 50$ candidates for each $t_{1j}$. The results show that both estimators had almost identical performance.

\begin{table}[ht]
  \centering
  \begin{tabular}{l|cccc|cccc|cccc}
  \hline
  \hline
  \# nonzero & \multicolumn{4}{c}{5} & \multicolumn{4}{c}{50} & \multicolumn{4}{c}{500} \\
  $\mu$ & 3 & 4 & 5 & 7 & 3 & 4 & 5 & 7 & 3 & 4 & 5 & 7  \\
  \hline
  \hline

  GMLEB & 39 & 34 & 23 & 11 & 157 & 105 & 58 & 14 & 459 & 285 & 139 & 18\\
  Proposed & 37 & 32 & 21 & 11 & 158 & 110 & 56 & 14 & 460 & 289 & 133 & 21 \\
  \hline
  \end{tabular}
  \caption{\label{tab:nonintegrative}Average total squared errors for the classical normal means problem without side information. GMLEB: the $g$-modeling method of \citet{jiang2009general}.}
\end{table}

\subsection{Settings for normal means problem with side information}
The primary data $\mathbf{X}_{\cdot 1} = (X_{11}, \ldots, X_{n1})$ were generated in four different ways, for three dense and one sparse configuration of their means $\boldsymbol{\theta}_{\cdot 1}$. To generate dense $\boldsymbol{\theta}_{\cdot 1}$, values of $\theta_{i1}$ were independently drawn from either a N$(0,1)$, a Unif$(-2, 2)$, or an Exp$(1)$ distribution. To generate the sparse configuration, 10\% of the coordinates of $\boldsymbol{\theta}_{\cdot 1}$ were set to 1.5 and the rest were set to 0. The observed primary data were generated as $X_{i1} = \theta_{i1} + \epsilon_{i1}$, where the $\epsilon_{i1}$ were independently drawn from standard normal random variables. The $\theta_{i1}$ were fixed across all replications.

For each of these four settings, the auxiliary data $\mathbf{X}_{\cdot 2} = (X_{12}, \ldots, X_{n2})$ were generated in three different ways, to model different degrees of informativeness of $\boldsymbol{\theta}_{\cdot 2}$. First define $e_i$ to be independent draws from a Unif$(-4, 4)$. To generate strongly, weakly, and non-informative side information, $\theta_{i2}$ was set to be either $2 \theta_{i1}^2$, $\theta_{i1}^2 + e_i$, or $e_i$, respectively. The observed auxiliary data were generated as $X_{i2} = \theta_{i2} + \epsilon_{i2}$, where the $\epsilon_i$ were again independently drawn from standard normal random variables. The $\theta_{i2}$ were fixed across all replications.

The proposed integrative normal means estimator \eqref{eq:proposed} was compared to two existing approaches that can incorporate side information. One was the procedure of \citet{banerjee2018adaptive}. The other was estimator (1) of \citet{kou2017optimal}, defined as
\[
\frac{\lambda}{\lambda + \sigma_1^2} X_{i1} + \frac{\sigma_1^2}{\lambda + \sigma_1^2} h(X_{i2})
\]
for some function $h(x)$ known up to a finite number of parameters. These unknown parameters, as well as $\lambda$, are chosen by minimizing an unbiased estimate of the risk of this estimator. This estimator is motivated by the regression model $\theta_{i1} = h(X_{i2}) + e_i$ for some error terms $e_i$. However, it can be difficult to choose the correct regression function $h(x)$. For example, in some of the present simulation settings, the true relationship between the primary and auxiliary data is $\theta_{i2} = 2 \theta_{i1}^2 + e_i$, which is difficult to translate into a regression model of $\theta_{i1}$ on $X_{i2}$. When implementing the method of \citet{kou2017optimal}, these simulations used both the nonlinear model $\theta_{i1} = \beta_0 + \beta_1 \vert X_{i2} \vert^{1/2} + e_i$ and the linear model $\theta_{i1} = \beta_0 + \beta_1 X_{i2} + e_i$.

Finally, two additional estimators for $\boldsymbol{\theta}_{\cdot 1}$ were also implemented to provide performance baselines. The first was the oracle \eqref{eq:oracle}, which attains the lowest possible risk of any separable decision rule that incorporates side information. The second was the $g$-modeling method of \citet{jiang2009general}, which can asymptotically achieve the optimal risk of any separable rule that does not use side information.

\subsection{Results for normal means problem with side information}
Figure \ref{fig:sims} illustrates the average losses, over 200 simulations, achieved by the competing methods for N$(0,1)$, Unif$(-2, 2)$, Exp$(1)$, or sparse $\boldsymbol{\theta}_{\cdot 1}$ and non-informative, weakly informative, or strongly informative $\boldsymbol{\theta}_{\cdot 2}$. Comparing the performances of the oracle rule \eqref{eq:oracle} and the method of \citet{jiang2009general} shows that including auxiliary data does not degrade estimation accuracy asymptotically when $\boldsymbol{\theta}_{\cdot 2}$ is non-informative, and can greatly improve it when $\boldsymbol{\theta}_{\cdot 2}$ is informative.

\begin{figure}[h!]
  \centering

  \includegraphics[width = \textwidth]{figs1.pdf}

  \includegraphics[width = \textwidth]{figs2.pdf}

  \includegraphics[width = \textwidth]{figs3.pdf}

  \includegraphics[width = \textwidth, trim = 0 20 0 0]{figs4.pdf} \par
  \caption{\label{fig:sims}Average losses for four different configurations of $\boldsymbol{\theta}_{\cdot 1}$ and three degrees of informativeness of $\boldsymbol{\theta}_{\cdot 2}$. GMLEB: method of \citet{jiang2009general}; KY, linear: method of \citet{kou2017optimal} with model $\theta_{i1} = \beta_0 + \beta_1 X_{i2} + e_i$; KY, nonlinear: method of \citet{kou2017optimal} with model $\theta_{i1} = \beta_0 + \beta_1 \vert X_{i2} \vert^{1/2} + e_i$; ASUS: method of \citet{banerjee2018adaptive}.}
\end{figure}

The performance of the proposed data-driven estimator $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ \eqref{eq:proposed} indeed appeared to converge to the oracle performance as the number of observations $n$ increased, consistent with Theorem \ref{thm:opt_risk}. Unlike the oracle, however, incorporating non-informative $\mathbf{X}_{\cdot 2}$ in $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ resulted in worse performance compared to the other methods when $n$ was small. This is expected, as non-informative $X_{i2}$ add extra noise without decreasing bias, and the data-driven method requires enough samples to learn that the $X_{i2}$ are not useful. In contrast, $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ regained its competitiveness for larger $n$, and when the auxiliary $\mathbf{X}_{\cdot 2}$ were at least weakly informative, it frequently achieved the lowest risk among all methods. These results suggest that incorporating $\mathbf{X}_{\cdot 2}$ using the proposed method is highly effective when $\mathbf{X}_{\cdot 2}$ is informative, and does not do too much harm when it is not.

The proposed $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ was sometimes outperformed by the two different implementations of the procedure of \citet{kou2017optimal}, for example when the $\theta_{i1}$ were generated from Exp$(1)$. This may be because this setting was particularly difficult for the proposed method. Out of the four configurations of $\boldsymbol{\theta}_{\cdot 1}$, the maximum value of $\vert \theta_{i1} \vert$ was largest under the Exp$(1)$ configuration, and Assumption \ref{a:theta} makes it clear that restricting this maximum value is important for the good performance of $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$. On the other hand, when $n = 1,000$ $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ had essentially the same risk as the methods of \citet{kou2017optimal}, and for other configurations of $\theta_{i1}$, $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ could perform significantly better.

Finally, the proposed rule performed extremely well with sparse $\boldsymbol{\theta}_{\cdot 1}$, even though it was not designed for this scenario. When the auxiliary data were strongly informative, it achieved the lowest risks among all implemented methods when $n \geq 200$. It would be interesting to explore extensions of the proposed procedure to estimate sparse normal means.

\section{\label{sec:data}Data analysis}
High-dimensional classification is an important problem in genomics. \citet{shi2010microarray} studied the effectiveness of using gene expression microarray data to develop classification rules for various phenotypes. This section focuses on classification of two of these phenotypes: estrogen receptor status and treatment response status in breast cancer patients. The training and validation datasets they used are publicly available from the Gene Expression Omnibus \citep{edgar2002gene} under accession number GSE20194.

Integrating auxiliary data may help improve classification accuracy. \citet{wang2005gene} developed a gene expression signature for distant metastasis-free survival in estrogen receptor-positive and -negative breast cancer patients. It may be possible to leverage data from \citet{wang2005gene}, publicly available under accession number GSE2034, to more accurately classify the two outcomes from \citet{shi2010microarray}. However, it is not clear how to best integrate these auxiliary data.

The normal means estimation problem using side information, studied in this paper, provides one approach. \citet{greenshtein2009application} showed that minimizing the squared error risk in the normal means problem is closely connected to minimizing the misclassification rate in high dimensional classification. Let $\bar{G}_i^Y$ denote the average expression level of gene $i$ in across all training subjects in class $Y = 0, 1$ and $\hat{s}^Y_i$ denote its estimated standard deviation. \citet{greenshtein2009application} considered classifying an observed gene expression vector $(G_1, \ldots, G_n)$ using
\begin{equation}
  \label{eq:classifier}
  I\left( \sum_{i = 1}^n \hat{\theta}_i G_i / \hat{s}_i \geq c \right)
\end{equation}
for some cutoff $c$, where $\hat{s}_i = \{ (\hat{s}^1_i)^2 / n_1 + (\hat{s}^0_i)^2 / n_0 \}^{1/2}$ and $\hat{\theta}_i$ is an estimate of the expected value of $Z_i = (\bar{G}^1_i - \bar{G}^0_i) / \hat{s}_i$. They showed that using the $f$-modeling procedure of \citet{brown2009nonparametric} to obtain $\hat{\theta}_i$  can lead to more accurate classification compared to simply using $\hat{\theta}_i = Z_i$.

Combined with ideas in this paper, this framework leads to a natural integrative classifier. Let $X_{i1}$ equal $Z_i$ calculated for either estrogen receptor status or treatment response status from the \citet{shi2010microarray} study, and let $X_{i2}$ be the differential expression $Z$-score of the $i$th gene with respect to either estrogen receptor status or distant metastasis-free survival from the \citet{wang2005gene} study. Integrating $X_{i2}$ into the estimate $\hat{\theta}_{i1}$ should lead to more accurate classification.

This integrative classification was implemented using the proposed rule $\boldsymbol{\delta}^{\hat{\mathbf{t}}}_\rho$ \eqref{eq:proposed}, the method of \citet{kou2017optimal} using a model linear in $X_{i2}$, and the procedure of \citet{banerjee2018adaptive} for sparse normal means. These were compared to five classifiers that do not make use of auxiliary information: 1) the method of \citet{greenshtein2009application} but implemented using the $g$-modeling procedure of \citet{jiang2009general}, 2) the naive Bayes classifier, 3) logistic lasso using the R package \verb|glmnet| \citep{friedman2010regularization}, 4) random forest using the R package \verb|ranger| \citep{wright2017ranger}, and 5) the regularized optimal affine discriminant analysis of \citet{fan2012road} using the R package \verb|TULIP| \citep{pan2019tulip}. Tuning parameters for lasso and the method of \citet{fan2012road} were chosen using 10-fold cross-validation while random forest was run using default parameters.

The integrative, naive Bayes, and \citet{greenshtein2009application} classifiers all assume that the $X_{id}$ are independent across $i$. For these procedures, screening was thus first performed to ensure that the magnitude of the correlation between every pair of genes in the training data was small, similar to what was done in \citet{dicker2016high}. Specifically, genes were sorted from most to least significantly associated with the outcome in the training data, with $p$-values calculated using the R package \verb|limma| \citep{smyth2005limma}. Starting from the most significant gene, any other gene with correlation greater than 0.2 in magnitude was removed from the dataset. No screening was performed for lasso, random forest, or the method of \citet{fan2012road}.

\begin{figure}
  \includegraphics[width = \textwidth]{errors_02.pdf}
  \caption{\label{fig:data}Average misclassification errors for treatment response status or estrogen receptor (ER) status from \citet{shi2010microarray}. GP: method of \citet{greenshtein2009application}; KY: method of \citet{kou2017optimal}; ASUS: method of \citet{banerjee2018adaptive}; ROAD: method of \citet{fan2012road}. ``+ ER status/survival'': using differential expression with respect to either ER status or distant metastasis-free survival from \citet{wang2005gene} as auxiliary data.}
\end{figure}

Misclassification rates for estrogen receptor and treatment response status were assessed using the same training and testing datasets used in \citet{shi2010microarray}, and classification was also repeated after swapping the roles of the training and testing data. The averages of the two resulting misclassification rates for the different methods are displayed in Figure \ref{fig:data}.

The results suggest that integrative classification can be a useful strategy. Intuitively, the survival results from \citet{wang2005gene} should be most informative for predicting treatment response, while the ER status data from \citet{wang2005gene} should be most useful for predicting ER status. Indeed, the proposed integrative classifier using survival $Z$-scores to predict treatment response gave the lowest misclassification rate among all methods. The proposed method integrating ER status $Z$-scores to predict ER status performed better than every method except random forest and lasso.

\section{\label{sec:disc}Discussion}
This paper assumes that the primary data $\mathbf{X}_{\cdot 1}$ and the auxiliary data $\mathbf{X}_{\cdot 2}$ are statistically independent. However, in some practical settings $X_{i1}$ and $X_{i2}$ may be correlated for each $i$, for example if $\mathbf{X}_{\cdot 1}$ and $\mathbf{X}_{\cdot 2}$ arise from case-control studies with shared controls \citep{zaykin2010p}. The ideas proposed in this paper can be naturally extended to this correlated setting. Assuming $(X_{i1}, X_{i2})$ were bivariate normal with a known correlation, the oracle integrative rule would be similar to \eqref{eq:oracle} and is given in \eqref{eq:oracle_corr} in the Appendix. An asymptotically risk-optimal data-driven estimator could then be constructed by minimizing an unbiased risk estimate derived using Stein's lemma.

As pointed out by one referee, this setting is especially interesting because when $X_{i1}$ and $X_{i2}$ are correlated, the $\mathbf{X}_{\cdot 2}$ provides useful information for estimating $\boldsymbol{\theta}_{\cdot 1}$ even when $\boldsymbol{\theta}_{\cdot 2}$ and $\boldsymbol{\theta}_{\cdot 1}$ are completely unrelated. This is not true when $X_{i1}$ and $X_{i2}$ are independent. This is verified by Figure \ref{fig:sims_corr} in the Appendix, where the oracle integrative rule performed better than the oracle non-integrative rule when $\vert cor(X_{i1}, X_{i2}) \vert = 0.9$ even though $\boldsymbol{\theta}_{\cdot 2}$ was generated to be non-informative for $\boldsymbol{\theta}_{\cdot 1}$. Thus rules such as \eqref{eq:oracle} can therefore take full advantage of information about $\boldsymbol{\theta}_{\cdot 1}$ contained the auxiliary $\mathbf{X}_{\cdot 2}$, whether that information comes in the form of informative $\theta_{i2}$, correlated $X_{i2}$, or both.

This paper considered only a single sequence of auxiliary data, but it is straightforward to extend the proposed procedure to multiple auxiliary sequences. However, this would result in theoretical and computational difficulties, for given $D - 1$ auxiliary datasets, Assumption \ref{a:theta} would require $\vert \theta_{id} \vert \leq n^{1/(2D) - \eta}$ for $d = 1, \ldots, D$, and the proposed procedure would require optimizing over $Dn$ parameters. It would be of great interest to study whether there exists a convex surrogate of the unbiased risk estimate \eqref{eq:sure}. An alternative approach might be to use parametric or semiparametric methods, such as those proposed by \citet{kou2017optimal}, but to endow them with data-driven model selection capabilities.

It would be interesting to extend data integration ideas to other variants of the classical normal means problem, such as heteroscedastic sequences, sparse sequences, and non-normal observed data. It would also be interesting to consider broader applications of the compound decision framework beyond the simultaneous estimation of a mean vector, such as the integrative high-dimensional classification problem in Section \ref{sec:data}.

Though this paper studied the highly stylized problem of normal means estimation with side information, its results reveal several general principles of integrative analysis. First, auxiliary data can be useful even if they are statistically independent of, and have no clearly expressible functional relationship with, primary data. The two datasets need only be related in the sense discussed in Section \ref{sec:oracle}. Second, in principle, integrating auxiliary data can only help and not harm the primary analysis. This is because it is possible to learn from the data themselves the degree to which the auxiliary data are informative, and thus the degree to which they should influence inference in the primary data. Third, nonparametric methods, such as the proposed \eqref{eq:proposed}, can asymptotically achieve ideal performance.

\section*{Acknowledgments}
The author thanks Professors Jiaying Gu, Gourab Mukherjee, and Roger Koenker for helpful discussions. This research was supported in part by NSF grant DMS-1613005.

\bibliographystyle{abbrvnat}

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Greo takes us through the journey so far in "Thank You". He raps about how he started and where fate has taken him to and appreciates everyone who has supported his cause. The Choco Jay produced song samples Betty Wright's "Thank You for The Thing's You have done" had Greo waxing lyricism with storytelling and He pulls it off effortlessly. Listen Up.


text:
To Fight Zika, Texas Medicaid Will Pay for Mosquito Repellent
KUT 90.5 | By Edgar Walters | Texas Tribune
Published August 3, 2016 at 1:54 PM CDT
With federal funding to fight Zika stuck in Congress, and amid news that mosquito-to-human transmission of the virus has reached the continental U.S., Texas officials announced on Wednesday they would allow Medicaid to pay for mosquito repellent for women, in the hopes of preventing the disease.
The decision comes two months after the Obama administration first told states they could allow Medicaid, the federal-state health insurance program for the poor and disabled, to pay for Zika prevention services, including mosquito repellent and family planning.
Women who are pregnant or are between the ages of 10 and 45 are eligible so long as a doctor prescribes the mosquito repellent and the patient picks it up at a pharmacy. State officials say women should not schedule an office visit to receive mosquito repellent unless it is medically necessary.
"The best way to protect yourself from Zika is by avoiding mosquito bites," said Charles Smith, executive commissioner of the Texas Health and Human Services Commission, in a statement. "There is currently no vaccine to prevent the disease or drug to treat it, making it essential that people use mosquito repellent to protect women and unborn babies."
Eligible women can get up to two cans of repellent per month through Oct. 31, according to state officials.
Researchers have linked the Zika virus in pregnant women to microcephaly, a condition causing babies to be born with abnormally small brains and skulls. A small proportion of Zika infections may also trigger Guillain-Barré syndrome, an illness that targets the nervous system, scientists say.
State and local officials in Texas have long clamored for more funding to help combat the Zika virus, though the state's public health agency recently said there was no reason to panic over the spread of Zika to mosquito populations in South Florida.
In June, Gov. Greg Abbott asked federal health officials to provide Texas with about $11 million to help pay for Zika surveillance and laboratory infrastructure.
Despite urgings from health experts and the White House, Congress failed to pass a Zika funding bill before lawmakers left Washington for their summer recess earlier this month. One proposed bill would have allocated $1.1 billion to fighting the virus, but it stalled in the Senate under Democratic opposition. Republicans had included a part of the bill that would have stripped funding from Planned Parenthood.
This story was produced by .
Copyright 2020 KUT 90.5. To see more, visit .
Preparing for Zika in Texas
Edgar Walters | Texas Tribune
Edgar Walters is a reporter with the Texas Tribune.
See stories by Edgar Walters | Texas Tribune

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Herd Fest, a local music festival curated by Buffalo-based indie music website, buffaBLOG will take over numerous venues in Buffalo from June 16th – 19th showcasing the talents of local, regional, and national bands of all genres.
The lineup for the third annual music festival aims to gather all of Western New York's up-and-coming bands into one hustled weekend that showcases the city's growing music scene. This year's festival will include performances from 67 bands out of WNY and beyond in seven different venues including Record Theatre, Mohawk Place, DBGB's, Nietzsche's, Milkie's, Mr. Goodbar, and Dreamland.
The group has also developed a Herd Fest app for both Google and Apple IOS where festival goers can go to acquaint themselves with the schedule, view band profiles, and create their own schedule for personal customization and curation.
All shows are $5 each or all-access bracelets are available to purchase for $20 at buffaBLOG's Big Cartel page. View the full schedule/lineup is at herdfest.com and download the app here.


text:
Home > Quiz > SOC 32519
SOC 32519
Archaeology: Theories-- Methods-- and Practice 6th Edition
Colin Renfrew
Hearths are very easy to find and recognize in early prehistoric campsites.
Maya "emblem glyphs" can be linked to specific sites, which helps archaeologists to
understand the political organization of the region.
The idea that the origins of great civilizations lie in the large-scale irrigation of the
alluvial plains of great rivers is called the hydraulic hypothesis.
Although a subject of debate for many years, carbon dating performed in 1988 by
laboratories in Tucson, Oxford, and Zurich all agreed that the Shroud of Turin dated to
the time for Christ: the 1st century AD.
Unfortunately, the oral traditions of non-literate societies are just too unreliable to be of
any use at all to archaeologists.
According to Marshall Sahlins, balanced reciprocity takes place among those well
known to each other in a definite social context. Negative reciprocity operates between
strangers or those socially distant from one another.
A key point in archaeology is the idea of association. For example, if objects are found
in the same, sealed, archaeological deposit, it is probable that they were buried at the
In order to be of use, an archaeological typology must reflect the way ancient people
would have classified those same objects.
Although scientific techniques for characterization are very helpful to archaeological
studies of material sources, in most cases visual examination is sufficient.
Tertiary cultural deposits are those that have been completely removed from their
original context and have been reused.
Recent experimental work indicates that the hand axes of the Lower Paleolithic could
not serve as useful and versatile butchery tools, they could only have been used for
chopping wood.
The ancient Maya are known to have intentionally and irreversibly deformed their
children's skulls.
Although interesting historically, the Maya calendar was not very precise.
Diatoms are useful for identifying whether ancient bodies of water were fresh, brackish
or salt, but not for finer grained analytical questions, such as the level of water pollution
or where ancient shorelines were located.
The remains of Great Zimbabwe near Masvingo in modern Zimbabwe were established
by Phoenician sea-traders, and represent the most distant evidence for their settlement.
The study of ancient dung contributes to the debate over large animal extinctions.
The earliest known glass furnace was found at Tell el-Amarna, Egypt, dated to about
1350 bc.
Unlike other animals, which have useful hides to make leather out of, birds have only
been hunted for their meat.
Written records form probably the greatest single source of information for ancient state
societies; they directly relate knowledge that requires no interpretation.
Although pollen grains are of little use in studies of plant domestication, molecular
genetics are now beginning to make a contribution to the question.
However interesting they may be from a cultural point of view, neither art nor literature
can provide any evidence about what people ate in the past.
From a single leg bone, one can reasonably estimate the sex, age, height and general
size of an individual.
Large mammals are not generally good indicators of vegetation since herbivores thrive
in a wide range of environments and eat a variety of plants.
In some mammals, annual rings of cementum (a mineralized deposit) form around the
tooth roots below the gumline; a thin-section of these indicates alternating seasons of
stress or plenty.
Hammering copper makes the metal brittle, but by heating the metal, some pliability
can be restored.
The Rosetta Stone was discovered during Pitt Rivers' military expedition to India
Arid conditions encourage decay through the shortage of water, which allows
micro-organisms to thrive.
The earliest dated (using AMS) plant remains in Mesoamerica is maize and was found
at Guil Naquitz.
The reason that archaeology touches on issues of an ethical nature is that it is associated
with questions of identity.
Excavations at Pompeii during the 17th century consisted of some of the earliest, proper
methodological techniques applied to an ancient site.
Thanks to modern scientific methods, we know that no fake antiquities are being
displayed in the world's major museums.
Some graves at Khok Phanom Di contained white fabric shrouds, made of both beaten
bark and asbestos.
Ancient fecal matter often contains evidence for parasites, but hunter-gatherers who
lived in open country in temperate latitudes were frequently parasite free.
In the systems approach, when a change in the output stimulates further growth and
change, this is known as
a) negative feedback
b) positive feedback
c) induction
d) positivism
e) stimulation
In a new examination of John Yellen's ethnographic study of the ______________,
archaeologist Todd Whitelaw used computer modeling, producing a map based on
kinship distance that closely matched the actual original camp plan
a) !Kung San of the Kalahari Desert
b) Sherpa of the Tibetan Plateau
c) Inuit of Northern Greenland
d) Mopan Maya of souther Belize
e) Y nomam of Amazonian Brazilaa
In order to understand the sequence of creating the various elements in Upper
Paleolithic cave paintings, Alexander Marshack has used ____________ to see through
red ochre so that other pigments are visible
a) infrared film
b) thermoluminescence
c) ultraviolet light
d) fluxgate gradiometer
The research objectives of Kent Flannery and Joyce Marcus in the Oaxaca region
a) studying the effects of the Spanish conquest
b) elucidating stone tool development
c) verifying the pottery chronology for the area
d) the examination of the origins of agriculture
e) an investigation of the Secondary Products Revolution
The formation of a dynamic model to help understand change through time is called:
a) systems thinking
b) critical theory
c) morphogenesis
d) simulation
e) the multivariate approach
An approach to the analysis of decorative style that claims that patterns can be divided
into distinct groups or classes, and that these arrangements within a culture are not
random, is known as
a) symmetry analysis
b) systems thinking
c) synchronic analysis
d) cognitive analysis
e) all of the above
Non-artifactual organic and environmental remains, or ecofacts, include items such as
a) animal bones, plant remains, pottery fragments
b) soils, sediments, animal bones, plant remains
c) stone tools, weapons, pottery
d) soils sediments, animal bones, stone tools, weapons, pottery
The act of burial of human remains implies, at the very least,
a) some kind of respect of feeling for the deceased
b) a belief in a superior being
c) a desire to preserve the body for as long as possible
d) ancestor worship
__________ is difficult to recognize in the archaeological record: features such as
pottery decoration or differences in toolkits do not necessarily separate different cultural
a) Age
b) Genetic relation
c) Artistic talent
d) Ethnicity
e) Craft-making ability
Bernard Knapp and Michael Given designed a survey of northern Cyprus to investigate
the relationship between the production and distribution of agricultural and
metallurgical resources through time. Their survey strategy included
a) walking transects and entering the data daily into a GIS program
b) stratified areas according to vegetation, with random sampling squares in each
c) random squares, determined with a random numbers table, for intensive survey
d) nearly 100% survey coverage, with data entered into a GIS program
Scientific analysis of DNA sequences may indicate the relationships between members
of royal families and determine whether or not members of a dynasty practiced
a) incest
b) cannibalism
c) exogamy
d) polygamy
e) necrophagy
Relying on a single dominant explanatory factor, or "prime mover," to account for a
cultural change is known as a(n)
a) idealist explanation
b) multivariate explanation
c) hypothetico-deductive explanation
d) monocausal explanation
e) postprocessual explanation
At Ozette, the extraordinary preservation of organic materials such as wooden
containers, baskets, weaving equipment, and fishing and hunting equipment, was the
a) a glacier covering the village
b) a mudslide covering the settlement
c) a volcano burying the settlement in ash
d) a lava flow burying the village
Frequency seriation relies upon measuring
a) the width of tree growth rings
b) aligning radiocarbon dates from earliest to latest
c) changes in the proportional abundance of artifact attributes
d) pollen zone sequences
After 10,000 years ago, as the farming way of life took root in different parts of the
world, the introduction of new agricultural methods such as plowing, terracing, and
irrigation, as well as the use of poorer quality land, suggest that there was a gradual
a) intensification of food production
b) domination of state organizations
c) spread of egalitarian societies
d) intensification of non-religious behavior
e) increase in rainfall
Archaeology has been popularized through
a) books
b) television
c) the internet
e) a and b only
The process of cutting out a piece of bone from the skull, probably to alleviate pressure
on the brain caused by a skull fracture dates back at least 7000 years and is called:
a) boring
b) probing
c) trepanation
d) osteonization
e) scalping
Ethics is the science of:
a) emotion
b) feeling
c) conflict
d) cooperation
e) morals
Interpretations which stress that human actions are guided by beliefs and symbolic
concepts and that underlying these are structure of thought which find expression in
various forms are known as ___________
a) functionalist
b) Marxist
c) scientistic
d) processualist
e) structuralist
Chemical and infrared spectroscopic analyses of a yellowish residue found inside a
pottery jar from the Neolithic site of Hajji Firuz Tepe, Iran, was identified as tartaric
acid. Dating to about 5400"5000 BC, this represents the earliest evidence for
a) beer
b) a malt beverage
c) resinated wine
d) opium
e) milk
Proton-induced X-ray emission (PIXE) has similar analytical possibilities as scanning
electron microscope (SEM) analysis, but is better for analyses of
a) small areas of light materials (such as paper, pigments, or soldering of alloys)
b) samples in the form of a prepared or pressed powder or glass pellet
c) alloys and major components of pottery, faience, glass, and glazes
In North America, the influential anthropologist Julian Steward emphasized the
importance of environmental adaptation in cultural change. This approach was termed
a) environmentalism
b) cultural ecology
c) eco-archaeology
d) diffusionism
e) processualism
Due to ethnic tensions in ____________, the Old Bridge at Mostar, was constantly
shelled and was eventually destroyed
a) the former Rhodesia
b) the former Yugoslavia
c) the former Soviet Union
d) the former Zaire
e) the former Siam
Although the Spanish razed the standing structure of the Great Temple of the Aztecs in
1521, ______________ were preserved underneath
a) parts of Colonial Mexico City
b) Maya sites
c) earlier temples that were in that spot
d) Olmec houses
_________________ is a particularly useful method used in distinguishing between
different sources of amber
a) Electron probe microanalysis
b) Infrared absorption spectroscopy
c) Fall-off analysis
d) Geochemical analysis
e) X-ray diffraction analysis
Although palynology is very useful for the reconstruction of past vegetation in forested
regions, grass pollens are typically indistinguishable from one another. In grassy
environments an alternative is the study of the protective layer of the skin or epidermis
of leaves or blades of grass that retain the characteristic shapes of the underlying
epidermal cells known as
a) phytoliths
b) diatoms
c) fossil cuticles
d) plant DNA
e) coprolites
The cargo of over 350 copper four-handled ingots from the Uluburun shipwreck
appears to have originated in Cyprus. This was determined using
a) side scanning sonar
b) lead isotope analysis
c) infrared absorption spectroscopy
d) atomic absorption spectrometry
e) optical emission spectrometry
A disadvantage of some geophysical techniques based on magnetism is that their use in
urban areas may be subject to distortion resulting from power lines and metals in the
immediate vicinity. One method that can be used in such areas relies on soil moisture
and is known as
a) proton precession magnetometry
b) ground penetrating radar (GPR)
c) fluxgate magnetometry
d) electrical resistivity
e) metal detectors
Immanuel Wallerstein used the term __________ to designate an economic unit
articulated by trade networks that extended far beyond the boundaries of each
individual political unit, linking them together in a larger functioning unit
a) reciprocal emulation
b) kula network
c) globalization
d) potlatch system
e) world system
Historical accounts show that some of the Upper Mangrove Creek area's hill tops were
maker traveling routes and that people often came together across great distances to
________ as part of male initiation ceremonies
a) fight
b) trade
c) participate in tattooing
d) take wives
e) learn to hunt
by Colin Renfrew


text:
Biology's Ola Olapade Travels the Globe to Study the World of Microbes
When your research and teaching focuses on microbes, you can work anywhere—and Dr. Ola Olapade, professor of biology, has covered a lot of ground. His own research has taken him to the confluence of the Indian and Atlantic oceans in South Africa, Mayan ruins in Mexico and Belize (where he studied the impacts and contributions of microbes to the degradation and deterioration of stone monuments), Morocco, the Great Lakes and more.
Dr. Richard Youle, '74, Finds a Path Forward in the Treatment of Parkinson's
Dr. Richard Youle, '74, came to Albion College with the hope of teaching one day. But when the research bug bit him, he embarked on a career as an NIH researcher that has spanned more than four decades. This year, his research into Parkinson's Disease has earned him the prestigious Breakthrough Prize in Life Sciences.
Dr. Gregory Eastwood, '62, Examines End-of-Life Issues in New Book
Gregory Eastwood knew since he was young that he wanted to be a doctor. Now, a career that has included work as a gastroenterologist and a bioethicist has led him to write a book about the issues involved with the end of life. "I think it is important to be aware of the process of dying and how it plays out in contemporary America," he says.
Mackenzie Moore, '19, Has Studies, Time Management Down to a Science
Biology major and Britons volleyball hitter Mackenzie Moore, '19, will graduate in three years and pursue a master's degree in public health. She hopes her academic journey will include medical school. Moore, from Quincy, Mich., was dual-enrolled at Kellogg Community College for her final two years of high school and arrived at Albion with a semester and a half of college credit. "I'm excited for the future and the opportunities I will have," says Moore, a member of the Wilson Institute for Medicine and the Brown Honors Program.
Read more at GoBrits.com ...
Coral, Conch and College Credit: Annika Markovich, '19, Studies in Turks and Caicos
Molly Hancock, '18, Reflects on Her Summer Research Experience
First-Year Britons Runner Forges His Own Academic Path
Heitman's Horse Research Project Making a Lot of Scents
Biology's Dale Kennedy Honored by Wilson Ornithological Society
Global Medical Brigades Provides Hands-On Experience


Anthony Wile: Hello, Mark. It's been over a year since we last interviewed you. Give us an update on your recent work.
Mark Thornton: I was asked to debate the War on Drugs at Oxford Union in June of last year. I was happy to see such a political elitist institution interested in debating the War on Drugs. Maybe the new requirement for paying tuition is having a positive effect or it could be just part of the worldwide trend against the War on Drugs and political institutions in general.
Recently, I was asked to speak to the Atlanta Economic Club at the Federal Reserve Bank of Atlanta. I spoke about the Skyscraper Curse, a topic that I was criticized about by the Economist magazine. It's a topic that I have been talking about often these days as we look out at a pending Skyscraper Signal in Saudi Arabia and the collapse of the world economy.
Anthony Wile: We continue to follow the cannabis trend at The Daily Bell, and given your focus on "the economics of prohibition" over the years, would like to talk with you about several recent developments, especially in regard to two areas – the economics of prohibition and tariffs and price controls. When we talked with you in March 2014 you talked about "UNGASS 2016 and the Enormous Benefits of Legalizing Marijuana." Given the progress to date, what do you expect to come from UNGASS 2016? And how would you assess the working sessions being hosted around the world in preparation for the UN General Assembly Special Session (UNGASS) on Drugs, in April 2016?
Mark Thornton: Well, I have heard about the leaks and those leaks seem quite promising. I believe they basically confirmed what we hoped and expected from UNGASS 2016. It is complete reversal from 1998. In 1998 the report said that policy could achieve a drug-free world. That is an insane position to take, like no more storms, no more automobile accidents. It is the Pollyanna delusion. In 2016 we expect the UNGASS to issue a harm reduction recommendation where illicit drug consumption and possession would not be criminal acts that could lead to a prison sentence. This is a sane policy position. Prison sentences for illicit drug users are clearly harmful to the consumer (who now has a criminal record) and to government (cost of courts and prisons, etc.) and it does not deter consumption, or crime, or other negative consequences.
It is not the correct position of full legalization, but decriminalization has achieved good results in Portugal and Ireland has announced its intention to follow Portugal's lead. With UNGASS 2016 recommendations we can expect many more counties to follow suit with legalization of marijuana and decriminalization of the hard drugs. Many governments have already turned a blind eye towards illicit drug consumers in anticipation of UNGASS 2016 and are ready to enact harm reduction policies such as decriminalization, marijuana legalization, and needle exchange programs. So I am hopeful that we will see a windfall of policy liberalization in many countries and then political change in countries that do not follow suit.
There will be some pushback from countries like the US, Iran, and Saudi Arabia. The for-profit prison companies in the US will spend lots of money trying to defeat public opinion. Someone should keep an eye on them and their political and lobbying spending. If politicians knew that anyone accepting for-profit prison money would be outed as an ally of crony capitalism, it could be helpful.
Anthony Wile: What about individual response from member nations? Many have already implemented new regulations but I've stated my expectation that many changes will likely need to be made to at least parts of those regulatory regimes post-UNGASS, in order to come in line with treaty agreements.
Mark Thornton: Yes, it will be a messy process. Government always is a mess. Some will stick with the War on Drugs, some will adjust within the new framework, and others will venture outside the new guidelines and treaty agreements. As I mentioned, Ireland has already jumped the gun in preparing for full decriminalization. There will be experimentation and lots of reviewing of the evidence. I am sure there will be reform efforts that fail, but overall I am not worried about the long term because we have solid evidence from Portugal.
Medical marijuana legalization has been a big success. Recreational marijuana legalization has been a big success, rather than the disaster that was predicted, for Colorado and Washington. Marijuana and hemp are going to be big businesses in areas as diverse as medicine, textiles, chemicals, building materials, and fuel. They can be grown in diverse climates without herbicides, pesticides, fertilizers, and without much labor.
It is a "master ingredient" like petroleum, except it's renewable. So it's appealing to our progressive and environmentalist friends and it's the type of thing that will turn the current and future generations of entrepreneurs to create a giant leap in human progress. I think this potential will force regulators to innovate and fix problems in their systems.
Anthony Wile: I've also cautioned excited investors to take a deep breath and wait a bit for the market to shake out, for this very reason. Comment?
Mark Thornton: It is wise to caution investors. Any new product or ingredient experiences a great deal of chaos in the early years. I expect thousands of new and existing businesses to get involved in cannabis and hemp, everything from retailing recreational marijuana, to new cutting-edge medical applications, to competing with the petrol-chemical industry. Initially there could be hundreds of firms in each "product space," but over time each will come to be dominated by a small number of firms. Think about automobiles, soda drinks, and personal computers. They started out with more than a hundred entrants and ended up with just a couple of primary domestic producers. Therefore, it is very difficult to pick winners at this early stage. By the way, there are no Tweets with #marijuanaprofits yet!
Anthony Wile: An argument is made that conflates legalization efforts with promoting the "normalization" of drug use. We suggest, rather, that ending prohibition simply accepts human nature for what it is and enables any adult person seeking to utilize the cannabis plant for either medicinal or adult use recreational purposes in a safe and maturely managed manner. We suggest maturity prevail in this conversation. Recently, the pro-legalization campaign in Ohio featured "Buddie," a marijuana mascot on street corners and public areas in view of children and families, in essence promoting cannabis use. This is quite reminiscent of the "Joe Camel" tobacco advertising that was dangerous and offensive, long ago relegated to the dustbin of history as an unsuitable figure for big tobacco. This kind of promotion is certainly not consistent with public values. Perhaps that's part of why they lost the vote.
Would you agree that the main objective is to remove the black market from the equation but not to encourage drug use per se or to focus on expanding the usage of cannabis? Does anyone really think a costumed marijuana bud is going to fly?
Mark Thornton: Yes, I completely agree. I argue that the policy of government prohibition causes us all much harm and provides no benefits. Legalization creates opportunities for many economic benefits. However, when it comes to using marijuana for recreational use I am not an advocate. It may be less dangerous than alcohol, but that does not mean I recommend consuming it. It is a drug after all.
The situation in Ohio is important to discuss. The recreational legalization measure on the ballot was recently defeated and properly so. It would have changed the state constitution and put 10 monopoly marijuana growers into the constitution! These 10 monopolists were the ones that wrote and paid for the measure to be put on the ballot. They are the ones that paid for "Buddie" to be on the streets of Ohio. Naturally, crony capitalist/monopolists want to encourage consumption of their product. It is just shameful.
It is important to note that the defeated ballot measure was opposed by many advocates of legalization. Another measure to prevent any monopoly from being written into the state constitution was also passed by the voters. Hopefully, legalization will come in the next round of voting in Ohio. Overall, I consider the ballot results a great victory.
PLEASE CONTINUE READING THIS INTERVIEW AT LINK BELOW!


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Home Blogs Kurt's Hikes Picture Postcards from Down Under: New Zealand, Part 2
Mountains on the Other Side of the World: New Zealand, Part 1
Parakeets, Pigeons and a Volcanic Island: New Zealand, Part 3
Picture Postcards from Down Under: New Zealand, Part 2
by Kurt Rheinheimer
Lake Matheson
Gail's best shot of the trip? It's over Lake Matheson, looking toward Mt. Cook and Mt. Tasman.
One happy guy on a hike on the Coromandel Peninsula.
Matapouri walk
Part of the Matapouri walk was walking through a narrow pass to emerge - upon the Pacific!
Near Cape Reinga, on Te Werahi Beach, the Tasman Sea is COLD.
Ace guide Darrell
Ace guide Darrell (note flower in hiking stick) expounds on New Zealand plant life (no, that's not Kiwi marijuana).
Gail Rheinheimer in New Zealand
The Greatest Day Hiker Of Them All in a tight cold spot on Franz Joseph Glacier.
Hilli Maas-Barrett hams it up with helper Vanessa.
Just outside the Kai Kart on Stewart island, owner Hilli Maas-Barrett hams it up with helper Vanessa. (Kai is the Maori word for food; the fish and chips were great.)
Hikes, February 17-March 6, 2010. Up, down and around all three islands of New Zealand, in mostly short hikes. About 40 miles total: Part 2: April 24-27. Over our nearly three glorious weeks in New Zealand (a gift to me – AND The Greatest Day Hiker Of Them All – from Leisure publisher Richard Wells as a 25-years' service pin!), we walked every day – through cities, along beaches and exploring small towns, probably covering 75 miles or more over the 18 days. The brief descriptions to follow are only of the more defined hikes – through the bush, up volcanoes, onto a glacier and to some of the most amazing vistas in the whole wide world.
February 24. Guided hike in the bush of Abel Tasman National Park, along the coastline. About 5 miles.
Guide Darrell was the quintessential Kiwi neo-hippie – walking barefoot, sticking a flower in his wooden hiking pole – and also the quintessential Kiwi Plant Guy too, identifying and explaining most the trees and flowers along our leisurely walk. He also had several good lines, prime among which was spoken when two of the six people in our group expressed concerns over crossing a swinging bridge. "This one's good," Darrell said, "though there was one that fell down with 30 people on it in another part of the park. But this one's good; that one was built with nails on Friday."
We'd begun the day with Darrell with a kayak paddle out to an island in the Tasman Sea to see sea lions on the rocks surrounding it. On the way back in to Medlands Beach, where the walk began, we also got a brief glimpse of a blue penguin swimming by. The walk itself was leisurely and oft-interrupted with explanations; and also by many passing and crossing hikers, as the Abel Tasman Track is among the most popular in New Zealand. One more good bit of esoteric backcountry information from Darrell came in the form of a detailed verbal appreciation on his part for a few hundred yards after we'd crossed a particularly fine specimen, which Darrell identified by its Latin name: Wonderous pusherupperous, or something like that.
February 25. Truman Track and short walk to blowholes, Pancake Rocks and other formations near Punakaiki. About 2 miles total.
The Truman Track takes you through thick rainforest for a short distance before emerging through a "meadow" of New Zealand flax and then down onto the beach (when the tide is out, as it was for us). We waited out a shower under overhanging formations, part of the carving of the edge-of-mountains shoreline by the sea. The other walk – to another clash of mountain and sea – is also short, but offers not only the limestone formation of rocks looking like giant stacks of pancakes, but also several spots where the sea, as the tide comes in, sends waves crashing against, under and through rock formations.
February 26. Franz Joseph Glacier guided hike. About 4 miles.
From the little town of Franz Joseph, the bus takes you about a mile to a parking lot along the river that flows – whitish and fast – out of the glacier. At this point – on the little trail to get to the wide, bouldery riverbed with the water flowing on only one narrow strip – the glacier is hard to understand visually, hard to judge as to size and nature. And even as we walked across the rocks toward it, it was slow to reveal itself in terms of what was to come.
At some point our two guides stopped the 25 or so people and suggested we break into a slow group and a faster group. Very quickly, all of the 30-and-under-somethings clustered on one side and those who appeared to be on the north side of 50 on the other. The lone exception was of course The Greatest Day Hiker Of Them All, who motioned me to come along to the fast group. "I'm not going with a bunch of geezers," she muttered to me under her breath.
But alas, when it was announced that the fast group was a little too big, she looked around and surrendered; "Yeah, I guess it's only right for us to go over here."
Up close but still not on it, the glacier remains hard to figure, except that you can see rocks and debris on it – explained later as having been shot out from under by water.
Before the beginning of the climb – a steep, slick section where you use a pull-rope – we get lessons in crampon; how to attach them to your boots and how to keep your feet straight ahead instead of angled as you sort of naturally want to do.
Once up on the ice, you begin to understand the magnitude of the glacier. It extends upward into the clouds, with occasional glimpses of blue beyond. As we pick our way slowly upward, people pause to put on long sleeves and then, a little higher up, windbreakers. Even young, cheeky guide Ben finally dons a jacket, shortly before he leads us through a steep, narrow crevice where we are "only the second group to go through," and where we again need a rope and also get our shoulders wet as we rub against the blue blue ice. We climb further – until 6 p.m. on an excursion that began at 3 – and then start our way back down.
"Trust your crampons," Ben repeats, as we all work to overcome the temptation to side-saddle it on the steep slick ice. "Keep you toes pointing down the ice and the spikes will hold."
February 27. Lake Matheson loop walk. About 2 miles.
This easy, flat, warm, gentle walk was the antithesis of the one the day before, though its highlight holds a hint of the cold of the day before: Near the mid-point of the walk is a stunning view – out over the clear blue lake with Mt. Cook (New Zealand's highest) and Mt. Tasman standing tall and snowy against the sky and also reflecting themselves ideally in the lake. The photos we took here are perhaps the "most postcard" of all we took, to the point that we placed one as the screensaver on the home computer. And not too many days after one 20-something son commented that he'd bet you could flip the photo and no one would notice, a two-something grandson sat at the computer with his little sippy cup in one hand and somehow, for the second time in his short life, FLIPPED the screen 180 degrees. And no, you can't tell the image is upside down, though the icons are harder to read and the mouse is impossible to use.
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Rhum Rhum is a collaboration between Luca Gargano of Velier and Master Distiller Vittorio Capovilla. Using small copper pot stills located near to the Bielle distillery on Marie-Galante. The rhum was distilled in a copper alembic pot still, specifically designed by Capovilla and made by a the German artisan Muller.


text:
Anglesey footballing greats: Eifion Williams
PICTURE by Sky Sports.
TODAY'S spotlight is one of the greatest footballers to ever have come out of Anglesey. Although born in Bangor, Eifion Williams was raised in Llangoed on Anglesey and laser went on to become a legend of Wales' top flight.
Having been an apprentice at Wolverhampton Wanderers, where things didn't quite work out for him, Williams returned to live on Anglesey and signed for Gwynedd side Caernarfon Town, where he would first make a name for himself between 1995 and 1997.
Just as nowadays, back then the Canaries were a League of Wales (now JD Cymru Premier) side. Whilst playing there, Williams was also working as a builder.
And having laid some great goal-scoring foundations by scoring 63 goals in just 76 appearances for Caernarfon, bigger clubs came calling for his signature and he made the switch to the formidable Barry Town ahead of the 1997/98 campaign.
Back then, Barry were the only professional side in the League of Wales and it told as they won four successive titles between 1995 and 1999. When they signed Williams it was for a then-record fee of £25,000. And he went on to more than prove his worth to the club, even netting a UEFA Champions League qualifying goal for the club against Dynamo Kiev.
But it was in the league of Wales that the lethal forward would prove to be at his very best for Barry Town. He once scored four goals in one game to help the club record its biggest ever victory – a 9-0 thrashing of Conwy United.
Williams also netted a ridiculous 40 goals in just 37 matches as Barry went a whole season undefeated to claim the League of Wales title in 1997/98. The year after he notched up 28 goals in 22 matches, taking his overall tally in Wales' top flight to an impressive 131 goals in 135 matches.
After amassing more than 100 League of Wales goals by March 1999, the English League came swooping in for Williams and he made the switch to Torquay United who broke their record to sign him for £70,000.
And if anyone thought he'd struggle upon crossing the border, they were wrong. Williams mad ehis league debut against Hartlepool United and bagged a hat-trick – becoming the first Torquay player to do so on his debut since Peter Laraman against Barnsley in 1962. This led to now-famous goalkeeper Neville Southall, who was a team-mate of Williams' at Torquay at the time, stating that the latter could 'make it all the way to the Premiership'.
Wycombe's Sam Stockley (right) and Hartlepool's Eifion Williams battle for the ball during the Coca-Cola Football League Two match at Causeway Stadium, High Wycombe. (Photo by PA Images via Getty Images)
But such pressure on his shoulders following such a great display may or may not have led to a downfall in Williams' career in English football from there on out.
Injuries later affected his form, causing him to play with a broken toe for part of 2000–01, although he still managed nine goals in what was a dismal season for the club. He was placed on the transfer list on 2 November 2001 by mutual consent after finding himself playing mainly as a substitute and left to join Hartlepool United in March 2002 for a fee of £30,000.
Williams made his Hartlepool debut against Bristol Rovers and by the end of the season he had managed five goals in 10 appearances to help Pools make the play-offs. The following season saw Williams play in 45 of Hartlepool's 46 league matches. Due to the injury of Gordon Watson he became Hartlepool's first choice striker and finished as Hartlepool's top scorer, scoring 15 goals in 45 games, helping Hartlepool get automatic promotion in the process.
William's first season in League One saw him repeat the success of the previous as he participated in 41 games and scored 13 times despite playing right wing for a large majority of the season. These performances led to him being called up to the Welsh national team – then led by Mark Hughes, although he never managed to earn an actual cap. He did however play for Wales' B team and also for the Ynys Mon mens team once upon a time.
2004/05 saw him spend most of the season on the wing again for Hartelpool, reducing his goal-scoring record greatly as he scored just five goals in 41 appearances. However, he did have yet another big say in their season as he equalise for Hartlepool in their play-off final against Sheffield Wednesday in front of a crowd of 59,808 at the Millenium Stadium in Cardiff before they eventually lost out 4–2 after extra time.
The 2005–06 season saw Williams once again playing up front due to injuries to team-mates and he managed to score eight goals.
Williams was released from his Hartlepool United contract at the end of the 2006/07 season, which saw Williams scoring Hartlepool's goal of the season with a sublime strike in a 3–0 win over arch-rivals Darlington.
Williams would then return to North Wales, signing for League Two side Wrexham in June 2007. He scored two goals on his début in a 3–2 home defeat in a friendly against Liverpool. However, his only official goal for Wrexham came in a 1-0 win over Bury on December 4, 2007 – and he injured his hamstring whilst scoring it, with now Aberystwyth Town captain Marc Williams coming on to replace him.
This turned out to be his final appearance as a professional footballer and he retired during the 2007/08 season.
He briefly came out of retirement in 2008 to play for the now-defunct Jarrow Roofing FC, based near Sunderland. Whilst playing part-time for the Northern League side he began working for Hartlepool Youth Offending Service.
Williams, who turns 45 years old on the 15th of this month, is said to currently still reside in Hartlepool, where he is a P.E. teacher at Dyke House College.
Undoubtedly one of Anglesey's finest who went on to achieve the dream of playing in the English Football League. It's just a shame that injuries hampered what could have been an even better career for Williams I'm sure!
Williams' full career record was as follows;
Caernarfon Town (1995-1997): 63 goals in 75 appearances
Barry Town United (1997-1999): 68 goals in 60 appearances
Torquay United (1999-2002): 24 goals in 107 appearances
Hartelpool United (2002-2007): 50 goals in 208 appearances
Wrexham (2007-2008) 1 goal in 13 appearances
Total: 206 goals in 463 appearances
Posted in Cymru PremierTagged Barry Town United, Caernarfon Town, Eifion Williams, football, soccer, WalesBy Arron EvansLeave a comment
← AFE's JD Cymru Premier Player of the Month for October 2020 is … Jack Wilson (Haverfordwest)
The five times Wales' top scorer also finished as the top scorer in Europe →


Peter Neumann is Professor of Security Studies at the War Studies Department, and serves as Director of the International Centre for the Study of Radicalisation (ICSR, www.icsr.info), which he founded in early 2008.
Prof. Neumann has authored or co-authored five books, including Old and New Terrorism, published by Polity Press in 2009; and The Strategy of Terrorism (with MLR Smith), published by Routledge in 2008. He is the author of numerous peer-reviewed articles, dealing with different aspects of terrorism and radicalization, especially 'homegrown' radicalization in Western countries.
As part of ICSR, he has led research projects and written influential policy reports about issues such as online radicalization, prison-based de-radicalization programs, and terrorist recruitment in Europe.The most recent, "Preventing Violent Radicalization in America," was published in June 2011 by the Bipartisan Policy Center in Washington DC, where Prof. Neumann served as visiting scholar (www.bipartisanpolicy.org).
Prof. Neumann is a member of the editorial boards of two leading, peer-reviewed journals, Studies in Conflict and Terrorism and Democracy and Security, and serves as investigator for the National Consortium for the Study of Terrorism and Responses to Terrorism (START) at the University of Maryland. He is an Affiliate of the European Commission's European Network of Experts on Radicalisation, a member of the German Federal Criminal Office's (BKA) European Expert Network on Terrorism Issues, and sits on the advisory boards of numerous other think-tanks and institutions, including the Club de Madrid, the association of former Presidents and Prime Ministers.
At the Department of War Studies, Prof. Neumann co-directs the MA program in Terrorism, Security and Society, and supervises six research students. He has taught courses on terrorism, counterterrorism, intelligence, radicalization and counter-radicalization at King's College London and the School of Foreign Service at Georgetown University, where he continues to serve as adjunct professor.
Prof. Neumann holds an MA in Political Science from the Free University of Berlin, and a PhD in War Studies from King's College London. Before becoming an academic, he worked as a radio journalist in Germany.