标题： 直接密度比估计量的存在性 显示英文标题

标题： Existence of Direct Density Ratio Estimators

摘要： 许多两样本问题需要比较来自指数族的两个分布。 密度比估计方法通过直接估计自然参数的差异来解决这些问题。 术语“直接”表示避免估计两个边缘分布。 在此背景下，我们考虑Kullback--Leibler重要性估计过程（KLIEP），该方法最近成为差分网络研究的主题。 我们的主要结果表明，KLIEP估计量的存在性由一个样本的平均充分统计量是否属于第二个样本中所有数据点的充分统计量集合的凸包来决定。 对于高维问题，通常通过添加调参和参数差异向量范数的乘积来对KLIEP损失进行正则化。 我们证明，正则化KLIEP估计量的存在性要求调参不小于平均充分统计量与凸包之间的基于对偶范数的距离。 这些存在性问题的含义在差分网络分析的应用中得到了探讨。 显示英文摘要

摘要： Many two-sample problems call for a comparison of two distributions from an exponential family. Density ratio estimation methods provide ways to solve such problems through direct estimation of the differences in natural parameters. The term direct indicates that one avoids estimating both marginal distributions. In this context, we consider the Kullback--Leibler Importance Estimation Procedure (KLIEP), which has been the subject of recent work on differential networks. Our main result shows that the existence of the KLIEP estimator is characterized by whether the average sufficient statistic for one sample belongs to the convex hull of the set of all sufficient statistics for data points in the second sample. For high-dimensional problems it is customary to regularize the KLIEP loss by adding the product of a tuning parameter and a norm of the vector of parameter differences. We show that the existence of the regularized KLIEP estimator requires the tuning parameter to be no less than the dual norm-based distance between the average sufficient statistic and the convex hull. The implications of these existence issues are explored in applications to differential network analysis.