In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is embedded in the wider theory of divergences and distances between distributions which includes Kullback-Leibler, Jensen-Shannon, Jeffreys-Bregman divergence and Bhattacharyya distance. A general construction is given based on defining a directional derivative of a function $ϕ$ from one distribution to the other whose concavity or strict concavity influences the properties of the resulting divergence. For the normal distribution these divergences can be expressed as matrix formula for the (multivariate) means and covariances. Optimal experimental design criteria contribute a range of functionals applied to non-negative, or positive definite, information matrices. Not all can distinguish normal distributions but sufficient conditions are given. The k-th order simplicial distance is revisited from this aspect and the results are used to test empirically the identity of means and covariances.
基于密度幂发散和对数密度幂发散的最小发散程序在生成推理程序方面非常流行和成功，该推理程序将高度的模型效率与强离群值稳定性相结合。在实际情况中，这样的程序总是比以效率的主要代价实现其鲁棒性或高效但鲁棒性差的程序更优选。Basu等人的密度功率发散（DPD）家族。（1998）和Jones et al.（2001）提供了灵活的发散类别，其中效率和鲁棒性之间的调整由单个实非负参数控制。这两个分歧族在统计推断中的有用性使得以同样的精神寻找其他相关的分歧族是有意义的。DPD族是Bregman分歧类的一员，而LDPD族是通过对DPD族中分歧的不同部分进行对数变换而得到的。DPD族和LDPD族在作为调节参数$α\rightarrow 0$的极限情况下都导致Kullback-Leibler发散。在本文中，我们详细研究了这种关系，并证明，这种日志转换只能是有意义的上下文中的DPD（或凸生成函数的DPD）内的一般倍布雷格曼分歧，给我们一个限制的范围内搜索有用的分歧可能是成功的。
A Dimensionality Reduction Method for Finding Least Favorable Priors with a Focus on Bregman Divergence
New Jersey Institute of Technology（新泽西理工学院）；Bremen University of Applied Sciences（不来梅应用技术大学）；Princeton University（普林斯顿大学）；Technion - Israel Institute of Technology（以色列理工学院）
Bregman-divergence-guided Legendre exponential dispersion model with finite cumulants (K-LED)
School of Liberal Arts, Korea University of Technology and Education（韩国技术教育大学文理学院）
Exponential dispersion model is a useful framework in machine learning and statistics. Primarily, thanks to the additive structure of the model, it can be achieved without difficulty to estimate parameters including mean. However, tight conditions on cumulant function, such as analyticity, strict convexity, and steepness, reduce the class of exponential dispersion model. In this work, we present relaxed exponential dispersion model K-LED (Legendre exponential dispersion model with K cumulants). The cumulant function of the proposed model is a convex function of Legendre type having continuous …
Unbiased Estimating Equation on Inverse Divergence and Its Conditions
Media Center Toyohashi University of Technology Toyohashi, Aichi, Japan Email；Department of Computer Science；Engineering Toyohashi University of Technology Toyohashi, Aichi, Japan Email
