This paper focuses on the Bregman divergence defined by the reciprocal function, called the inverse divergence. For the loss function defined by the monotonically increasing function $f$ and inverse divergence, the conditions for the statistical model and function $f$ under which the estimating equation is unbiased are clarified. Specifically, we characterize two types of statistical models, an inverse Gaussian type and a mixture of generalized inverse Gaussian type distributions, to show that the conditions for the function $f$ are different for each model. We also define Bregman divergence as a linear sum over the dimensions of the inverse divergence and extend the results to the multi-dimensional case.

翻译：本文聚焦于由倒数函数定义的Bregman散度，即逆散度。针对单调递增函数$f$与逆散度所定义的损失函数，明确了统计模型及函数$f$满足估计方程无偏性的条件。具体而言，我们刻画了两类统计模型——逆高斯型分布与广义逆高斯型混合分布，并证明函数$f$的条件因模型而异。此外，我们将Bregman散度定义为逆散度在各维度上的线性组合，并将相关结论拓展至多维情形。