When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous results on robust VI with location-scale families under target symmetries. We derive sufficient conditions guaranteeing exact recovery of the mean when using the forward Kullback-Leibler divergence and $α$-divergences. We further show how and why optimization can fail to recover the target mean in the absence of our sufficient conditions, providing initial guidelines on the choice of the variational family and $α$-value.

翻译：在用变分推断（VI）近似难以处理的密度时，变分族通常被选为简单的参数族，这很可能不包含目标密度。这就引出一个问题：在模型设定错误的情况下，我们能否恢复目标密度的特征？在本文中，我们推广了先前在目标对称性下使用位置-尺度族进行稳健变分推断的结果。我们推导出充分条件，确保在使用前向Kullback-Leibler散度和α-散度时能够精确恢复均值。我们进一步展示了在缺乏充分条件的情况下，优化为何以及如何无法恢复目标均值，从而为变分族和α值的选择提供了初步指导。