Despite the successes of probabilistic models based on passing noise through
neural networks, recent work has identified that such methods often fail to
capture tail behavior accurately, unless the tails of the base distribution are
appropriately calibrated. To overcome this deficiency, we propose a systematic
approach for analyzing the tails of random variables, and we illustrate how
this approach can be used during the static analysis (before drawing samples)
pass of a probabilistic programming language compiler. To characterize how the
tails change under various operations, we develop an algebra which acts on a
three-parameter family of tail asymptotics and which is based on the
generalized Gamma distribution. Our algebraic operations are closed under
addition and multiplication; they are capable of distinguishing sub-Gaussians
with differing scales; and they handle ratios sufficiently well to reproduce
the tails of most important statistical distributions directly from their
definitions. Our empirical results confirm that inference algorithms that
leverage our heavy-tailed algebra attain superior performance across a number
of density modeling and variational inference tasks.

该研究提出了一种系统性的方法来分析随机变量的尾部，建立了一个基于广义 Gamma 分布的代数，该代数能够在各种操作下区分不同尺度的亚高斯函数，可以直接从定义中重现大部分重要的统计分布，并通过利用重尾代数的推理算法，实现了在多个密度建模及变分推理任务上实现了卓越的性能。