Normalizing flows model a complex target distribution in terms of a bijective transform operating on a simple base distribution. As such, they enable tractable computation of a number of important statistical quantities, particularly likelihoods and samples. Despite these appealing properties, the computation of more complex inference tasks, such as the cumulative distribution function (CDF) over a complex region (e.g., a polytope) remains challenging. Traditional CDF approximations using Monte-Carlo techniques are unbiased but have unbounded variance and low sample efficiency. Instead, we build upon the diffeomorphic properties of normalizing flows and leverage the divergence theorem to estimate the CDF over a closed region in target space in terms of the flux across its \emph{boundary}, as induced by the normalizing flow. We describe both deterministic and stochastic instances of this estimator: while the deterministic variant iteratively improves the estimate by strategically subdividing the boundary, the stochastic variant provides unbiased estimates. Our experiments on popular flow architectures and UCI benchmark datasets show a marked improvement in sample efficiency as compared to traditional estimators.
翻译:以简单的基分布法操作的双向变形模式将复杂的流动模式目标分布标准化。 因此, 它们能够对一些重要的统计数量, 特别是可能性和样本进行可移植的计算。 尽管有这些有吸引力的特性, 计算复杂地区( 如多管区) 的累积分布功能( CDF) 仍然具有挑战性。 使用蒙特- 卡洛 技术的传统 CDF 近似值是公正的, 但却没有限制差异和低采样效率。 相反, 我们利用正常流动的二面形特性, 并利用差异性标语来估计目标空间中封闭区域的 CDF 。 我们用正态流流流的实验和 UCI 基准数据集, 显示样本与传统估量器相比的效率显著提高 。