Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

翻译：截断密度模型的估计较为困难，因为这类模型具有难以处理的归一化常数且边界条件不易满足。分数匹配方法可应用于解决截断密度估计问题，但需要定义一个在边界处为零、其余区域为正的连续权重函数。评估此类权重函数（及其梯度）通常需要截断边界的封闭表达式，并求解复杂的优化问题。本文提出近似斯坦因类，进而推导出适用于截断密度估计的松弛斯坦因恒等式。我们开发了一种新的差异度量——截断核化斯坦因差异（TKSD），该度量无需预先设定权重函数，仅需利用边界样本即可评估。通过最小化TKSD的拉格朗日对偶项来估计截断密度模型。实验结果表明，即使在不掌握边界显式函数形式的情况下，本方法的精度仍优于现有工作。