We present an exact, analytic and deterministic method for sampling densities whose Cumulative Distribution Functions (CDFs) cannot be inverted analytically. Indeed, the inverse‐CDF method is often considered the way to go for sampling non‐uniform densities. If the CDF is not analytically invertible, the typical fallback solutions are either approximate, numerical, or non‐deterministic such as acceptance‐rejection. To overcome this problem, we show how to compute an analytic area‐preserving parameterization of the region under the curve of the target density. We use it to generate random points uniformly distributed under the curve of the target density and their abscissae are thus distributed with the target density. Technically, our idea is to use an approximate analytic parameterization whose error can be represented geometrically as a triangle that is simple to cut out. This triangle‐cut parameterization yields exact and analytic solutions to sampling problems that were presumably not analytically resolvable.

中文翻译：

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不能反转 CDF？曲线下区域的三角切割参数化

我们提出了一种精确的、分析的和确定性的采样密度方法，其累积分布函数 (CDF) 不能通过分析反演。事实上，逆 CDF 方法通常被认为是采样非均匀密度的方法。如果 CDF 在解析上不可逆，则典型的回退解决方案要么是近似的、数值的，要么是非确定性的，例如接受-拒绝。为了克服这个问题，我们展示了如何计算目标密度曲线下区域的解析面积保留参数化。我们用它来生成均匀分布在目标密度曲线下的随机点，因此它们的横坐标随目标密度分布。从技术上讲，我们的想法是使用近似解析参数化，其误差可以在几何上表示为易于切割的三角形。这种三角形切割参数化为采样问题提供了精确的解析解，这些问题可能无法通过解析解析。