Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The usual description, which we will call the Ulam-Galerkin method, is in terms of projection onto basis functions that are characteristic functions supported over a fine grid of rectangles. In these terms, the usual Ulam-Galerkin approach can be understood as density estimation by the histogram method. Here we show that the problem can be recast in statistical density estimation formalism. This recasting of the classical problem, is a perspective that allows for an explicit and rigorous analysis of bias and variance, and therefore toward a discussion of the mean square error. Keywords: Transfer Operators; Frobenius-Perron operator; probability density estimation; Ulam-Galerkin method;Kernel Density Estimation.

翻译：从数据推断传递算子常被表述为一个依赖于Ulam方法的经典问题。通常的描述（我们称之为Ulam-Galerkin方法）涉及投影到支撑于精细矩形网格上的特征函数基函数。在此框架下，常规的Ulam-Galerkin方法可理解为基于直方图方法的密度估计。本文表明，该问题可重新表述为统计密度估计的形式。这种对经典问题的重构视角，使得能够对偏差与方差进行明确且严格的分析，进而讨论均方误差。关键词：传递算子；Frobenius-Perron算子；概率密度估计；Ulam-Galerkin方法；核密度估计。