Sequential neural posterior estimation (SNPE) techniques have been recently proposed for dealing with simulation-based models with intractable likelihoods. They are devoted to learning the posterior from adaptively proposed simulations using neural network-based conditional density estimators. As a SNPE technique, the automatic posterior transformation (APT) method proposed by Greenberg et al. (2019) performs notably and scales to high dimensional data. However, the APT method bears the computation of an expectation of the logarithm of an intractable normalizing constant, i.e., a nested expectation. Although atomic APT was proposed to solve this by discretizing the normalizing constant, it remains challenging to analyze the convergence of learning. In this paper, we propose a nested APT method to estimate the involved nested expectation instead. This facilitates establishing the convergence analysis. Since the nested estimators for the loss function and its gradient are biased, we make use of unbiased multi-level Monte Carlo (MLMC) estimators for debiasing. To further reduce the excessive variance of the unbiased estimators, this paper also develops some truncated MLMC estimators by taking account of the trade-off between the bias and the average cost. Numerical experiments for approximating complex posteriors with multimodal in moderate dimensions are provided.

翻译：近年来，序贯神经后验估计（SNPE）技术被提出用于处理具有难处理似然的基于模拟的模型。该类方法利用基于神经网络的密度估计器，通过自适应地选择模拟数据来学习后验分布。作为SNPE技术的一种，Greenberg等人（2019）提出的自动后验变换（APT）方法表现突出，可扩展至高维数据。然而，APT方法需要计算关于一个难处理归一化常数的对数期望，即嵌套期望。尽管原子APT通过离散化归一化常数得以解决该问题，但学习过程的收敛性分析仍具挑战。本文提出一种嵌套APT方法，转而估计其中涉及的嵌套期望，从而为收敛性分析建立基础。由于损失函数及其梯度的嵌套估计存在偏差，我们采用无偏的多层蒙特卡洛（MLMC）估计器进行去偏。为降低无偏估计器的过度方差，本文还通过权衡偏差与平均计算代价，开发了截断式MLMC估计器。数值实验验证了该方法在中等维度多模态复杂后验逼近中的有效性。