利用嵌套多层蒙特卡洛进行不可处理似然下的序贯神经后验估计
Leveraging Nested MLMC for Sequential Neural Posterior Estimation with Intractable Likelihoods
AI总结：
本文针对序贯神经后验估计中的APT方法，将其重构为嵌套估计问题，并提出多种多层蒙特卡洛估计器以高效估计损失及梯度，同时给出收敛性分析，数值实验验证了方法的有效性。
摘要：
由于序贯神经后验估计（SNPE）技术在具有不可处理似然的基于模拟的模型中具有优势，对其的研究兴趣日益增长。这些方法旨在利用基于神经网络的条件密度估计器，从自适应提出的模拟中学习后验分布。作为SNPE技术之一，由Greenberg等人（2019）提出的自动后验变换（APT）方法表现良好，并能扩展到高维数据。然而，APT方法需要计算不可处理归一化常数的对数的期望，即嵌套期望。尽管使用原子提议使得归一化常数具有解析形式，但分析学习的收敛性仍然具有挑战性。在本文中，我们将APT重新表述为一个嵌套估计问题。在此基础上，我们为损失函数及其梯度构建了多种多层蒙特卡洛（MLMC）估计器以适应不同场景，包括两个无偏估计器，以及一个有偏估计器，该估计器以较小的偏差换取方差减小和可控的运行时间及内存使用。我们还提供了随机梯度下降的收敛性结果，以量化梯度估计器的偏差和方差之间的相互作用。我们提供了在中等维度下逼近具有多模态的复杂后验分布的数值实验，以检验所提出方法的有效性。
英文摘要：
There is a growing interest in studying sequential neural posterior estimation (SNPE) techniques due to their advantages for simulation-based models with intractable likelihoods. The methods aim to learn the posterior from adaptively proposed simulations using neural network-based conditional density estimators. As an SNPE technique, the automatic posterior transformation (APT) method proposed by Greenberg et al. (2019) performs well and scales to high-dimensional data. However, the APT method requires computing the expectation of the logarithm of an intractable normalizing constant, i.e., a nested expectation. Although atomic proposals were used to render an analytical normalizing constant, it remains challenging to analyze the convergence of learning. In this paper, we reformulate APT as a nested estimation problem. Building on this, we construct several multilevel Monte Carlo (MLMC) estimators for the loss function and its gradients to accommodate different scenarios, including two unbiased estimators, and a biased estimator that trades a small bias for reduced variance and controlled runtime and memory usage. We also provide convergence results of stochastic gradient descent to quantify the interaction of the bias and variance of the gradient estimator. Numerical experiments for approximating complex posteriors with multimodality in moderate dimensions are provided to examine the effectiveness of the proposed methods.
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