一种用于加性和乘性噪声下非线性逆问题的分层似然模型
A Hierarchical Likelihood Model for Non-linear Inverse Problems under Additive and Multiplicative Noise
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中文总结 AI 辅助
针对加性和乘性噪声下的非线性逆问题，提出通用分层贝叶斯模型及高效MCMC算法。该方法无需校准近似模型超参数，更通用。通过天文学合成数据评估，在逐点估计和计算成本上达最优，为从业者选模型提供指导。
中文摘要 AI 辅助
不适定逆问题在众多应用中出现，可能具有高度非线性正向模型、加性和乘性噪声源以及删失数据。在缺乏真实值时，不确定性量化对评估估计可靠性至关重要，这促使使用贝叶斯模型和马尔可夫链蒙特卡罗算法等随机推理方法。结合所有这些挑战的问题通常导致复杂且可能多峰的后验分布，难以实际处理。文献中提出了近似方法，要么忽略噪声源，要么使用似然函数的易处理近似。这些方法要么导致模型不准确，要么可能需要对近似似然进行复杂校准。本文提出用通用分层贝叶斯模型和高效MCMC算法解决此类问题。所提出的公式绕过了校准近似模型超参数的需要且更通用。在所提出的方法在天文学中遇到的具有挑战性场景下，使用合成数据在各种噪声和删失配置下进行评估。与两个基线和适用于此背景的最新方法进行比较。所提出的方法是通用的，在逐点估计和计算成本方面产生了最新结果，具有卓越的预测性能。补充材料中的结果进一步完善了这个全面且严格的模型研究。这项工作可为从业者根据其特定应用选择最佳似然模型提供指导。
英文摘要
Ill-posed inverse problems are encountered in numerous applications, possibly characterized by a highly non-linear forward model, both additive and multiplicative sources of noise, and censored data. In the absence of ground truth, uncertainty quantification is crucial to assess estimation reliability. This motivates the use of a Bayesian model and stochastic inference methods such as Markov Chain Monte Carlo algorithms. Problems combining all these challenges often lead to a complex and potentially multimodal posterior distribution, difficult to handle in practice. Approximate approaches have been proposed in the literature by either neglecting a source of noise or by using a tractable approximation of the likelihood function. These approaches either lead to an inaccurate model, or may require a complex calibration of the approximate likelihood. This paper proposes to tackle such problems with a general hierarchical Bayesian model and an efficient MCMC algorithm. The proposed formulation bypasses the need for calibrating the hyperparameters of an approximate model and is more versatile. The proposed method is assessed on a challenging scenario encountered in astronomy using synthetic data, in a variety of noise and censoring configurations. Comparisons are conducted against two baselines and a state-of-the-art method applicable in this context. The proposed approach is general and yields state-of-the-art results in terms of point-wise estimates and computing costs, with superior predictive performance. Results in the supplementary material further complete this comprehensive and rigorous model study. This work can serve as a guide for practitioners to select the best likelihood model according to their specific application.
