混合噪声下非对数凹后验分布的高效采样
Efficient sampling of non log-concave posterior distributions with mixture of noises
AI总结：
本文针对复杂黑箱反问题中非对数凹且多模态的后验分布，提出结合改进PMALA与MTM核的MCMC采样算法，实现参数估计与不确定性量化，并在天文学反问题上验证其有效性。
摘要：
本文聚焦于应用中经常遇到的一类具有挑战性的反问题。前向模型是一个复杂的非线性黑箱，可能非单射，其输出幅度跨越多个数量级。观测数据被假设同时受到加性噪声、乘性噪声以及删失的影响。与许多应用中的需求一致，本工作的目标是在参数估计的基础上提供不确定性量化。由此产生的对数似然难以处理，且可能非对数凹。本文提出一种适配的贝叶斯方法，以同时提供可信区间和点估计。针对多模态后验分布，提出一种MCMC算法，即使在不存在全局Lipschitz常数（或该常数非常大）的情况下也能有效处理。该算法结合了两个核：改进版的预条件Metropolis调整Langevin算法（PMALA）核和多重尝试Metropolis（MTM）核。当目标函数光滑时，其梯度的Lipschitz常数过大，无法在推断过程中加以利用。该采样器解决了复杂似然形式所带来的所有挑战。所提方法在经典的多模态测试分布以及一个具有挑战性的、现实的天文学反问题上进行了验证。
英文摘要：
This paper focuses on a challenging class of inverse problems that is often encountered in applications. The forward model is a complex non-linear black-box, potentially non-injective, whose outputs cover multiple decades in amplitude. Observations are supposed to be simultaneously damaged by additive and multiplicative noises and censorship. As needed in many applications, the aim of this work is to provide uncertainty quantification on top of parameter estimates. The resulting log-likelihood is intractable and potentially non-log-concave. An adapted Bayesian approach is proposed to provide credibility intervals along with point estimates. An MCMC algorithm is proposed to deal with the multimodal posterior distribution, even in a situation where there is no global Lipschitz constant (or it is very large). It combines two kernels, namely an improved version of (Preconditioned Metropolis Adjusted Langevin) PMALA and a Multiple Try Metropolis (MTM) kernel. Whenever smooth, its gradient admits a Lipschitz constant too large to be exploited in the inference process. This sampler addresses all the challenges induced by the complex form of the likelihood. The proposed method is illustrated on classical test multimodal distributions as well as on a challenging and realistic inverse problem in astronomy.
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