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.
翻译:本文侧重于在应用中经常遇到的具有挑战性的一系列反面问题。 前方模型是一个复杂的非线性黑盒,有可能是非定向的,其产出覆盖了数十年的振幅。 观测应该同时受到添加性和倍增性噪音和检查的破坏。 许多应用中, 这项工作的目的是在参数估计之上提供不确定性的量化。 由此产生的日志相似性是棘手的, 并且可能不是log- conculve。 一种经调整的巴耶西亚方法建议提供可信度间隔和点估计。 一种MCMC算法建议处理多式远端分布, 即使在没有全球利普西茨常数( 或极大) 的情况下也是如此。 它将两种核心结合起来, 即改进版的( 有条件的Metropoliis 调整 Langevin) PMA 和多端Terit Metropolis ( MMM) 内核。 它的梯度在平坦度中, 只要简单, 就可以在引力过程中使用一个大而无法被利用的利普西茨常数。 这个采样器处理所有由复杂、 度的摩天体形式所展示的方法。