基于温控贝叶斯后验的似然信息维度约简
Likelihood-informed dimension reduction across tempered Bayesian posteriors
AI总结：
本文提出了一种通用的方法，用于在温控贝叶斯后验中进行似然信息维度约简，通过理论支持构建部分信息空间并改进在数据有限和噪声大的实际应用中的性能。
摘要：
科学计算模拟无法在现实应用中代表所有尺度。为了弥合模型-数据差距，参数被注入模型并利用贝叶斯反演进行约束。为了减少模拟器评估次数（可能达到10^5次以上），现代方法结合了维度约简和正向映射的建模。由于模型评估和数据稀缺，这种维度约简对后验采样性能至关重要。最近的工作利用似然信息子空间（LIS）通过优化信息损失的界限来截断到信息性方向，尽管在数学上适合采样，但在实践中往往受限。本文证明了该方法可以推广到α-温控（即退火、功率后验）分布，其中α ∈ [0,1]。我们提供理论来构建称为α-LIS的部分信息空间。我们展示了α < 1可以经常产生近最优的空间。此外，我们专注于将α-LIS应用于实际案例，其中可用数据严重受限且存在噪声。我们提出并测试了利用整个分布序列α_0 < ... < α_k的数据扩展方法，并使用简单的模型梯度近似，使得我们的方法可用于混沌或随机系统的正向映射建模，其中导数不可用或无信息。在实验中，我们的累积方法在这些具有挑战性的条件下比理论上最优的α=1方法更加稳健。
英文摘要：
Scientific computer simulations cannot represent all scales in realistic applications. To bridge this model-data gap, parameters are injected into models and constrained with noisy data using Bayesian inversion. To reduce the number of simulator evaluations, which can be 10^5 or more, modern approaches employ dimension reduction in conjunction with emulation of the forward map (that contains the simulator). Due to scarcity of model evaluations and data, this dimension reduction becomes very important for posterior sampling performance. Recent work on likelihood-informed subspaces (LIS) truncates to informative directions by optimizing bounds on information loss, and though mathematically well-adapted to sampling, they are often restrictive in practice. In this work, we provably generalize this methodology to facilitate application to $α$-tempered (i.e., annealed, power-posterior) distributions for $α$ in [0,1]. We provide theory to build partially-informed spaces termed $α$-LIS. We show how $α$ < 1 can often produce near-optimal spaces. In addition, we focus on applying $α$-LIS to practical cases, where the available data is severely limited and noisy. We propose and test extensions for utilizing data from the entire sequence of distributions $α$_0 < ... < $α$_k, and use simple approximations of model gradients so that our approach can be used for emulation of forward maps for chaotic or stochastic systems where derivatives are unavailable or uninformative due to noise. In experiments, our accumulated approach is much more robust to these challenging circumstances than the theoretically optimal $α$ = 1.
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