认证的贝叶斯最优实验设计：从期望信息增益到信噪比

中文摘要 AI 辅助

最优实验设计通常被表述为期望信息增益（EIG）的最大化，EIG衡量观测数据后关于感兴趣参数的不确定性的预期减少。对于非线性前向模型和非高斯先验，计算EIG通常需要代价高昂的嵌套蒙特卡洛估计器或复杂的密度近似技术。在本工作中，我们利用维数对数Sobolev不等式推导了EIG的新上下界。所提出的界以信噪比（SNR）表示，保留了线性-高斯设置中EIG的简单矩阵结构，且仅依赖于协方差矩阵和Fisher信息矩阵。我们引入了两类界：保守界，旨在保留全部信息内容；以及增量界，优先考虑顺序实验设计中的早期信息获取。当模型梯度可用时，我们进一步推导了Fisher信息项的可计算近似，仅需少量梯度评估，从而大幅节省计算成本。这些界的紧致性由前向模型的非线性控制，在线性-高斯设置中，这些界恢复EIG的精确值。通过消除嵌套采样瓶颈，所提出的框架为大规模贝叶斯实验设计提供了易于处理且理论上有依据的直接EIG估计替代方案，同时保留了基于SNR的界的可解释性。

英文摘要

Optimal experimental design is often formulated as the maximization of the Expected Information Gain (EIG), which measures the expected reduction in uncertainty about a parameter of interest after observing the data. For nonlinear forward models and nonGaussian priors, computing the EIG typically requires costly nested Monte Carlo estimators or sophisticated density-approximation techniques. In this work, we derive new upper and lower bounds on the EIG using dimensional logarithmic Sobolev inequalities. The proposed bounds are expressed in terms of signal-to-noise ratios (SNR), preserving the simple matrix-based structure of the EIG in the linear-Gaussian setting and depending solely on covariance matrices and Fisher information matrices. We introduce two families of bounds: conservative bounds, designed to retain the entire information content, and incremental bounds, that prioritize early information acquisition for sequential experimental design. When model gradients are available, we further derive computable approximations of the Fisher-information terms that require only a small number of gradient evaluations, yielding substantial computational savings. The tightness of these bounds is controlled by the nonlinearity of the forward model, and in the linear-Gaussian setting, these bounds recover the exact value of the EIG. By eliminating the nested-sampling bottleneck, the proposed framework offers a tractable and theoretically grounded alternative to direct EIG estimation for large-scale Bayesian experimental design, while retaining the interpretability of the SNR-based bounds.

发表机构

• Inria（法国国家数字技术研究所）

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