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.

翻译：暂无翻译