Extending Cross Entropy Based Importance Sampling for Bayesian Updating (CEBU) with Empirical Priors and Kolmogorov-Smirnov Based Convergence Diagnostics

Bayesian inverse problems in Earth sciences often ask for inversion techniques capable of handling high-dimensional nonlinear forward models, and prior information that is neither Gaussian nor analytically representable. This contribution focuses on the methodological developments underlying our application of cross entropy based importance sampling for Bayesian updating (CEBU) to Antarctic tidal grounding line migration based upon Sentinel-1 line of sight offsets. In particular, we highlight how the algorithm is extended to incorporate empirical, hence, nonparametric priors, how its sequential structure enables detailed convergence diagnostics, and how its evidence estimate can support filtering and model selection.

The grounding line marks the transition from grounded ice to floating ice shelf in Antarctica’s marine-terminating glaciers. The underlying elastic beam model simulating the bending of the ice in response to tidal deflection is, among others, based on an ice thickness parameter. Its prior shall be defined by the values from a dataset of a previous study. This prior exhibits non‑Gaussian structure and parameter dependencies that cannot be captured by standard parametric assumptions. Hence, we extend the CEBU framework by introducing an isoprobabilistic transform that maps the empirical ensemble into the standard normal space in which the update is performed. The extension allows CEBU to operate directly on empirical prior information, thereby embedding physical knowledge into the Bayesian update in a fully nonparametric manner.

After the initial transformation to standard normal space, CEBU proceeds through a sequence of tempered intermediate distributions that gradually introduce the likelihood. This sequential structure provides a transparent view of convergence behavior: we introduce the Kolmogorov–Smirnov distance between each intermediate importance sampling density and the prior as a measure of information gain and respective parameter importance. This quantity provides a nonparametric and interpretable metric of which components of the parameter vector are most informed by our observations and which remain dominated by prior uncertainty. The difference of information gained per step determines the respective importance of a parameter at a particular tempering step. Hence, by the distance metric introduced, CEBU intrinsically provides a convergence curriculum used to attain the posterior distribution.

After convergence, CEBU yields a Bayesian model evidence estimate. It quantifies the conceptual fit of the data observed and the model used. Accordingly, this evidence can be used for filtering the results, e.g., if the observation data of a particular inverse problem is too noisy, i.e., does not follow the measurement error model. Further, that quantity may be used for Bayesian model selection, offering a principled mechanism for evaluating competing forward models or prior assumptions. For example, that setting can be used to decide between multiple empirical priors, and thus between competing studies.

From a computational perspective, all forward evaluations and likelihood computations are embarrassingly parallelizable. That makes the approach well suited for large‑scale inference tasks on modern high performance clusters and cloud infrastructures.