Eulerian Parameter Inference: a probabilistic change of variables for model-based inference with high-variability data sets

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Advances in measurement technology have led to the generation of increasingly large data sets across various scientific fields. Data that capture the variability of the underlying system or process, such as single-cell or imaging data, are particularly interesting. However, calibrating computational models to explain this type of data remains challenging.We interpret the model calibration as a Stochastic Inverse Problem (SIP), where the measurements are interpreted as probabilistic samples. Our new SIP solution approach, Eulerian Parameter Inference (EPI), only requires data with large empirical variance and a deterministic simulation model that maps model parameters to simulation results. EPI solves the SIP through a change of variables. This computationally efficient approach allows for direct and point-wise evaluation of parameter densities without ever inverting the simulation model. The resulting parameter distribution captures data variability completely, enabling full data reconstruction. Further, estimating multivariate model parameters can be reduced to a series of one-dimensional problems under the assumption of stochastically independent parameters. This property is vital for complex simulation models and involved problems.With EPI, we explain 1) annual average temperature data of 3168 weather stations around the world with an arithmetic model, 2) German district-specific COVID-19 infection data through a differential equation model, and 3) artificial data generated via a partial differential equation model with independent parameters. With a robust mathematical foundation and an easily accessible Python package, we provide a versatile solution framework for integrating data with a high variance into simulation models that promotes application to various research questions.