Bayesian Inference for Construction of Inverse Models from Data

This talk considers the inverse problem y=f(x), where x and y are observable parameters, in which we wish to recover the model f. Examples include dynamical systems and combat models with y=dx/dt and x=parameter(s), water catchments with y=streamflow and x=rainfall, and groundwater vulnerability with y=pollutant concentration(s) and x=hydrological parameter(s). Historically, these have been solved by many methods, including regression or sparse regularization for dynamical system models, and various empirical correlation methods for rainfall-runoff and groundwater vulnerability models. These can instead be analyzed within a Bayesian framework, using the maximum a posteriori (MAP) method to estimate the model parameters, and the Bayesian posterior distribution to estimate the parameter variances (uncertainty quantification). For systems with unknown covariance parameters, the joint maximum a-posteriori (JMAP) and variational Bayesian approximation (VBA) methods can be used for their estimation. These methods are demonstrated by the analysis of a number of dynamical and hydrological systems.