Hierarchical Bayesian Approach to Fitting Discretized Partial Differential Equation Models to Spatial-temporal Data: The Effects of Numerical Instability on Parameter Estimates

Hierarchical Bayesian (HB) models provide a flexible framework for modeling spatiotemporal data and processes. In cases where the latent dynamics can be modeled with partial differential equations (PDEs), one approach is to parameterize a dynamic linear model by a discretized PDE model, often using forward differencing. The forward Euler discretization is computationally appealing (i.e., there are no matrix inversions to compute) but it can suffer from numerical instability causing the numerical solution to diverge from the true solution. While such a model might still be useful for prediction, this divergence can affect the posterior parameter estimates. In this paper we use a synthetic dataset to demonstrate how numerical instability in forward Euler-based PDE HB models can bias posterior parameter estimates, and how using other finite difference schemes allow unbiased estimates of HB model parameters.