A Sandwich with Water Bayesian & Frequentist Inference Under Model Misspecification

In this paper we review basic elements of Frequentist inference, specifically maximum likelihood (ML) and M-estimation, to point out a critical flaw of Bayesian methods for hydrologic model training and uncertainty quantification. Under model misspecification, the sensitivity matrix Âₙ and variability matrix B̂ₙ of the ML model parameter estimates θ̂ₙ provide conflicting information about the observed Fisher information Îₙ of the data ω₁,…,ωₙ for θ = (θ₁,…,θ_d)ᵀ. As a result, the estimated ML parameter covariance matrix does not simplify to the inverse of the observed Fisher information, Îₙ⁻¹, as suggested by naïve ML estimators and Bayesian methods, but instead corresponds to the so-called sandwich matrix Ĝₙ⁻¹ = n⁻¹ Âₙ⁻¹ B̂ₙ Âₙ⁻¹, where the observed Godambe information Ĝₙ is the fundamental currency of data informativeness under model misspecification. The sandwich matrix is a metaphor for a meat matrix B̂ₙ between two bread matrices Âₙ and yields asymptotically valid “robust standard errors” even when the likelihood function Lₙ(θ) is incorrectly specified. The implications of the sandwich variance estimator are demonstrated in three case studies involving modeling of soil water infiltration, watershed hydrologic fluxes, and rainfall–discharge transformation. First and foremost, our analytic and numerical results demonstrate that the sandwich variance estimator substantially increases hydrologic model parameter and predictive uncertainty. The sandwich estimator is invariant to likelihood stretching as practiced by the GLUE method as a remedy for over-conditioning and requires magnitude and/or curvature adjustments to the likelihood function to yield asymptotically valid sandwich parameter estimates and inference via Monte Carlo simulation.