A Sandwich With Water: Bayesian/Frequentist Uncertainty Quantification under Model Misspecification

Abstract

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 $\widehat{\mathbf{A}}_{n}$ and variability $\widehat{\mathbf{B}}_{n}$ matrices of the ML model parameter values $\widehat{\bm{\uptheta}}_{n}$ provide conflicting information about the observed Fisher information $\widehat{\boldsymbol{\mathcal{I}}}\vphantom{\overline{\widehat{\boldsymbol{\mathcal{I}}}}}_{n}$ of the data $\omega_{1},\ldots,\omega_{n}$ for $\bm{\uptheta} = (\theta_{1},\ldots,\theta_{d})^{\top}$. As a result, the ML parameter covariance matrix, $\Var(\widehat{\bm{\uptheta}}_{n})$, does not simplify to the matrix inverse of the observed Fisher information, $\widehat{\boldsymbol{\mathcal{I}}}\vphantom{\overline{\widehat{\boldsymbol{\mathcal{I}}}}}_{n}$, as suggested by naive ML estimators and Bayesian MCMC methods but amounts instead to the so-called sandwich matrix $\Var(\widehat{\bm{\uptheta}}_{n}) = \widehat{\boldsymbol{\mathcal{G}}}\vphantom{\overline{\widehat{\boldsymbol{\mathcal{G}}}}}_{n}^{-1} = \fracn \widehat{\mathbf{A}}_{n}^{-1}\widehat{\mathbf{B}}^{\vphantom{-1}}_{n}\widehat{\mathbf{A}}_{n}^{-1}$, where the observed Godambe information $\widehat{\boldsymbol{\mathcal{G}}}\vphantom{\overline{\widehat{\boldsymbol{\mathcal{G}}}}}_{n}$ is the fundamental currency of data informativeness under model misspecification. The \textit{sandwich} matrix is a metaphor for a \textit{meat} matrix $\widehat{\mathbf{B}}_{n}$ between two \textit{bread} matrices $\widehat{\mathbf{A}}_{n}$ and yields asymptotically valid “robust standard errors” even when the likelihood function $L_{n}(\bm{\uptheta})$ (model) is incorrectly specified. The implications of the sandwich variance estimator are demonstrated in three case studies involving the modeling of soil water infiltration, watershed hydrologic fluxes and the rainfall-discharge transformation. First and foremost, our analytic and numerical results demonstrate that the sandwich variance estimator increases substantially hydrologic model parameter and predictive uncertainty. The sandwich estimator is invariant to likelihood stretching 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 MCMC simulation.

Authors 4

1. Jasper Alexander Vrugt corresponding

   University of California, Irvine

   Affiliation as printed

   University of California, Irvine

   Department of Civil and Environmental Engineering, University of California, Irvine, California, USA

2. Cees G. H. Diks

   University of California, Irvine · University of Amsterdam

   Affiliation as printed

   University of Amsterdam, the Netherlands

   Department of Civil and Environmental Engineering, University of California, Irvine, California, USA

3. Ramon de Punder

   University of Amsterdam

   Affiliation as printed

   University of Amsterdam, the Netherlands

   Faculty Economics and Business, University of Amsterdam, Amsterdam, The Netherlands

4. Peter Grünwald Aachen

   Leiden University · Centrum Wiskunde & Informatica · University of California, Irvine · University of Amsterdam

   Affiliation as printed

   CWI

   Centrum Wiskunde & Informatica, University of Amsterdam, Amsterdam, The Netherlands

   Department of Civil and Environmental Engineering, University of California, Irvine, California, USA

   Faculty Economics and Business, University of Amsterdam, Amsterdam, The Netherlands

   Leiden University, Leiden, The Netherlands