Maximum likelihood Bayesian averaging of uncertain model predictions

Abstract.Hydrologic analyses typically rely on a single conceptual-mathematical model. Yet hydrologic environments are open and complex, rendering them prone to multiple interpretations and mathematical descriptions. Adopting only one of these may lead to statistical bias and underestimation of uncertainty. A comprehensive strategy for constructing alternative conceptual-mathematical models of subsurface flow and transport, selecting the best among them, and using them jointly to render optimum predictions under uncertainty has recently been developed by Neuman and Wierenga (2003). This paper describes a key formal element of this much broader and less formal strategy that concerns rendering optimum hydrologic predictions by means of several competing deterministic or stochastic models and assessing their joint predictive uncertainty. The paper proposes a Maximum Likelihood Bayesian Model Averaging (MLBMA) method to accomplish this goal. MLBMA incorporates both site characterization and site monitoring data so as to base the outcome on an optimum combination of prior information (scientific knowledge plus data) and model predictions. A preliminary example based on real data is included in the paper.

[1]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[2]  Jesús Carrera,et al.  Subsurface Flow and Transport: Formulations and computational issues of the inversion of random fields , 1997 .

[3]  Keith Beven.  Uniqueness of place and non-uniqueness of models in assessing predictive uncertainty , 2000 .

[4]  Keith Beven,et al.  Prophecy, reality and uncertainty in distributed hydrological modelling , 1993 .

[5]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 3. Application to Synthetic and Field Data , 1986 .

[6]  George Christakos,et al.  On the assimilation of uncertain physical knowledge bases: Bayesian and non-Bayesian techniques. , 2002 .

[7]  G. Hornberger,et al.  Approach to the preliminary analysis of environmental systems , 1981 .

[8]  You‐Kuan Zhang.  Stochastic Methods for Flow in Porous Media: Coping with Uncertainties , 2001 .

[9]  Alberto Guadagnini,et al.  Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion , 2003 .

[10]  Leslie Smith,et al.  A Bayesian Approach to the quantification of the effect of model error on the predictions of groundwater models , 2001 .

[11]  D. Madigan,et al.  Model Selection and Accounting for Model Uncertainty in Graphical Models Using Occam's Window , 1994 .

[12]  B. Babbitt,et al.  METHODS AND GUIDELINES FOR EFFECTIVE MODEL CALIBRATION , 2001 .

[13]  George Christakos,et al.  Modern Spatiotemporal Geostatistics , 2000 .

[14]  Adrian E. Raftery,et al.  Bayesian Model Selection in Structural Equation Models , 1992 .

[15]  G Christakos.  Critical conceptualism in environmental modeling and prediction. , 2003, Environmental science & technology.

[16]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[17]  Keith Beven,et al.  The future of distributed models: model calibration and uncertainty prediction. , 1992 .

[18]  Hugh G. Gauch,et al.  Prediction, Parsimony and Noise , 1993 .

[19]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information , 1986 .

[20]  A. J. Desbarats,et al.  Subsurface Flow and Transport: A Stochastic Approach , 1998 .

[21]  S. P. Neuman,et al.  A comprehensive strategy of hydrogeologic modeling and uncertainty analysis for nuclear facilities a , 2003 .

[22]  Adrian E. Raftery,et al.  Accounting for Model Uncertainty in Survival Analysis Improves Predictive Performance , 1995 .

[23]  Hirotugu Akaike,et al.  On entropy maximization principle , 1977 .

[24]  Daniel M. Tartakovsky,et al.  NONLOCAL AND LOCALIZED ANALYSES OF CONDITIONAL MEAN TRANSIENT FLOW IN BOUNDED, RANDOMLY NONUNIFORM DOMAINS , 2001 .

[25]  Keith Beven,et al.  Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology , 2001 .

[26]  David Draper,et al.  Assessment and Propagation of Model Uncertainty , 2011 .

[27]  R. Taplin.  Robust Likelihood Calculation for Time Series , 1993 .

[28]  Rangasami L. Kashyap,et al.  Optimal Choice of AR and MA Parts in Autoregressive Moving Average Models , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  D. A. Zimmerman,et al.  A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow , 1998 .

[30]  George Christakos.  The role of conceptual frameworks in hydrological research and development , 2003 .

[31]  Alberto Guadagnini,et al.  Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains: 2. Computational examples , 1999 .

[32]  Adrian E. Raftery,et al.  Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors , 1999 .

[33]  A. Sahuquillo,et al.  Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data—I. Theory , 1997 .

[34]  Curtis M. Oldenburg,et al.  Linear and Monte Carlo uncertainty analysis for subsurface contaminant transport simulation , 1997 .

[35]  D. Madigan,et al.  Bayesian Model Averaging in Proportional Hazard Models: Assessing the Risk of a Stroke , 1997 .

[36]  M. Bhaskara Rao,et al.  Model Selection and Inference , 2000, Technometrics.