Sensitivity analysis and assessment of prior model probabilities in MLBMA with application to unsaturated fractured tuff

Previous application of maximum likelihood Bayesian model averaging (MLBMA, Neuman (2002, 2003)) to alternative variogram models of log air permeability data in fractured tuff has demonstrated its effectiveness in quantifying conceptual model uncertainty and enhancing predictive capability (Ye et al., 2004). A question remained how best to ascribe prior probabilities to competing models. In this paper we examine the extent to which lead statistics of posterior log permeability predictions are sensitive to prior probabilities of seven corresponding variogram models. We then explore the feasibility of quantifying prior model probabilities by (1) maximizing Shannon's entropy H (Shannon, 1948) subject to constraints reflecting a single analyst's (or a group of analysts') prior perception about how plausible each alternative model (or a group of models) is relative to others, and (2) selecting a posteriori the most likely among such maxima corresponding to alternative prior perceptions of various analysts or groups of analysts. Another way to select among alternative prior model probability sets, which, however, is not guaranteed to yield optimum predictive performance (though it did so in our example) and would therefore not be our preferred option, is a minimum-maximum approach according to which one selects a priori the set corresponding to the smallest value of maximum entropy. Whereas maximizing H subject to the prior perception of a single analyst (or group) maximizes the potential for further information gain through conditioning, selecting the smallest among such maxima gives preference to the most informed prior perception among those of several analysts (or groups). We use the same variogram models and log permeability data as Ye et al. (2004) to demonstrate that our proposed approach yields the least amount of posterior entropy (residual uncertainty after conditioning) and enhances predictive model performance as compared to (1) the noninformative neutral case in which all prior model probabilities are set equal to each other and (2) an informed case that nevertheless violates the principle of parsimony.