Revisiting the framework of (Barillas, Francisco, and Jay Shanken, 2018, Comparing asset pricing models, The Journal of Finance 73, 715–754). BS henceforth, we show that the Bayesian marginal likelihood‐based model comparison method in that paper is unsound : the priors on the nuisance parameters across models must satisfy a change of variable property for densities that is violated by the Jeffreys priors used in the BS method. Extensive simulation exercises confirm that the BS method performs unsatisfactorily. We derive a new class of improper priors on the nuisance parameters, starting from a single improper prior, which leads to valid marginal likelihoods and model comparisons. The performance of our marginal likelihoods is significantly better, allowing for reliable Bayesian work on which factors are risk factors in asset pricing models.

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The human health risk (HHR) assessment to dense non-aqueous phase liquids (DNAPLs) exposure has become an important part of groundwater environment management. Usually, DNAPL transport models are applied to simulate the concentration distribution of contaminant for HHR assessment. The present paper studied the influences of model uncertainties on the HHR assessment, and the metric of Incremental Lifetime Cancer Risk (ILCR) was used to quantify HHR. The impacts of permeability's heterogeneity and the structure of DNAPL transport model (e.g., the constitutive model) on HHR assessment were evaluated based on a synthetical DNAPL transport model. The results demonstrate that, compared with the low heterogeneity, the high heterogeneity leads to lower average ILCR value at the control planes near the source zone, and higher average ILCR value at the control planes far away from the source zone. In addition, the HHR assessments would be inconsistent for the two constitutive models, i.e., Stone-Parker (S-P) and Corey-van Genuchten (C-v) models. Compared with the HHR assessment depending on C-v model, the mean of ILCR's probability distribution produced by S-P model is larger at the control planes near the source zone, and smaller at the control planes far away from the source zone. Moreover, based on a sandbox experiment, the impact of parameter uncertainty of DNAPL transport model on HHR assessment was evaluated by Markov chain Monte Carlo (MCMC) simulation. The results show that it is infeasible and risky to assess HHR by the specific parameters of contaminant transport model and ignoring parameter uncertainty. The HHR assessment by incorporating Bayesian uncertainty analysis could provide more flexible information. In addition, the sparse grid (SG) surrogate is an effective way to reduce computation burden caused by the larger number of model executions in the MCMC based HHR assessment.

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