Generalized likelihood uncertainty estimation (GLUE) and approximate Bayesian computation: What's the connection?

There has been a recent debate in the hydrological community about the relative merits of the informal generalized likelihood uncertainty estimation (GLUE) approach to uncertainty assessment in hydrological modeling versus formal probabilistic approaches. Some recent literature has suggested that the methods can give similar results in practice when properly applied. In this note, we show that the connection between formal Bayes and GLUE is not merely operational but goes deeper, with GLUE corresponding to a certain approximate Bayesian procedure even when the generalized likelihood is not a true likelihood. The connection we describe relates to recent approximate Bayesian computation (ABC) methods originating in genetics. ABC algorithms involve the use of a kernel function, and the generalized likelihood in GLUE can be thought of as relating to this kernel function rather than to the model likelihood. Two interpretations of GLUE emerge, one as a computational approximation to a Bayes procedure for a certain error-free model and the second as an exact Bayes procedure for a perturbation of that model in which the truncation of the generalized likelihood in GLUE plays a role. The intent of this study is to encourage cross-fertilization of ideas regarding GLUE and ABC in hydrologic applications. The connection we outline suggests the possibility of combining a formal likelihood with a kernel based on a generalized likelihood within the ABC framework and also allows advanced ABC computational methods to be used in GLUE applications. The model-based interpretation of GLUE may also be helpful in partially illuminating the implicit assumptions in different choices of generalized likelihood. Citation: Nott, D. J., L. Marshall, and J. Brown (2012), Generalized likelihood uncertainty estimation (GLUE) and approximate Bayesian computation: What's the connection?, Water Resour. Res., 48, W12602, doi: 10.1029/2011WR011128.