So just why would a modeller choose to be incoherent?

This article provides an extended response to the criticisms of the GLUE methodology by Mantovan and Todini [Mantovan, P., Todini, E., 2006. Hydrological forecasting uncertainty assessment: incoherence of the GLUE methodology. J. Hydrol. 330, 368–381]. It is shown that the formal Bayesian identification of models is a special case of GLUE that can be used where the modeller is prepared to make very strong assumptions about the nature of the modelling errors. Under such assumptions, GLUE can be coherent in the sense of Manotvan and Todini. In real applications, however, with multiple sources of uncertainty including model structural error, their strong definition of coherence is shown to be inapplicable to the extent that the choice of a formal likelihood function based on a simple error structure may be an incoherent choice. It is demonstrated by some relatively minor modifications of their hypothetical example that misspecification of the error model and the non-stationarities associated with the presence of input error and model structural error in the Bayes approach will then produce well-defined but incorrect parameter distributions. This empirical result is quite independent of GLUE, but the flexibility of the GLUE approach may then prove to be an advantage in providing more coherent and robust choices of model evaluation in these cases and, by analogy, in other non-ideal cases for real applications. At the current time it is difficult to make a reasoned choice between methods of uncertainty estimation for real applications because of a lack of understanding of the real information content of data in conditioning models.