GLUE, introduced by Keith Beven and Andrew Binley in 1992, was developed to address a practical problem in environmental modeling: complex hydrological and ecological models have many parameters, nonlinear responses, and structural errors that make formal statistical inference difficult or impossible. Traditional likelihood-based methods assume the model is "correct" and that residuals follow a known statistical distribution. GLUE abandons these assumptions, instead defining a generalized likelihood measure that assesses how well each parameter set reproduces observed behavior, and using Monte Carlo sampling to explore the parameter space.
2. Run the model for each θᵢ, producing output Ŷᵢ
3. Compute a likelihood weight L(θᵢ | Y) for each based on fit to observations Y
4. Reject parameter sets below a threshold ("non-behavioral")
5. Rescale weights of "behavioral" sets to sum to 1
6. Construct weighted prediction intervals from behavioral simulations
Common Likelihood Measures Nash-Sutcliffe efficiency: NSE = 1 − Σ(Yᵢ − Ŷᵢ)² / Σ(Yᵢ − Ȳ)²
Inverse error variance: L(θ) ∝ σ⁻²ᴺ (informal likelihood)
Equifinality
The central concept motivating GLUE is equifinality: in complex models, many different combinations of parameter values can produce simulations that are equally consistent with the observed data. There is no single "best" parameter set — instead, there is a population of acceptable parameter sets, and the uncertainty in predictions arises from this population.
This perspective challenges the classical statistical paradigm of point estimation. Rather than seeking the maximum likelihood or posterior mode, GLUE embraces the full distribution of behavioral parameter sets and propagates this uncertainty through to predictions.
GLUE has been both praised as pragmatic and criticized as statistically informal. Mantovan and Todini (2006) and others have argued that GLUE's likelihood measures do not correspond to proper statistical likelihoods, that its uncertainty intervals lack formal coverage properties, and that properly formulated Bayesian methods would be preferable. Beven has responded that formal likelihoods require assumptions about model structural error that are themselves uncertain, and that GLUE's flexibility in defining the likelihood measure is a feature rather than a bug. The debate continues, with many practitioners using GLUE alongside formal Bayesian methods.
Applications
GLUE is most widely used in hydrology — calibrating rainfall-runoff models, predicting flood flows, estimating water quality — but has been applied in ecology, atmospheric science, geomorphology, and any field where complex simulation models must be calibrated against observations. Its appeal lies in its simplicity: any model that can be run forward with different parameters can be analyzed with GLUE, without requiring derivatives, adjoint models, or assumptions about error distributions.
Relationship to Bayesian Methods
GLUE can be viewed as an informal approximation to Bayesian inference. If the generalized likelihood measure is replaced by a proper statistical likelihood, the prior ranges become a proper prior distribution, and the rejection threshold is removed, GLUE converges to standard Bayesian inference via importance sampling. The gap between GLUE and formal Bayes lies in the treatment of model error and the definition of the likelihood — precisely the aspects that are most difficult and most contested in environmental modeling.
"Every model is a simplification of reality, and the recognition that many parameter sets may be equally valid — equifinality — is the starting point for honest uncertainty assessment." — Keith Beven and Andrew Binley, "The Future of Distributed Models" (1992)