Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. We propose to formulate the validation of a given model as an iterative construction process that mimics the often implicit process occurring in the minds of scientists. We offer a formal representation of the progressive build-up of trust in the model. Thus, we replace static claims on the impossibility of validating a given model by a dynamic process of constructive approximation. This approach is better adapted to the fuzzy, coarsegrained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the methodology first with the maturation of quantum mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, a multifractal random walk model for financial time series, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability.A t the heart of the scientific endeavor, model building involves a slow and arduous selection process, which can be roughly represented as proceeding according to the following steps: (i) start from observations and/or experiments; (ii) classify them according to regularities that they may exhibit: the presence of patterns, of some order, also sometimes referred to as structures or symmetries, is begging for ''explanations'' and is thus the nucleation point of modeling; (iii) use inductive reasoning, intuition, analogies, and so on, to build hypotheses from which a model is constructed (by model, we understand an abstract conceptual construction based on axioms and logical relations developed to extract logical propositions and predictions); (iv) test the model obtained in step iii with available observations, and then extract predictions that are tested against new observations or by developing dedicated experiments. The model is then rejected or refined by an iterative process, a loop going from i to iv. A given model is progressively validated by the accumulated confirmations of its predictions by repeated experimental and/or observational tests.The validation of models is becoming a major issue as humans are increasingly faced with decisions involving complex tradeoffs in problems with large uncertainties, as for instance in attempts to control the growing anthropogenic burden on the planet (1) within a risk-cost framework (2, 3) based on predictions of models. For policy decisions, federal, state, and local governments increasingly depend on computer models that are scrutinized by scientific agenci...
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