Building effective models from sparse but precise data: Application to an alloy cluster expansion model

A common approach in computational science is to use a set of highly precise but expensive calculations to parameterize a model that allows less precise but more rapid calculations on larger-scale systems. Least-squares fitting on a model that underfits the data is generally used for this purpose. For arbitrarily precise data free from statistic noise, e.g., ab initio calculations, we argue that it is more appropriate to begin with an ensemble of models that overfit the data. Within a Bayesian framework, a most likely model can be defined that incorporates physical knowledge, provides error estimates for systems not included in the fit, and reproduces the original data exactly. We apply this approach to obtain a cluster expansion model for the CaZr_1-xTiₓO₃ solid solution.