Quantifying the multi-objective cost of uncertainty

Investigating real-world systems and phenomena typically requires complex models that involve a large number of parameters. Even with sizeable amount of observation data, the high complexity of the model may render accurate parameter estimation impossible. While finding a reliable point estimate of the parameter vector may not be possible in such a case, it may be possible to identify the parameter ranges based on the available data and/or prior system knowledge, or in a more general setting, we may assume a joint distribution of the model parameters. Since different parameter values are possible, this gives rise to an uncertainty class of all possible models [1, 2]. Furthermore, this naturally places the model in a Bayesian framework, where the likelihood of every possible model in the uncertainty class is described by a prior that could be constructed from prior system knowledge and existing data [3, 4]. Given an uncertain model and its uncertainty class, how can one mathematically quantify the amount of uncertainty present in the model? Common approaches include estimating the variance or entropy of the uncertain parameters, as they both provide a simple and intuitive measure of the model uncertainty. However, they both have a critical downside from a practical perspective. In practical applications that involve mathematical modeling of a complex system, one cares about the model as it can serve as a vehicle for designing an effective operator (i.e., controller, classifier, filter) that can act on the system of interest or the data produced therefrom.