Why is uncertainty quantification of sloppy models challenging?*

Mathematical models are widely used in science. They summarize what we know about a physical system and probe domains that are experimentally challenging or impossible. Uncertainty quantification helps us determine how much we can trust the predictions a model makes. Multiparameter models are often sloppy, i.e., they have cost surfaces with long, narrow canyons and broad, flat plateaus. Typically, contours on these surfaces do not close; they extend to the limit of physically allowed parameter values such as zero or infinity. These features on the likelihood surface make uncertainty quantification of sloppy models challenging. To show this, I use both Bayesian (Markov Chain Monte Carlo) and Frequentist (profile likelihood) methods to quantify parametric uncertainty of two interatomic potentials, Lennard-Jones and Stillinger-Weber. I calibrate these models on energy and force data for several atomic configurations from the OpenKIM database. I demonstrate that these models have infinite uncertainty in some of their parameters and discuss challenges this poses for uncertainty quantification in sloppy models.