  Equation-of-state (EOS) models underpin numerical simulations at the core of
research in high energy density physics, inertial confinement fusion,
laboratory astrophysics, and elsewhere. In these applications EOS models are
needed that span ranges of thermodynamic variables that far exceed the ranges
where data are available, making uncertainty quantification (UQ) of EOS models
a significant concern. Model uncertainty, arising from the choice of functional
form assumed for the EOS, is a major challenge to UQ studies for EOS that is
usually neglected in favor of parameteric and data uncertainties which are
easier to capture without violating the physical constraints on EOSs. In this
work we introduce a new statistical EOS construction that naturally captures
model uncertainty while automatically obeying the thermodynamic consistency
constraint. We apply the model to existing data for $B_4C$\ to place an upper
bound on the uncertainty in the EOS and Hugoniot, and show that the neglect of
thermodynamic constraints overestimates the uncertainty by factors of several
when data are available and underestimates when extrapolating to regions where
they are not. We discuss extensions to this approach, and the role of GP-based
models in accelerating simulation and experimental studies, defining portable
uncertainty-aware EOS tables, and enabling uncertainty-aware downstream tasks.
