Bayesian mixture model approach to quantifying the empirical nuclear saturation point

The equation of state (EOS) in the limit of infinite symmetric nuclear matter exhibits an equilibrium density, ${n}_{0}\ensuremath{\approx}0.16\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$, at which the pressure vanishes and the energy per particle attains its minimum, ${E}_{0}\ensuremath{\approx}\ensuremath{-}16\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$. Although not directly measurable, the nuclear saturation point $({n}_{0},{E}_{0})$ can be extrapolated by density-functional theory (DFT), providing tight constraints for microscopic interactions derived from chiral effective-field theory (EFT). However, when considering several DFT predictions for $({n}_{0},{E}_{0})$ from Skyrme and relativistic mean field (RMF) models together, a discrepancy between these model classes emerges at high confidence levels that each model prediction's uncertainty cannot explain. How can we leverage these DFT constraints to rigorously benchmark nuclear saturation properties of chiral interactions? To address this question, we present a Bayesian mixture model that combines multiple DFT predictions for $({n}_{0},{E}_{0})$ using an efficient conjugate prior approach. The inferred posterior distribution for the saturation point's mean and covariance matrix follows a normal-inverse-Wishart (NIW) class, resulting in posterior predictives in the form of correlated, bivariate $t$ distributions. The DFT uncertainty reports are then used to mix these posteriors using an ordinary Monte Carlo approach. At the 95% credibility level, we estimate ${n}_{0}\ensuremath{\approx}0.157\ifmmode\pm\else\textpm\fi{}0.010\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ and ${E}_{0}\ensuremath{\approx}\ensuremath{-}15.97\ifmmode\pm\else\textpm\fi{}0.40\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ for the marginal (univariate) $t$ distributions. Combined with chiral EFT calculations of the pure neutron matter equation of state, we obtain bivariate normal distributions for the nuclear symmetry energy and its slope parameter evaluated at ${n}_{0}$: ${S}_{v}\ensuremath{\approx}32.0\ifmmode\pm\else\textpm\fi{}1.1\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ and $L\ensuremath{\approx}52.6\ifmmode\pm\else\textpm\fi{}8.1\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ (95%), respectively. Our Bayesian framework is publicly available, so practitioners can readily use and extend our results.