Bayesian model-data comparison incorporating theoretical uncertainties

TL;DR

This work addresses the problem of extracting physical parameters when theoretical models carry domain-limited validity by introducing a Bayesian framework that models theory error as a Gaussian-process discrepancy $\delta(x)$. By jointly inferring model parameters $\boldsymbol{\theta}$ and GP hyperparameters, the approach quantifies uncertainties from both data and theory, with two kernels (Kernel I and Kernel II) encoding prior knowledge about the theory's reliability across input space. Demonstrated on a ball-drop test and multi-stage heavy-ion simulations, incorporating model discrepancy yields more robust, accurate parameter estimates and predictions, improving systematically as more observables are included. The framework provides a principled, transferable method for robust model-data comparisons across complex systems, including temperature-dependent transport coefficients like $\eta/s(T)$, while offering guidance on kernel choice and error quantification for reliable inference.

Abstract

Accurate comparisons between theoretical models and experimental data are critical for scientific progress. However, inferred physical model parameters can vary significantly with the chosen physics model, highlighting the importance of properly accounting for theoretical uncertainties. In this Letter, we present a Bayesian framework that explicitly quantifies these uncertainties by statistically modeling theory errors, guided by qualitative knowledge of a theory's varying reliability across the input domain. We demonstrate the effectiveness of this approach using two systems: a simple ball drop experiment and multi-stage heavy-ion simulations. In both cases incorporating model discrepancy leads to improved parameter estimates, with systematic improvements observed as additional experimental observables are integrated.

Figures (7)

• Figure 1: Results of Bayesian inference for the ball drop experiment. Top row: Corner plots displaying the inferred posteriors for $g$ and $v_0$ (with median $\pm 68\%$ CI displayed on the diagonals) as additional observables (specified above the panels) are sequentially incorporated (from left to right) into the Bayesian inference. Bottom row: Model predictions based on the inferred parameters corresponding to the cases in the top row.
• Figure 2: Predictions based on inferred parameters when all observables are considered in calibration. The black curve shows the prediction from the true theory. The blue band shows the predictions of model + discrepancy GP with Kernel I, and green shows the same with kernel II. The red band shows the model predictions without accounting for model discrepancy (same as in the bottom-right panel of Fig. \ref{['fig:balldrop']})
• Figure 3: Results for the two-parameter hydrodynamic simulation using mock data generated with fixed $\eta/s=0.1$ and $\epsilon_{\rm sw}=0.2\,$GeV/fm$^3$. Top row: Corner plots showing the posteriors for the inferred $\eta/s$ and $\epsilon_{\rm sw}$ (with median $\pm 68\%$ CI displayed on the diagonals) as more observables are included in the Bayesian inference (from left to right). Bottom rows: Model predictions based on inference using all observables.
• Figure 4: Results for the five-parameter hydrodynamic simulation with parametrized $\eta/s$ using mock data generated with fixed $\eta/s=0.1$ and $\epsilon_{\rm sw}=0.2\,$GeV/fm$^3$. Top row: Plots display the $\eta/s$ posterior (median and $\pm 95\%$ CI) as a function of temperature, with the posterior for $\epsilon_{\rm sw}$ (median $\pm 68\%$ CI) shown in the inset, as additional observables are sequentially incorporated into the Bayesian inference (from left to right). Bottom rows: Model predictions based on inference using all observables.
• Figure 5: Results for the five-parameter hydrodynamic simulation with parametrized $\eta/s$ using mock data generated using a parametrization of $\eta/s$ with $T_{\rm kink}{\,=\,}0.18\,{\rm GeV}$, $a_{\rm low}{\,=\,}{-}1\,{\rm GeV}^{-1}$, $a_{\rm high}{\,=\,}1\,{\rm GeV}^{-1}$, and $(\eta/s)_{\rm kink}{\,=\,}0.1$, along with $\epsilon_{\rm sw}{\,=\,}0.2\,$GeV/fm$^3$. The plot layouts and legends are identical to those in Fig. \ref{['fig:5flat']}. The corner plot for all five model parameters, when all available observables are included, is provided in \ref{['sec:app_corner']}, Fig. \ref{['fig:5notflat_modelcorner']} .