Analytic Marginalization over Binary Variables in Physics Data

TL;DR

This work tackles the problem of marginalizing over binary nuisance variables in data analyses, which would otherwise explode the parameter space. It shows that exact binary marginalization yields a log-likelihood correction identical to the Ising-model partition function, enabling fast likelihood evaluation via Ising techniques. Two practical approximations are developed: a Paramagnetic scheme that ignores data correlations and a Mean-Field scheme based on Hubbard–Stratonovich transformation, both illustrated on toy data and Type Ia SN mass-step calibrations. The results demonstrate accurate parameter recovery and quantify how host-mass uncertainties affect cosmological inferences, finding negligible impact on $H_0$ in the SN case, while also highlighting the method’s potential extensions to Cepheid overtone classification and beyond. The framework is general, offering a principled and scalable way to incorporate discrete uncertainties across physics and other fields.

Abstract

In many data analyses, each measurement may come with a simple yes/no correction; for example, belonging to one of two populations or being contaminated or not. Ignoring such binary effects may bias the results, while accounting for them explicitly quickly becomes infeasible as each of the $N$ data points introduces an additional parameter, resulting in an exponentially growing number of possible configurations ($2^N$). We show that, under generic conditions, an exact treatment of these binary corrections leads to a mathematical form identical to the well-known Ising model from statistical physics. This connection opens up a powerful set of tools developed for the Ising model, enabling fast and accurate likelihood calculations. We present efficient approximation schemes with minimal computational cost and demonstrate their effectiveness in applications, including Type Ia supernova calibration, where we show that the uncertainty in host-galaxy mass classification has negligible impact on the inferred value of the Hubble constant.

Figures (8)

• Figure 1: Illustration of the correspondence between the data-analysis and Ising-model views. Upper panel: Data-analysis view showing the residuals $r_i$—defined as the difference between data and model before applying the binary corrections $\pm\Delta$—with the residuals indicated by the thick shaded arrows. Lower panel: Ising-model view where each data point corresponds to a dipole with spin $s_i = \pm1$ in a magnetic field $\mathbf{h}$. The field strength is proportional to the residual. The strength of the magnetic field shows how decisive the data are—strong fields drive the spins toward one orientation, whereas weak fields correspond to cases where either orientation is nearly equally plausible. Neighbor couplings $J_{ij}$, induced by data correlations, favor aligned ($J_{ij}>0$) or anti-aligned ($J_{ij}<0$) spins. Here, ferromagnetic couplings cause the final dipole to flip against the magnetic field, illustrating how correlations can override an individual data point's preference.
• Figure 2: Data distribution for a sample of $N = 200$ thermometers, an offset $\Delta = 0.2$, and measurement uncertainty $\sigma = 0.1$. It is clear that the paramagnetic model including the binary offset, Eq. \ref{['eq:ToyModel']}, is superior compared with the baseline model.
• Figure 3: Log-likelihood for the baseline and paramagnetic models. Shaded regions denote the $68 \, \%$ confidence intervals around the best-fit value. Here $\Delta / \sigma = 2$ and the baseline model is biased, here high, with respect to the true value. We also find that the baseline model underestimates the true uncertainty, as it ignores the additional variability introduced by the scatter between positive and negative offsets.
• Figure 4: Sample variability of the inferred temperature from the baseline and paramagnetic models. Here, $N = 200$, $\Delta = 1$, $\sigma = 0.1$, and $p = 0.5$. The baseline estimate is biased in the majority of realizations whereas the paramagnetic model consistently reproduces accurate estimates.
• Figure 5: Sample variability of the inferred temperature across 20 realizations with strongly correlated sensors (mean correlation coefficient $\rho \simeq 0.3$). The paramagnetic approximation corrects much of the bias present in the baseline, whereas the mean-field approximation provides a further improvement and is consistent with the true value.