The ILIUM forward modelling algorithm for multivariate parameter estimation and its application to derive stellar parameters from Gaia spectrophotometry

I introduce an algorithm for estimating parameters from multidimensional data based on forward modelling. In contrast to many machine learning approaches it avoids fitting an inverse model and the problems associated with this. The algorithm makes explicit use of the sensitivities of the data to the parameters, with the goal of better treating parameters which only have a weak impact on the data. The forward modelling approach provides uncertainty (full covariance) estimates in the predicted parameters as well as a goodness-of-fit for observations. I demonstrate the algorithm, ILIUM, with the estimation of stellar astrophysical parameters (APs) from simulations of the low resolution spectrophotometry to be obtained by Gaia. The AP accuracy is competitive with that obtained by a support vector machine. For example, for zero extinction stars covering a wide range of metallicity, surface gravity and temperature, ILIUM can estimate Teff to an accuracy of 0.3% at G=15 and to 4% for (lower signal-to-noise ratio) spectra at G=20. [Fe/H] and logg can be estimated to accuracies of 0.1-0.4dex for stars with G<=18.5. If extinction varies a priori over a wide range (Av=0-10mag), then Teff and Av can be estimated quite accurately (3-4% and 0.1-0.2mag respectively at G=15), but there is a strong and ubiquitous degeneracy in these parameters which limits our ability to estimate either accurately at faint magnitudes. Using the forward model we can map these degeneracies (in advance), and thus provide a complete probability distribution over solutions. (Abridged)

💡 Research Summary

The paper introduces ILIUM, an algorithm for estimating astrophysical parameters (APs) from multidimensional data by employing a forward‑modelling (generative) approach rather than the more common inverse‑mapping machine‑learning techniques. The key idea is to approximate the unknown physical mapping from APs to observed fluxes, g_i(φ), with a continuous, differentiable forward model f_i(φ) trained on a discrete grid of synthetic spectra (templates). Because the forward model is analytic, its partial derivatives with respect to each AP (the sensitivity matrix S, where s_ij = ∂p_i/∂φ_j) can be evaluated at any point in parameter space.

ILIUM proceeds iteratively, using a Newton‑Raphson scheme to minimise the residual between the observed spectrum p₀ and the forward‑model prediction p(φ). Starting from the nearest template, the algorithm computes the sensitivity matrix at the current AP estimate, forms the residual δp = p − p₀, and updates the AP vector via
 δφ = (SᵀS)⁻¹ Sᵀ δp,
repeating until convergence criteria (change in parameters or residual below a threshold) are met. This yields not only point estimates but also a full covariance matrix for the APs, providing rigorous uncertainty quantification and a goodness‑of‑fit statistic.

A practical difficulty arises when some APs (e.g., surface gravity log g or metallicity