  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)
