(Go here for a non-technical summary and some pictures)
Astrometric surveys provide the opportunity to measure the absolute magnitudes of large numbers of stars, but only if the individual line-of-sight extinctions are known. Unfortunately, extinction is highly degenerate with stellar effective temperature when estimated from broad band optical/infrared photometry. To address this problem, I introduce a Bayesian method for estimating the intrinsic parameters of a star and its line-of-sight extinction. It uses both photometry and parallaxes in a self-consistent manner in order to provide a non-parametric posterior probability distribution over the parameters. The method makes explicit use of domain knowledge by employing the Hertzsprung--Russell Diagram (HRD) to constrain solutions and to ensure that they respect stellar physics. I first demonstrate this method by using it to estimate effective temperature and extinction from BVJHK data for a set of artificially reddened Hipparcos stars, for which accurate effective temperatures have been estimated from high resolution spectroscopy. Using just the four colours, we see the expected strong degeneracy (positive correlation) between the temperature and extinction. Introducing the parallax, apparent magnitude and the HRD reduces this degeneracy and improves both the precision (reduces the error bars) and the accuracy of the parameter estimates, the latter by about 35%. The resulting accuracy is about 200K in temperature and 0.2mag in extinction. I then apply the method to estimate these parameters and absolute magnitudes for some 47 000 F,G,K Hipparcos stars which have been cross-matched with 2MASS. The method can easily be extended to incorporate the estimation of other parameters, in particular metallicity and surface gravity, making it particularly suitable for the analysis of the 10^9 stars from Gaia.
Note: I point out in section 4.3 of the paper an issue with anomalously high extinctions of low luminosity stars, but would like to provide a few more details here. The extinctions of many nearby cool stars are probably overestimated by the method as implemented here. The reason is that the grid used to fit the model only extends down to 4700K. It therefore cannot give reliable estimates for stars which are truly cooler, in particular late K and M stars. The absolute V band magnitude of a late K dwarf is around M_V=8.5, and almost all of the stars in my sample (due to the Hipparcos limiting magnitude) are brighter than V=12. As one can see from the data, the extinction estimates start to rise for d<50pc (strictly w>20mas), which is a distance modulus of 3.5 (for zero extinction). In other words, cool dwarfs which are poorly modelled in this data set will be within 50pc. Why do they get assigned high extinctions? Because the model assigns a Teff and A0 which are consistent with the colour (and also with the HRD abd parallax, in the pq-model). Thus by design it is forced to overestimate the Teff and, because of the Teff/A0 degeneracy, also overestimates the A0. (Some of these stars are removed as "exterior stars", as described in the paper, but that cleaning method will not catch everything.) We see a similar problem with the cool giants (M_V<1.0). The only properly consistent way to solve this is to rebuild the models using training grids to lower Teff and then re-estimating all the APs.