We present a method to simultaneously infer the interstellar extinction parameters A _ { 0 } and R _ { 0 } , stellar effective temperature T _ { eff } , and distance modulus \mu in a Bayesian framework .
Using multi-band photometry from SDSS and UKIDSS , we train a forward model to emulate the colour-change due to physical properties of stars and the interstellar medium for temperatures from 4 000 to \unit { 9 000 } { K } and extinctions from 0 to \unit { 5 } { mag } .
We introduce a Hertzsprung-Russel diagram prior to account for physical constraints on the distribution of stars in the temperature-absolute magnitude plane .
This allows us to infer distances probabilistically .
Influences of colour information , priors and model parameters are explored .
Residual mean absolute errors ( MAEs ) on a set of objects for extinction and temperature are \unit { 0.2 } { mag } and \unit { 300 } { K } , respectively , for R _ { 0 } fixed to 3.1 .
For variable R _ { 0 } , we obtain MAEs of \unit { 0.37 } { mag } , \unit { 412.9 } { K } and 0.74 for A _ { 0 } , T _ { eff } and R _ { 0 } , respectively .
Distance moduli are accurate to approximately \unit { 2 } { mag } .
Quantifying the precisions of individual parameter estimates with 68 \% confidence interval of the posterior distribution , we obtain \unit { 0.05 } { mag } , \unit { 66 } { K } , \unit { 2 } { mag } and 0.07 for A _ { 0 } , T _ { eff } , \mu and R _ { 0 } , respectively , although we find that these underestimate the accuracy of the model .
We produce two-dimensional maps in extinction and R _ { 0 } that are compared to previous work .
Furthermore we incorporate the inferred distance information to compute fully probabilistic distance profiles for individual lines of sight .
The individual stellar AP estimates , combined with inferred 3D information will make possible many Galactic science and modelling applications .
Adapting our method to work with other surveys , such as Pan-STARRS and Gaia , will allow us to probe other regions of the Galaxy .