  We present a method to simultaneously infer the interstellar extinction
parameters $A_0$ and $R_0$, stellar effective temperature $T_{\rm eff}$, 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 4000 to 9000 K and extinctions from 0 to 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 0.2 mag and 300 K,
respectively, for $R_0$ fixed to 3.1. For variable $R_0$, we obtain MAEs of
0.37 mag, 412.9 K and 0.74 for $A_0$, $T_{\rm eff}$ and $R_0$, respectively.
Distance moduli are accurate to approximately 2 mag. Quantifying the precisions
of individual parameter estimates with $68\%$ confidence interval of the
posterior distribution, we obtain 0.05 mag, 66 K, 2 mag and 0.07 for $A_0$,
$T_{\rm eff}$, $\mu$ 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.
