  We determine the period-magnitude relations of RR Lyrae stars in 13
photometric bandpasses from 0.4 to 12 {\mu}m using timeseries observations of
134 stars. The Bayesian formalism, extended from our previous work to include
the effects of line-of-sight dust extinction, allows for the simultaneous
inference of the posterior distribution of the mean absolute magnitude, slope
of the period-magnitude power-law, and intrinsic scatter about a perfect
power-law for each bandpass. In addition, the distance modulus and
line-of-sight dust extinction to each RR Lyrae star in the calibration sample
is determined, yielding a sample median fractional distance error of 0.66%. The
intrinsic scatter in all bands appears to be larger than the photometric
errors, except in WISE W1 (3.4 {\mu}m) and W2 (4.6 {\mu}m) where the
photometric error ($\sigma \approx 0.05$ mag) is to be comparable or larger
than the intrinsic scatter. Additional observations at these wavelengths could
improve the inferred distances to these sources further. As an application of
the methodology, we infer the distance to the RRc-type star RZCep at low
Galactic latitude ($b = 5.5^\circ$) to be $\mu=8.0397\pm0.0123$ mag
($405.4\pm2.3$ pc) with colour excess $E(B-V)=0.2461\pm0.0089$ mag. This
distance, equivalent to a parallax of $2467\pm14$ microarcsec, is consistent
with the published HST parallax measurement but with an uncertainty that is 13
times smaller than the HST measurement. If our measurements (and methodology)
hold up to scrutiny, the distances to these stars have been determined to an
accuracy comparable to those expected with Gaia. As RR Lyrae are one of the
primary components of the cosmic distance ladder, the achievement of sub-1%
distance errors within a formalism that accounts for dust extinction may be
considered a strong buttressing of the path to eventual 1% uncertainties in
Hubble's constant.
