A Probabilistic Approach to Fitting Period-Luminosity Relations and
  Validating Gaia Parallaxes
  Pulsating stars, such as Cepheids, Miras, and RR Lyrae stars, are important
distance indicators and calibrators of the "cosmic distance ladder", and yet
their period-luminosity-metallicity (PLZ) relations are still constrained using
simple statistical methods that cannot take full advantage of available data.
To enable optimal usage of data provided by the Gaia mission, we present a
probabilistic approach that simultaneously constrains parameters of PLZ
relations and uncertainties in Gaia parallax measurements. We demonstrate this
approach by constraining PLZ relations of type $ab$ RR Lyrae stars in
near-infrared W1 and W2 bands, using Tycho-Gaia Astrometric Solution (TGAS)
parallax measurements for a sample of $\approx100$ type $ab$ RR Lyrae stars
located within 2.5 kpc of the Sun. The fitted PLZ relations are consistent with
previous studies, and in combination with other data, deliver distances precise
to 6% (once various sources of uncertainty are taken into account). To a
precision of 0.05 mas ($1\sigma$), we do not find a statistically significant
offset in TGAS parallaxes for this sample of distant RR Lyrae stars (median
parallax of 0.8 mas and distance of 1.4 kpc). With only minor modifications,
our probabilistic approach can be used to constrain PLZ relations of other
pulsating stars, and we intend to apply it to Cepheid and Mira stars in the
near future.
