BANYAN. XI. The BANYAN $\Sigma$ multivariate Bayesian algorithm to identify members of young associations within 150 pc.

BANYAN $\Sigma$ is a new Bayesian algorithm to identify members of young stellar associations within 150 pc of the Sun. It includes 27 young associations with ages in the range ~1-800 Myr, modelled with multivariate Gaussians in 6-dimensional XYZUVW space. It is the first such multi-association classification tool to include the nearest sub-groups of the Sco-Cen OB star-forming region, the IC 2602, IC 2391, Pleiades and Platais 8 clusters, and the $\rho$ Ophiuchi, Corona Australis, and Taurus star-formation regions. A model of field stars is built from a mixture of multivariate Gaussians based on the Besan\c{c}on Galactic model. The algorithm can derive membership probabilities for objects with only sky coordinates and proper motion, but can also include parallax and radial velocity measurements, as well as spectrophotometric distance constraints from sequences in color-magnitude or spectral type-magnitude diagrams. BANYAN $\Sigma$ benefits from an analytical solution to the Bayesian marginalization integrals that makes it more accurate and significantly faster than its predecessor BANYAN II. A contamination versus hit rate analysis is presented and demonstrates that BANYAN $\Sigma$ achieves a better classification performance than other moving group classification tools, especially in terms of cross-contamination between young associations. An updated list of bona fide members in the 27 young associations, augmented by the Gaia-DR1 release, are presented. This new tool will make it possible to analyze large data sets such as the upcoming Gaia-DR2 to identify new young stars. IDL and Python versions of BANYAN $\Sigma$ are made available with this publication. (shortened)