Hierarchical inference of the relationship between Concentration and
  Mass in Galaxy Groups and Clusters
  Mass is a fundamental property of galaxy groups and clusters. In theory weak
gravitational lensing will enable an approximately unbiased measurement of
mass, but parametric methods for extracting cluster masses from data require
the additional knowledge of concentration. Measurements of both mass and
concentration are limited by the degeneracy between the two parameters,
particularly in low mass, high redshift systems where the signal-to-noise is
low. In this paper we develop a hierarchical model of mass and concentration
for mass inference we test our method on toy data and then apply it to a sample
of galaxy groups and poor clusters down to masses of $\sim$ 1e13 M$_\odot$. Our
fit and model gives a relationship among masses, concentrations and redshift
that allow prediction of these parameters from incomplete and noisy future
measurements. Additionally the underlying population can be used to infer an
observationally based concentration-mass relation. Our method is equivalent to
a quasi- stacking approach with the degree of stacking set by the data. We also
demonstrate that mass and concentration derived from pure stacking can be
offset from the population mean with differing values depending on the method
of stacking.
