  We present KaRMMa, a novel method for performing mass map reconstruction from
weak-lensing surveys. We employ a fully Bayesian approach with a physically
motivated lognormal prior to sample from the posterior distribution of
convergence maps. We test KaRMMa on a suite of dark matter N-body simulations
with simulated DES Y1-like shear observations. We show that KaRMMa outperforms
the basic Kaiser-Squires mass map reconstruction in two key ways: 1) our best
map point estimate has lower residuals compared to Kaiser-Squires; and 2)
unlike the Kaiser-Squires reconstruction, the posterior distribution of KaRMMa
maps are nearly unbiased in all summary statistics we considered, namely:
one-point and two-point functions, and peak/void counts. In particular, KaRMMa
successfully captures the non-Gaussian nature of the distribution of $\kappa$
values in the simulated maps. We further demonstrate that the KaRMMa posteriors
correctly characterize the uncertainty in all summary statistics we considered.
