  We present a Bayesian reconstruction algorithm that infers the
three-dimensional large-scale matter distribution from the weak gravitational
lensing effects measured in the image shapes of galaxies. The algorithm is
designed to also work with non-Gaussian posterior distributions which arise,
for example, from a non-Gaussian prior distribution. In this work, we use a
lognormal prior and compare the reconstruction results to a Gaussian prior in a
suite of increasingly realistic tests on mock data. We find that in cases of
high noise levels (i.e. for low source galaxy densities and/or high shape
measurement uncertainties), both normal and lognormal priors lead to
reconstructions of comparable quality, but with the lognormal reconstruction
being prone to mass-sheet degeneracy. In the low-noise regime and on small
scales, the lognormal model produces better reconstructions than the normal
model: The lognormal model 1) enforces non-negative densities, while negative
densities are present when a normal prior is employed, 2) better traces the
extremal values and the skewness of the true underlying distribution, and 3)
yields a higher pixel-wise correlation between the reconstruction and the true
density.
