  Neutrinoless double beta decay is the only realistic probe of the Majorana
nature of the neutrino. In the standard picture, its rate is proportional to
$m_{ee}$, the e-e element of the Majorana neutrino mass matrix in the flavor
basis. I explore minimally allowed $m_{ee}$ values within the framework of mass
matrix anarchy where neutrino parameters are defined statistically at low
energies. Distributions of mixing angles are well defined by the Haar
integration measure, but masses are dependent on arbitrary weighting functions
and boundary conditions. I survey the integration measure parameter space and
find that for sufficiently convergent weightings, $m_{ee}$ is constrained
between (0.01-0.4) eV at 90% confidence. Constraints from neutrino mixing data
lower these bounds. Singular integration measures allow for arbitrarily small
$m_{ee}$ values with the remaining elements ill-defined, but this condition
constrains the flavor structure of the model's ultraviolet completion. Bounds
below $m_{ee} \sim 5\times10^{-3}$ eV should indicate symmetry in the lepton
sector, new light degrees of freedom or the Dirac nature of the neutrino.
