  Cosmological measurements of structure are placing increasingly strong
constraints on the sum of the neutrino masses, $\Sigma m_\nu$, through Bayesian
inference. Because these constraints depend on the choice for the prior
probability $\pi(\Sigma m_\nu)$, we argue that this prior should be motivated
by fundamental physical principles rather than the ad hoc choices that are
common in the literature. The first step in this direction is to specify the
prior directly at the level of the neutrino mass matrix $M_\nu$, since this is
the parameter appearing in the Lagrangian of the particle physics theory. Thus
by specifying a probability distribution over $M_\nu$, and by including the
known squared mass splittings, we predict a theoretical probability
distribution over $\Sigma m_\nu$ that we interpret as a Bayesian prior
probability $\pi(\Sigma m_\nu)$. We find that $\pi(\Sigma m_\nu)$ peaks close
to the smallest $\Sigma m_\nu$ allowed by the measured mass splittings, roughly
$0.06 \, {\rm eV}$ ($0.1 \, {\rm eV}$) for normal (inverted) ordering, due to
the phenomenon of eigenvalue repulsion in random matrices. We consider three
models for neutrino mass generation: Dirac, Majorana, and Majorana via the
seesaw mechanism; differences in the predicted priors $\pi(\Sigma m_\nu)$ allow
for the possibility of having indications about the physical origin of neutrino
masses once sufficient experimental sensitivity is achieved. We present fitting
functions for $\pi(\Sigma m_\nu)$, which provide a simple means for applying
these priors to cosmological constraints on the neutrino masses or
marginalizing over their impact on other cosmological parameters.
