  We study the broken non-abelian X-ray transform in Minkowski space. This
transform acts on the space of Hermitian connections on a causal diamond and is
known to be injective up to an infinite-dimensional gauge. We show a stability
estimate that takes into account the gauge, leading to a new proof of the
transform's injectivity. Our proof leads us to consider a special type of
connections that we call light-sink connections. We then show that we can
consistently recover a light-sink connection from noisy measurement of its
X-ray transform data through Bayesian inversion.
