The broken non-abelian X-ray transform was introduced in the study of inverse problems for the Yang-Mills-Higgs equations. This transform acts on the space of Hermitian connections on a causal diamond and is known to be injective only up to the action of an infinite-dimensional gauge group. I will present a stability estimate that is invariant under the action of the gauge and leads one to consider a new type of connections that I call light-sink connections. I will then explain how to use light-sink connections to consistently recover the gauge orbit of a connection from noisy measurement of its X-ray transform data through Bayesian inversion.