We present AURA, a novel abstract interpretation for obtaining sound, precise bounds on the posterior distributions computed by probabilistic programs. AURA allows programmers to specify interval bounds that capture uncertainty or perturbations of the observed data. AURA abstractly computes the infinite set of posteriors that would result from performing inference for any possible data value in the specified perturbation range. AURA then certifies precise bounds on probabilistic queries over that set of posteriors. AURA’s precision stems from a novel gradient-based optimization leveraging the structure of probabilistic programs. Our evaluation across 11 benchmarks with data perturbation shows that AURA improves precision by an order of magnitude (12.8x on average) over the interval-based abstract interpreter, within a run time of 3.1 seconds (geomean), using a GPU parallel implementation.