<p>Making probabilistic comparisons about changes in multidimensional poverty at the small-area level over time and between municipalities is an essential goal. Statistical comparisons require information about both the mean and variance (uncertainty) of the estimated prevalence of poverty. The official municipal-level poverty figures in Mexico have been produced using an unconventional bottom-up approach designed to meet multiple information needs for the public administration. This approach, however, is at odds with contemporary small-area estimation standards in that it attempts to treat both individuals and municipalities like simultaneous estimation targets in a multivariate setting. Having multiple estimation goals has two significant side effects: high bias and a lack of uncertainty intervals for making proper statistical comparisons. Therefore, it is impossible to make statistical inferences on changes as only biased point estimates are reported. This paper employs hierarchical Bayesian estimation to produce multidimensional poverty prevalence rates at the municipal level for 2010, 2015, and 2020. These estimates reduce bias by half compared to the official figures and provide credible intervals to enable straightforward probabilistic comparisons of poverty changes between municipalities and over time.</p>