Multilevel regression with post-stratification (MrP) is the dominant method for estimating public opinion in subnational units. The method requires the full joint distribution of demographic variables in a subnational unit, which is often unavailable from public census tabulations. Existing approaches fill this gap by imposing assumptions to combine marginal distributions of census variables into a synthetic joint distribution. We propose sharp MrP bounds instead: the smallest interval containing all estimates of subnational public opinion consistent with the marginal distributions alone. The bounds solve a pair of linear programs and are computationally feasible for typical applications in political science. Applied to two prominent examples in the literature, the bounds are tight enough to be informative and often more accurate than MrP restricted to a coarser, observed joint distribution of census variables. The bounds serve as margin-consistent interval estimates and a diagnostic when researchers use a synthetic joint distribution to construct MrP estimates.

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