Agents receive private signals about an unknown state. The resulting joint belief distributions are complex and lack a simple characterization. Our key insight is that, when conditioned on the state, the structure of belief distributions simplifies: feasibility constrains only the marginal distributions of individual agents across states, with no joint constraints within a state. We apply this insight to multi-receiver persuasion, identifying new tractable cases and introducing optimal transportation and duality tools.Comment: subsumes the old paper titled "Persuasion as Transportation"