Equational reasoning for conditioning as disintegration

Conditional distributions are widely used for practical inference, even when the condition has zero probability (such as setting a continuous variable to an observed value). This popularity contrasts with the scary pitfalls (such as Borel’s paradox) that beset rigorous treatments of conditioning. In general, conditional expectations may arise that do not correspond to any conditional distribution at all. This gap between theory and practice has made it difficult to automate conditioning calculations soundly. In particular, given a generative model, we want to produce its conditional expectations mechanically, so as to optimize them numerically, simplify them symbolically, and so on.