We present a tool for conditioning distributions over R that are mixtures of discrete and continuous parts, as well distributions over disjoint sums and dependent products.

This is useful in several scenarios. The primary example is Russell’s GPA problem [Hay et al. 2017], where the nationality of a student – hypothesized using clamped normal distributions – is to be determined using their observed GPA. This involves conditioning on mixtures of continuous and discrete distributions over R. A different scenario emerges in the single-site proposal distribution for the Metropolis-Hastings MCMC sampler, where for any given transition we perturb at most one dimension of a multi-dimensional state. This involves conditioning on a dependent product of variables arising from mixture distributions. Finally, the reversible-jump MCMC sampler presents a third scenario, involving conditioning on a disjoint sum of expressions of varying dimensions. The conditioning tool applies uniform reasoning across all three kinds of applications.