DisCo Algebra: Why Disintegration and Combination Are Not Inverses

Abstract

We present DisCo algebra---a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not---the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.