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.

Probability (math.PR); Machine Learning (cs.LG); Mathematical Physics (math-ph); Category Theory (math.CT); Statistics Theory (math.ST)

arXiv:2610.15006 [math.PR]

(or arXiv:2610.15006v1 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2610.15006 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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