DisCo Algebra: Why Disintegration and Combination Are Not Inverses

The gist: DisCo algebra presents a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on bare probability measures, revealing an asymmetry where combination followed by disintegration is the identity, but disintegration followed by combination is not, exposing a fundamental hardness in representing high-dimensional dependence <ref:2610.15006#pg5>.

Objects: bare probability measures

The framework operates on the set of all Borel probability measures on I n denoted as Pn:= Prob([0, 1]n <ref:2610.15006#pg3>. No assumption on marginal uniformity, absolute continuity, or any external transformation is made <ref:2610.15006#pg4>. The only structure beyond a general finite measure is normalization to total mass 1 <ref:2610.15006#pg4>.

Coordinate partitions and block projections

A partition I = (I1,..., Ik) of the index set is fixed, defining block sizes mj = Ij and Pj mj = n <ref:2610.15006#pg5>. The block projection πI reorders coordinates such that πI(u) = (u(1),..., u(k)) <ref:2610.15006#pg5>. This pushforward yields the block-wise joint measure µI = (πI)∗µ, and each block margin is defined as µj:= (πIj)∗µ ∈ Pmj <ref:2610.15006#pg5>.

Block-wise CDF summaries: intrinsic, not external

To reduce dimension, a summary is created by compressing the mj-dimensional vector to a scalar using the block’s own joint cumulative distribution function Ψj (u(j)):= µj [0, u(j) 1] × · · · × [0, u(j) mj] <ref:2610.15006#pg6>. This summary is canonical because it depends only on the cumulative mass of µj and has probability semantics where the fiber Ψ−1 j (t) is the equiprobability surface “cumulative probability exactly t” <ref:2610.15006#pg6>. Composing these summaries yields a dimension-reduction map Π = Ψ1 × · · · × Ψk: I n −→ I k, resulting in Φk:= Π∗µ ∈ Pk <ref:2610.15006#pg6>.

Disintegration along block-wise CDF summaries

Lemma 5.1 establishes the existence of a unique regular conditional measure t 7→ µ(· t) supported on Π−1 (t) such that Z In f(u) dµ(u) = Z I k Z Π−1(t) f(u) dµ(u t)! dΦk(t). The fiber Π−1 (t) is characterized block-wise by Ψj (u(j)) = tj <ref:2610.15006#pg7>. Independently, for each block j, the measure µj admits a unique regular conditional measure u(j) 7→ µj (· tj) along its own CDF summary Ψj <ref:2610.15006#pg8>. This leads to the combination formula: µˆ:= Z I k (O k j=1 µj (· tj)) dΦk(t1,..., tk) (<ref:2610.15006#pg8>).

Probability cooperad and operad structures

The family P• carries a symmetric cooperad structure via the cocomposition map ∆I: Pn −→ O k j=1 Pmj ⊗ Pk, where ∆I(µ) = (µ1,..., µk; Φk). Coassociativity follows from iterated disintegration uniqueness and the additive compatibility of block CDF summaries. The family also carries a symmetric operad structure with composition γ Φk; µ1,..., µk:= Z I k (O k j=1 µj (· tj)) dΦk(t1,..., tk). The signature checks confirm that γ: Pk ⊗ Pm1 ⊗ · · · ⊗ Pmk −→ Pn is closed on bare measures.

The asymmetry: not inverses

Proposition 8.1 states that combination then disintegration is the identity on admissible inputs, as ∆I γ(Φk; µ1,..., µk) = (µ1,..., µk; Φk) holds for every admissible input. However, disintegration then combination is not the identity on Pn in general, as γ ∆I(µ) ≠ µ whenever µ /∈ im(γ). This discrepancy arises because the integrand in γ uses N j µj (· tj)—the conditionally independent product—as its integrand, which differs from the true conditional distribution µ(· t) when residual cross-block dependence exists.

Information-theoretic quantification

Theorem 9.1 quantifies the information loss of disintegration as DKLµ∥ µ1 ⊗ · · · ⊗ µk = DKLΦk∥ Φ ind k + X k j=1 Et∼Φk DKLµj (· tj). The residual loss of the disintegration–combination round trip is Et∼Φk h DKL µ(· t) O k j=1 µj (· tj) i ≥ 0, with equality iff µ ∈ im(γ). This measures precisely what is lost: the conditional dependence between blocks given their scalar summaries.

Why this is not a defect

The asymmetry is a hardness result about high-dimensional probability, where combination is a lossless embedding and disintegration is lossy compression. The image im(γ) ⊊ Pn (for k < n) represents the class of distributions whose cross-block dependence is exactly captured by the block-wise CDF summary Φk. This illustrates that dependence is layered, and low-dimensional summaries transmit only one layer, meaning the DisCo format has an intrinsic expressive ceiling.

Relation to copula theory and prior work

DisCo generalizes Sklar’s theorem by allowing blocks to group coordinates, where the skeleton lives on block-level summaries. The classical copula density formula is the degenerate special case where blocks are not further subdivided. DisCo algebra belongs to the stream addressing how a joint distribution factorizes into interfaces and conditioned parts, distinct from categorical approaches like the Giry monad.

Conclusion

Joint probability is derived in DisCo algebra by choosing a partition, disintegrating into block margins and a coupling skeleton, and then combining. The central lesson is that compression is never the inverse of construction; disintegration followed by combination fails to recover arbitrary distributions because the format cannot hold what it was not designed to hold.

References

[1] A. Sklar. Fonctions de r´epartition a n dimensions et leurs marges, Publications de l’Institut de Statistique de l’Universit´e de Paris, 8:229–231, 1959.

[2] O. Kallenberg. Foundations of Modern Probability, 2nd ed. Probability and its Applications, Springer, 2002.

[3] M. Giry. A categorical approach to probability theory, In B Banaschewski (ed.), Categorical Aspects of Topology and Analysis, Lecture Notes in Mathematics 915, Springer, 68–85, 1982.

[4] T. Leinster. An operadic introduction to entropy, n-Category Caf´e, 2011; see Chapter 12 of [5].

[5] T. Leinster. Entropy and Diversity: The Axiomatic Approach, Cambridge University Press, 2021.

1. DisCo algebra provides a framework for constructing complex joint probability distributions by explicitly separating lossy compression from lossless embedding, allowing AI systems to manage information trade-offs during modeling. The combination followed by disintegration is the identity on admissible inputs property ensures that if an AI system learns a representation through combination, it can perfectly reconstruct the original structure when decomposed back into block margins and summaries.

2. AI systems can utilize block-wise CDF summaries as intrinsic, measure-determined 1D summaries to achieve lossy compression. This allows for efficient dimensionality reduction in high-dimensional data (like image features or large datasets) by compressing the joint distribution into a manageable scalar summary before performing analysis.

3. The asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries, enabling AI to identify and quantify the irreducible complexity of high-dimensional data that cannot be represented by simple block-level summaries. This helps in understanding the inherent limitations of feature engineering or low-rank approximations in complex models.

4. AI models can employ the KL decomposition theorem to precisely quantify the residual loss incurred during compression, stating that The residual loss of the disintegration–combination round trip is exactly quantifiable. This allows for adaptive model design where one can tune the trade-off between model complexity and information retention based on a measurable error bound.

5. AI systems can leverage the operad structure to define composition rules through conditional kernels are not stored in the objects; they are invoked inside the definition of γ, enabling modular AI architectures where complex conditional dependencies are dynamically computed during inference rather than being pre-stored as static parameters.

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 P n= 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.