The cost of factorizing
relational structure
The whole is sometimes more than the sum of its parts. Circulatory Fidelity measures when that's true, and by how much, across … scientific domains.
Most of the time, treating a system as independent parts works fine. Sometimes the connections carry more than the parts do. When that happens, splitting the system apart throws away structure you can't get back from the pieces alone.
Circulatory Fidelity is a way to tell those two cases apart, and to put a number on the difference.
The Problem
Factorization has a cost,
and that cost is measurable
Across the sciences, we routinely approximate a system as independent parts: we treat the whole as the sum of its parts. In inference, that's the mean-field approximation, q(x, z) ≈ q(x) · q(z). It usually works well, and organisms evolved to do it cheaply. But sometimes the connections carry more than the parts do, and then factorizing throws away what you can't recover from the parts alone. Circulatory Fidelity measures how much.
The Factorization Assumption
Mean-field variational inference, the workhorse of modern Bayesian computation, approximates joint distributions as products of independent factors: q(z) = ∏i qi(zi). This explicitly discards all relational structure between variables.
When does this approximation fail?The Hidden Cost
There's no free lunch in factorization. Every time we treat coupled variables as independent, we pay an information cost, one the factorized analysis can't see. Where the coupling is strong, that cost can dominate the result.
How do we measure what we've lost?Beyond Pairwise Structure
Even when we check for correlations, standard pairwise analysis misses a deeper kind of structure. XOR-like dependencies, where three variables are tightly coupled but any two look independent, are invisible to marginal observation.
IC₂ = 0 does not mean independenceThe Metabolic Constraint
Biological observers evolved under severe energy budgets. Pairwise observation is metabolically cheap; higher-order observation is expensive. Whether such hidden structure is genuinely rare or just expensive to observe is an open question, and the reason a higher-order test is needed.
Higher-order structure can hide from cheap observationThe Diagnostic
One question, asked precisely
The four problems above come down to a single measurement. Inference Coupling reports how much of a system's behavior lives in the relationships rather than the parts. Read it, and one question answers itself.
Is the relational structure load-bearing?
Keep the relationships. Factorizing pays a cost you can now see and measure, and the structure it discards won't come back from the parts.
Factorize freely. Treating the system as independent parts is adequate here, and CF is what tells you so. Reductionism has its place; this marks where.