  The causal compatibility question asks whether a given causal structure graph
-- possibly involving latent variables -- constitutes a genuinely plausible
causal explanation for a given probability distribution over the graph's
observed variables. Algorithms predicated on merely necessary constraints for
causal compatibility typically suffer from false negatives, i.e. they admit
incompatible distributions as apparently compatible with the given graph. In
[arXiv:1609.00672], one of us introduced the inflation technique for
formulating useful relaxations of the causal compatibility problem in terms of
linear programming. In this work, we develop a formal hierarchy of such causal
compatibility relaxations. We prove that inflation is asymptotically tight,
i.e., that the hierarchy converges to a zero-error test for causal
compatibility. In this sense, the inflation technique fulfills a longstanding
desideratum in the field of causal inference. We quantify the rate of
convergence by showing that any distribution which passes the $n^{th}$-order
inflation test must be $O\left(n^{-1/2}\right)$-close in Euclidean norm to some
distribution genuinely compatible with the given causal structure. Furthermore,
we show that for many causal structures, the (unrelaxed) causal compatibility
problem is faithfully formulated already by either the first or second order
inflation test.
