The inflation technique solves completely the classical inference problem

The causal inference problem consists in determining whether a probability distribution over a set of observed variables is compatible with a given causal structure. In [arXiv:1609.00672], one of us introduced a hierarchy of necessary linear programming constraints which all the observed distributions compatible with the considered causal structure must satisfy. In this work, we prove that the inflation hierarchy is complete, i.e., any distribution of the observed variables which does not admit a realization within the considered causal structure will fail one of the inflation tests. More quantitatively, we show that any distribution of measurable events satisfying the $n^{th}$ inflation test is $O\left(\frac{1}{\sqrt{n}}\right)$-close in Euclidean norm to a distribution realizable within the given causal structure. In addition, we show that the corresponding $n^{th}$-order relaxation of the dual problem consisting in maximizing a $k^{th}$ degree polynomial on the observed variables is $O\left(\frac{k^2}{n}\right)$-close to the optimal solution.