Title: The inflation technique solves completely the classical inference problem,
Abstract: 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.