Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling

Abstract

Numerous statistical applications require the evaluation of the probability content of a convex polyhedron. We demonstrate for a given polyhedron in $R^d$ that there is a depth d inclusion-exclusion identity for its indicator function, which is a linear combination of indicator functions of intersections of at most d half-spaces. Terms in the identity are determined by the incidence of the facets of the polyhedron, which can be found using linear programming. This identity can be truncated at any depth to give a lower or upper bound. In addition, the resulting inequalities lead to importance sampling schemes for evaluating the probability content, and these methods tend to be more efficient than the naive hit-or-miss Monte Carlo method. These results arise in a more general setting which we introduce. An abstract tube consists of a pair $(\mathsf{A}, \mathsf{S})$ where $\mathsf{A} = {A_1, \dots, A_n}$ is a collection of sets, $\mathsf{S}$ is a simplicial complex, and where each subcomplex $\mathsf{S}(x) = {F \epsilon \mathsf{S}: x \epsilon \bigcap_{i \epsilon F} A_i}$ is contractible whenever $x \epsilon \bigcup_{i=1}^n A_i$. The notion presented here is stronger than the one introduced earlier by Naiman and Wynn. Several examples are given and key consequences are demonstrated. In particular, arrangements of points and half-spaces in $R^d$ give rise to abstract tubes via Voronoi decompositions and their associated Delauney dual complexes. Every abstract tube is shown to give rise to an inclusion-exclusion identity for $I_{\bigcup_{i=1}^n A_i}$, and upper and lower bounds are obtained by truncating the identity at an even or an odd depth. This property is analogous to the truncation inequality property of the classical inclusion-exclusion identity, which may be viewed as a special case. The notion of an abstract subtube is introduced, and it is shown that if $(\mathsf{A}, \mathsf{S}_1)$ is a subtube of $(\mathsf{A}, \mathsf{S}_2)$ then the truncation inequality gotten from the depth m truncation for $(\mathsf{A}, \mathsf{S}_1)$ is at least as sharp as the corresponding inequality from $(\mathsf{A}, \mathsf{S}_2)$. As a consequence, the generalized inclusion-exclusion inequalities are always at least as sharp as their classical counterparts.