Lower Bounds for the Probability of a Union via Chordal Graphs

We establish new Bonferroni-type lower bounds for the probability of a union of finitely many events where the selection of intersections in the estimates is determined by the clique complex of a chordal graph.


Introduction
The classical Bonferroni inequalities state that for any finite collection of events where P * (V ) denotes the set of non-empty subsets of V . Numerous variants of these inequalities are known, see, e.g., Galambos and Simonelli [6] for a survey.
In this paper, we establish a new variant where the selection of intersections in the probability bounds is determined by a chordal graph. Recall that a graph is chordal if it contains no cycles of length four or any higher length as induced subgraphs.
We refer to [2] for terminology on graphs. We write G = (V , E) to denote that G is a graph having vertex-set V , which we assume to be finite, and edge-set E, consisting of two-element subsets of V . We use C (G) to denote the clique complex of G, that is, the abstract simplicial complex consisting of all non-empty cliques of G.
A recent connection between Bonferroni inequalities and chordal graphs is the chordal graph sieve, which states that in the classical Bonferroni upper bound, P * (V ) may be replaced by the clique complex of some arbitrary chordal graph on V . We recall the chordal graph sieve in the following proposition. and Remark 1.2. Inequality (2) interpolates between Boole's inequality (G edgeless) and the sieve formula (G complete). Moreover, it generalizes a well-known result of Hunter [9] on trees. Note that for any tree G = (V , E), the clique complex of G consists of the vertices and edges of G. Remark 1.3. The case r = 2 in (3) was independently found by Boros and Veneziani [1] by linear programming techniques; see also [16] for a discussion on this bound.

Main result
In the sequel, we use c(G) resp. α(G) to denote the number of connected components of G resp. the independence number of G. Our main result, which is a lower bound analogue of Proposition 1.1, reads as follows. and Remark 2.2. As an immediate consequence of (2), and in contrast to the classical Bonferroni inequalities, the lower bound in (4) is guaranteed to be non-negative.
Remark 2.3. Inequality (4) becomes an equality if G is complete. In this case, (4) agrees with the sieve formula, while (5) coincides with the usual Bonferroni lower bounds.
The proof of Theorem 2.1 is facilitated by a lemma and a proposition. The lemma contains a well-known combinatorial identity.
Proof. It suffices to show that (7) holds for any connected chordal graph G, that is, with equality if r ≥ |V |/2. If G has at most one vertex, then this statement trivially holds. We proceed by induction on the number of vertices of G. If G has at least two vertices, then since G is chordal, G contains a simplicial vertex s, that is, a vertex whose where the equality in (9) follows from (6). Note that if r ≥ |V |/2, then by the induction hypothesis, the first inequality in (9) becomes an identity. The same applies to the second inequality, since for r ≥ |V |/2 the binomial coefficient |N G (s)|−1 2r−1 vanishes.
Remark 2.6. Eq. (8) can also be proved by combining the following topological results: 1. The clique complex of any connected chordal graph is contractible ( [5]).
We are now ready to proceed with the proof of Theorem 2.1.
Proof of Theorem 2.1. It suffices to prove (5), since (4) follows from (5) by choosing some arbitrary r ≥ |V |/2. For any I ⊆ V , I ∅, define Note that the B I 's form a partition of v∈V A v , that is, .
Remark 2.7. In view of the preceding proof, the bounds can be sharpened by replacing (10). In particular, if the underlying graph matches the structure of the events in such a way that G[J] is connected whenever B J is non-empty, then α ′ (G) = 1 and the bounds in (2) and (4) coincide. The resulting identity is well-known in abstract tube theory [3].
Remark 2.8. The requirement that G is chordal cannot be omitted from Theorem 2.1. Consider the non-chordal graph G = (V , E) depicted in Figure 1, and consider events A v , v ∈ V , with Pr(A v ) = 1 for any v ∈ V . The clique complex of this graph consists of 8 cliques of size 1, 20 cliques of size 2, and 16 cliques of size 3. Since α(G) = 3, the first inequality in Theorem 2.1 gives the non-valid bound 1 ≥ 1 3 (8 − 20 + 16) = 4 3 . Remark 2.9. The graph G in the preceding counterexample was obtained as a subgraph of G 3 -the first graph in an infinite sequence (G k ) k=3,5,... of non-chordal graphs for which the hypothesis of Theorem 2.1 does not hold. Each G k in this sequence is the join of k disjoint copies of the edgeless graph on three vertices. It turns out that G k has independence number 3, and that the clique complex of G k has Euler characteristic 1 + 2 k . By considering events A v , v ∈ V , with Pr(A v ) = 1 for any vertex v of G k , the first inequality in Theorem 2.1 becomes 1 ≥ 1 3 (1 + 2 k ), which is false for k = 3, 5, . . . .
Proof. In view of the proof of Theorem 2.1, it suffices to show that for any J ⊆ V , J ∅,

Particular cases
In case of a tree, Theorem 2.1 gives a lower bound analogue of Hunter's inequality [9].
Hunter's upper bound on Pr ( v∈V A v ) coincides with the bracketed term in (11).
Remark 3.2. Since α(G) ≤ n − 1 for any tree G on n > 1 vertices, we conclude from (11) that, in this case, To obtain a maximum lower bound of this type, we adapt Hunter's method [9] for computing a minimum upper bound: Any minimum spanning tree of the complete weighted graph on V with edge weights Pr(A v ∩ A w ) for any v, w ∈ V , v w, maximizes the right-hand side of the preceding inequality. Such a minimum spanning tree can be found in polynomial time, e.g., by applying Kruskal's algorithm [10]. Remark 3.3. If G is a path on n vertices, then α(G) = ⌈n/2⌉ and hence, In order to find a path on V that maximizes the lower bound in (12), consider the complete graph on V with edge lengths Pr(A v ∩ A w ) for any v, w ∈ V , v w. Any minimum length Hamiltonian path in this graph maximizes the right-hand side of (12). Unfortunately, the problem of finding a minimum length Hamiltonian path is NP-hard [8].
Remark 3.5. Inequality (13) may be considered as a lower bound analogue of Kwerel's inequality [11], which states that Pr Our next corollary provides a lower bound analogue of Seneta's inequality [15]. The upper bound in Seneta's inequality coincides with the bracketed term that follows.
By taking the average over all distinct j, k = 1, . . . , n in the preceding inequality, a lower bound analogue of another result due to Kwerel [12] is obtained: Let A 1 , . . . , A n be a finite collection of events where n ≥ 3. Then, The following corollary generalizes Corollary 3.7. Its upper bound analogue [3,4] is related to an inequality of Galambos and Xu [7].
Proof. For any M ⊆ {1, . . . , n} let G M denote the join of the complete graph on M and the edgeless graph on the complement of M. It follows by induction on |M| that each G M is chordal. By averaging the right-hand side of (4) over all G M with |M| = m we obtain Now follow the lines of the proof of Theorem 4.4.7 in [3, pp. 42-43], where it is shown that the bracketed terms in (14) and (15) are equal. Thus, the corollary is proved.

Open problems
Normally, when applying Bonferroni inequalities, joint probabilities of up to k events are given, and reasonable estimates on the probability of their union are wanted. To obtain a minimum upper bound, respectively maximum lower bound on this probability based on the inequalities in (2-5), a chordal graph having clique number k or less is sought which minimizes, respectively maximizes the bound on the right-hand side of these inequalities. This seems to be an intrinsic computational problem. Even in the particular case of a tree, where Hunter's method [9] gives a polynomial time algorithm for obtaining a minimum upper bound, no efficient algorithm is known for maximizing the bound in (11). We leave these computational problems for future research.

Appendix: Application to Network Reliability
Consider the directed network N in Figure 2(a), in which all nodes are perfectly reliable, and the arcs are subject to random and independent failure where all failure probabilities are assumed to be known. We are interested in lower bounds on its sourceto-terminal reliability R st (N ), which is the probability that there is a directed path from s and t consisting of non-failing arcs only. Obviously, this is the case if all arcs in the directed paths 15, 136, 245, 26 are operating. For this ordering of s, t-paths, let A i denote the event that all arcs in the ith path are operating (i = 1, . . . , 4). Then, we have If we assume that the reliability of each arc is p, then by the sieve formula, R st (N ) = 2p 2 + 2p 3 − 5p 4 + 2p 5 .
In order to compare these bounds with the classical Bonferroni bound of degree 2, we put r = 1 in (1). In this way, we obtain