Improved bounds on the probability of the union of events some of whose intersections are empty

Abstract

We formulate a linear program whose optimal objective function value can be used in other formulations to yield improved upper and lower bounds on the probability of the union of events if we know some empty intersections of small numbers of events. The LP relaxation of an extension of the maximum independent set problem provides an upper bound on the largest number of events that have a nonempty intersection. We present numerical experiments demonstrating the effectiveness of our formulation.

Introduction

Computing the probability of the union of events is important in reliability theory, stochastic programming, and other sciences concerned with stochastic systems. In network reliability, consider a communication network with nodes and links, each with a probability of failure. The two-terminal reliability of a pair of nodes is the probability of the union of events, each of which occurs when a path between the two nodes consists of links without failure. The all-terminal reliability is the probability of the union of events, each of which occurs when a spanning tree of the network consists of links without failure. In probabilistic constrained stochastic programming, a joint probabilistic constraint for random variables ${\xi }_1,\dots ,{\xi }_n$ specifies a lower bound on $\mathbb{P}({\xi }_1\le x_1\cap \cdots \cap {\xi }_n\le x_n)=1-\mathbb{P}({\xi }_1>x_1\cup \cdots \cup {\xi }_n>x_n)$, which involves the probability of the union of events. Although it is very hard to compute the exact probability of the union of a large number of events, we can compute its approximation by using the probabilities of individual events and intersections of a small number of events.

Let $\{A_1,\dots ,A_n\}$ be any set of $n(\in {\mathbb{Z}}_{>0})$ events in an arbitrary probability space and

$$N≔\{1,\dots ,n\}$$

be the set of all positive integers at most $n$. For any finite set $I$, we designate its cardinality (i.e., the number of elements in $I$) by $\left|I\right|$. For any subset $I\subseteq N$, we introduce a notation for the probability of the intersection of events $A_i$ for all $i\in I$ as follows:

$$p_I≔\mathbb{P}\left(\bigcap _{j\in I}{A}_j\right)\text{.}$$

We introduce binomial moments $S_i$ for all $i\in N$ as follows:

$$S_i≔\sum _{I\subseteq N:\left|I\right|=i}{p}_I\text{.}$$

Let $\nu$ denote the random number of events among $A_1\dots ,A_n$ that occur. Then we have the following relation for all $i\in N$:

$$\mathbb{E}\left[\left(\genfrac{}{}{00}{}{\nu }{i}\right)\right]=\sum _{j=0}^n\left(\genfrac{}{}{00}{}{j}{i}\right)\mathbb{P}(\nu =j)=S_i\text{,}$$

where we employ the extended definition of binomial coefficients, in which $\left(\genfrac{}{}{00}{}{j}{i}\right)=0$ for all $i,j\in {\mathbb{Z}}_{\ge 0}$ such that $i>j$. The value $S_i$ is called the $i$th binomial moment of $\nu$.

The classical inclusion–exclusion principle  [10] (see also Prékopa  [21]) gives the probability of the union of events by using binomial moments $S_i$ for all $i\in N$ as follows:

$$\mathbb{P}(A_1\cup \cdots \cup A_n)=S_1-S_2+\cdots +{(-1)}^{n-1}{S}_n\text{.}$$

However, this formula is impractical if the number of events $n$ is large, in which case the calculation of $S_i$ is intractable unless $i$ is small enough (close to 1) or large enough (close to $n$) since we need $\left(\genfrac{}{}{00}{}{n}{i}\right)$ sums in (1). We can still calculate $S_i$ for a few small $i$ and compute lower and upper bounds on the probability. Let $m\in N$ be the largest number of events the probability of whose intersection is used in approximating the probability of the union and

$$M≔\{1,\dots ,m\}$$

be the set of all positive integers at most $m$; we use only $p_I$ for all $I\subseteq N$ such that $\left|I\right|\in M$, from which $S_1,\dots ,S_m$ can be calculated by (1). In practice, we usually consider a small $m\ll n$, mostly such as $m=2,3,4$. The well-known Bonferroni inequalities (or bounds) [2] state that for any $m\in N$,

$$\mathbb{P}(A_1\cup \cdots \cup A_n)\left\{\begin{array}{c}\hfill \ge \\ \hfill \le \end{array}\right\}{S}_1-S_2+\cdots +{(-1)}^{m-1}{S}_m\{\begin{array}{cc}\hfill & \text{if }m\text{ is even}\hfill \\ \hfill & \text{if }m\text{ is odd .}\hfill \end{array}$$

These bounds are usually very weak, often out of [0,1]. Then the best possible (also called sharp) bounds using $S_1,\dots ,S_m$ for a small $m$ have been found. In the case $m=2$, the sharp lower and upper bounds expressed as closed forms in terms of $n,S_1,S_2$ are obtained by Dawson and Sankoff  [8] and Kwerel  [15], respectively (see also Prékopa  [18] and Boros and Prékopa  [3]). In the case $m=3$, the sharp bounds expressed as closed forms in terms of $n,S_1,S_2,S_3$ are obtained by Boros and Prékopa  [3]. In the case $m=4$, the sharp upper bound expressed as a closed form in terms of $n,S_1,S_2,S_3,S_4$ is obtained by Boros and Prékopa  [3]. For a general $m$, Prékopa  [18] observed that all these sharp bounds are optimal objective function values of certain linear programs, known as binomial moment problems. Furthermore, sharp bounds on the probabilities that exactly/at least $r$ events occur are given in Prékopa  [19], and the bounds on the probabilities and expectations of convex functions of discrete random variables are given in Prékopa  [20]. A binomial moment $S_i$ carries an aggregated information over event probabilities $p_I$, using every one of which without aggregation we can obtain better bounds. Hailperin  [14] formulated linear programs with an exponential number of variables, known as Boolean probability bounding problems, which give sharp bounds using disaggregated event probabilities.

A pair of binomial moment problems (also called the aggregated LP problems, for probability bounds of the union of events) is formulated as follows  [18]:

$$\text{min/max}\sum _{j\in N}{x}_j$$$$\text{subject to}\sum _{j\in N}\left(\genfrac{}{}{00}{}{j}{i}\right)x_j=S_i\text{for }i\in M$$$$x_j\ge 0\text{for }j\in N\text{.}$$

The optimal objective function values of these minimization and maximization problems are the sharp lower and upper bounds, respectively, on the probability of the union of $n$ events that can be computed using only the aggregated information $S_i$ for all $i\in M$.

A pair of Boolean probability bounding problems (also called the disaggregated LP problems, for probability bounds of the union of events) is formulated as follows  [14]:

$$\text{min/max}\sum _{J\subseteq N:\left|J\right|\in N}{x}_J$$$$\text{subject to}\sum _{J\subseteq N:\left|J\right|\in N}{a}_{IJ}{x}_J=p_I\text{for }I\subseteq N:\left|I\right|\in M$$$$x_J\ge 0\text{for}J\subseteq N:\left|J\right|\in N\text{,}$$

where we define

$$a_{IJ}≔\{\begin{array}{cc}1& \text{if }I\subseteq J\\ 0& \text{otherwise .}\end{array}$$

The optimal objective function values of these minimization and maximization problems are the sharp lower and upper bounds, respectively, on the probability of the union of $n$ events that can be computed using only the disaggregated information $p_I$ for all $I\subseteq N$ such that $\left|I\right|\in M$. Since such a set of event probabilities is almost always the most detailed information we can use, the bounds are the best possible we can expect in general. These problems are, however, impractical owing to an exponential number $(2^n-1)$ of the decision variables $x_J$.

Although we cannot solve the disaggregated LP problems for a large number of events $n$ in practice, we may extract useful information from disaggregated event probabilities without dealing with an exponential size and use it with their aggregations to obtain improved bounds.