Recursive computation for evaluating the exact $p$-values of temporal and spatial scan statistics

Let $V$ be a finite set of indices, and let $B_i$, $i=1,ldots,m$, be subsets of $V$ such that $V=bigcup_{i=1}^{m}B_i$. Let $X_i$, $iin V$, be independent random variables, and let $X_{B_i}=(X_j)_{jin B_i}$. In this paper, we propose a recursive computation method to calculate the conditional expectation $Ebigl[prod_{i=1}^mchi_i(X_{B_i}) ,|, Nbigr]$ with $N=sum_{iin V}X_i$ given, where $chi_i$ is an arbitrary function. Our method is based on the recursive summation/integration technique using the Markov property in statistics. To extract the Markov property, we define an undirected graph whose cliques are $B_j$, and obtain its chordal extension, from which we present the expressions of the recursive formula. This methodology works for a class of distributions including the Poisson distribution (that is, the conditional distribution is the multinomial). This problem is motivated from the evaluation of the multiplicity-adjusted $p$-value of scan statistics in spatial epidemiology. As an illustration of the approach, we present the real data analyses to detect temporal and spatial clustering.