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Let $V$ be a finite set of indices, and let $B_i$, $i=1,\ldots,m$, be subsetsof $V$ such that $V=\bigcup_{i=1}^{m}B_i$. Let $X_i$, $i\in V$, be independentrandom variables, and let $X_{B_i}=(X_j)_{j\in B_i}$. In this paper, we proposea recursive computation method to calculate the conditional expectation$E\bigl[\prod_{i=1}^m\chi_i(X_{B_i}) \,|\, N\bigr]$ with $N=\sum_{i\in V}X_i$given, where $\chi_i$ is an arbitrary function. Our method is based on therecursive summation/integration technique using the Markov property instatistics. To extract the Markov property, we define an undirected graph whosecliques are $B_j$, and obtain its chordal extension, from which we present theexpressions of the recursive formula. This methodology works for a class ofdistributions including the Poisson distribution (that is, the conditionaldistribution is the multinomial). This problem is motivated from the evaluationof the multiplicity-adjusted $p$-value of scan statistics in spatialepidemiology. As an illustration of the approach, we present the real dataanalyses to detect temporal and spatial clustering.

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