Journal of Data Science 6(2008), 261-268 A Bayesian Approach to Zero-Numerator Problems Using Hierarchical Models Zhongxue Chen and Monnie McGee Southern Methodist University Abstract: The rule of three gives 3/n as the upper 95% bound for the success rate of the zero-numerator problems. However, this bound is usu- ally conservative although it is useful in practice. Some Bayesian methods with beta distributions as priors have been studied. However, choosing the parameters for the priors is subjective and can severely impact the corre- sponding posterior distributions. In this paper, some hierarchical models are proposed, which provide practitioners other options for those zero-numerator problems. Key words: Bayesian hierarchical model, binomial model, Markov chain Monte Carlo, rare events. 1. Introduction 1.1 Introduction Suppose we want to know the probability of the occurrence of a certain kind of event when in n independent trials, the event never occurs. This situation is referred as the zero-numerator problem. A probability model for this issue can be built by a binomial distribution with sample size n and the probability p, which is usually very small. Based on this binomial model, the point estimate of p by the maximum likelihood estimator is p = x/n = 0 since here x = 0. This estimate is not accurate and may not useful in practice. Although this event is rare, it may occur on occasion based on our previous experience. 1.2 Frequentist method and the rule of three Louis (1981) gave a (1 − α) × 100 percent confidence interval for p: [0,p n ], p n =1 − α 1/n = S b /n, (1.1) where S n can be considered as a number of successes in a future experiment of the same size. By taking limit as n →∞, then we have