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Motivated by the analysis of extreme rainfall data, we introduce a general Bayesian hierarchical model for estimating the probability distribution of extreme values of intermittent random sequences, a common problem in geophysical and environmental science settings. The approach presented here relaxes the asymptotic assumption typical of the traditional extreme value (EV) theory, and accounts for the possible underlying variability in the distribution of event magnitudes and occurrences, which are described through a latent temporal process. Focusing on daily rainfall extremes, the structure of the proposed model lends itself to incorporating prior geo-physical understanding of the rainfall process. By means of an extensive simulation study, we show that this methodology can significantly reduce estimation uncertainty with respect to Bayesian formulations of traditional asymptotic EV methods, particularly in the case of relatively small samples. The benefits of the approach are further illustrated with an application to a large data set of 479 long daily rainfall historical records from across the continental United States. By comparing measures of in-sample and out-of-sample predictive accuracy, we find that the model structure developed here, combined with the use of all available observations for inference, significantly improves robustness with respect to overfitting to the specific sample.