Next-generation sequencing affords an efficient analysis of transposon insertion libraries, which can be used to identify essential genes in bacteria. To analyse this high-resolution data, we present a formal Bayesian framework for estimating the posterior probability of essentiality for each gene, using the extreme-value distribution to characterize the statistical significance of the longest region lacking insertions within a gene. We describe a sampling procedure based on the Metropolis-Hastings algorithm to calculate posterior probabilities of essentiality while simultaneously integrating over unknown internal parameters.