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This paper develops a Bayesian gamma mixture model approach to automatic target recognition (ATR). The specific problem considered is the classification of radar range profiles (RRPs) of military ships. However, the approach developed is relevant to the generic discrimination problem. We model the radar returns (data measurements) from each target as a gamma mixture distribution. Several different motivations for the use of mixture models are put forward, with gamma components being chosen through a physical consideration of radar returns. Bayesian formalism is adopted and we obtain posterior distributions for the parameters of our mixture models. The distributions obtained are too complicated for direct analytical use in a classifier, so Markov chain Monte Carlo (MCMC) techniques are used to provide samples from the distributions. The classification results on the ship data compare favorably with those obtained from two previously published techniques, namely a self-organizing map and a maximum likelihood gamma mixture model classifier.