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Based on the observation that design and inference are both generalized inverse problems, we devise a new approach that uses the Bayesian inference framework for the automated design of linear antenna arrays. Compared to the optimization-based techniques that are widely used for automated antenna design, this newly-developed method has a prominent advantage, which is the capability to determine automatically the number of antenna elements required to satisfy design requirements and specifications. Three broadside array design problems, which include the null-controlled pattern, the sector beam pattern, and the Chebyshev pattern, along with an end-fire array design problem, are presented as examples. The obtained results demonstrate the advantages of using the Bayesian inference framework for the design of linear antenna arrays.