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Gene arrays demonstrate a promising ability to characterize expression levels across the entire genome but suffer from significant levels of measurement noise. We present a rigorous new approach to estimate transcript levels and ratios from one or more gene array experiments, given a model of measurement noise and available prior information. The Bayesian estimation of array measurements (BEAM) technique provides a principled method to identify changes in expression level, combine repeated measurements, or deal with negative expression level measurements. BEAM is more flexible than existing techniques, because it does not assume a specific functional form for noise and prior models. Instead, it relies on computational techniques that apply to a broad range of models. We use Affymetrix yeast chip data to illustrate the process of developing accurate noise and prior models from existing experimental data. The resulting noise model includes novel features such as heavy-tailed additive noise and a gene-specific bias term. We also verify that the resulting noise and prior models fit data from an Affymetrix human chip set.