The Bayesian statistical theory of measurement uncertainty, recently developed to form a mathematical foundation of international recommendations, is applied to the problem of deciding whether or not two measurement results y1 and y2 of the same measurand conform with one another, taking into account the uncertainties associated with the measurement results. Four conformity criteria proposed within the framework of the theory are discussed and compared with three commonly used criteria based on conventional statistics and the least-squares method. All the criteria turn out to be essentially equivalent to one another. They all yield the conformity condition mod y1-y2 mod <or= beta s but with different suitable numbers beta in the range from 1 to 3. s is a standard deviation expressing the uncertainty corresponding to the difference y1-y2.