The Guide to the Expression of Uncertainty in Measurement (GUM) is self-consistent when Bayesian statistics is used for the Type A evaluations. We present the case that there are limitations on the kind of Bayesian statistics that can be used for the Type A evaluations of input quantities of the measurement function. The GUM recommends that the (central) measured value should be an unbiased estimate of the corresponding (true) quantity value. Also, the GUM uses the expected value of state-of-knowledge probability distributions as the (central) measured value for both the Type A and the Type B evaluations of input quantities. It turns out that the expected value of a Bayesian posterior distribution used as a Type A (central) measured value for an input quantity can be unbiased only when a non-informative prior distribution is used for that input quantity. Metrologically, this means that only the current observations without any additional information should be used to determine a Type A (central) measured value for an input quantity.
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