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A fundamental logical problem in the Bayesian inference of a series system's failure probability is described, a practical means for its mitigation is discussed, and its application to space launch vehicles is illustrated. The problem is the ldquoBayesian Anomaly,rdquo the difference in the system's failure probability per operation inferred from prior estimates, and test or operational experience applied at a lower-level of the system; and from the convolution of the same priors, and of the same experience applied at the system level (or any level above the first). In particular, unlike in a classical inference, the mean estimates differ critically. Although it is not possible to entirely resolve the problem, a practical procedure for mitigating it, establishing consistency among the mean and variance estimates at the two levels, is delineated.