The main advantages the Bayesian approach has over the others are (in addition to the non-negligible fact that it is able to treat problems on which the others fail):
When employed on the problem of measurement errors, as a special application of conditional probabilities, it allows all possible sources of uncertainties to be treated in the most general way.
When the problems get complicated and the general method becomes too heavy to handle, it is often possible to use approximate methods based on the linearization to evaluate average and standard deviation of the distribution, while the central limit theorem makes the final distributions approximately Gaussian. Nevertheless, there are some cases in which the linearization may cause severe problems, as shown in Section . In such cases one needs to go back to the general method or to apply other kinds of approximations which are not just blind use of the covariance matrix.
Many conventional (frequentistic) methods can be easily recovered, like maximum likelihood or fitting procedures, as approximation of Bayesian methods, when the (implicit) assumptions on which they are based are reasonable.