Bayesian reasoning is a valid way to update our estimate of the probability of project success. We believe that the objections to its use are excuses rather than reasons.
As the project progresses, we continually recompute the probability of success based on new performance-against-plan data. Bayes' Theorem provides us with the required mathematical framework. A key ingredient is the quality of the new data, based on two easily computed numbers that measure its validity. Binary tests usually have asymmetric characteristics, in that a positive result may boost an estimate more than a negative result depresses it, or vice-versa. Calculations involve a 2x2 matrix of historical results, and one nomogram.
The accuracy of the initial probability estimate becomes less and less important with more and more subsequent tests. Very weak tests don't move the needle; a single soft result doesn't count much against all the previous knowledge implicit in the current estimate. On the other hand, a very strong test can decisively tip the scales. Applied iteratively, the predicted probability usually converges to 0% or 100%, with the rate of convergence dependent on the strength of the tests.
The three principal advantages of the method are its use of historical data, the employment of binary tests rather than murky assessments at the milestones, and the removal of as much subjectivity from the process as possible. The caveat that needs to be added is that the organization must be relatively stable, because the analysis depends on historical data. If past performance is but a weak indicator of future outcomes, then the method will reflect that weakness. Judgment is always required, but reasoning from numbers can make the arguments more cogent. Bayes can be a powerful tool with good data, a stable organization, and intelligent application.