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Cyan comments on What is Bayesianism? - Less Wrong
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If it helps, I think this is an example of a problem where they give different answers to the same problem. From Jaynes; see http://bayes.wustl.edu/etj/articles/confidence.pdf , page 22 for the details, and please let me know if I've erred or misinterpreted the example.
Three identical components. You run them through a reliability test and they fail at times 12, 14, and 16 hours. You know that these components fail in a particular way: they last at least X hours, then have a lifetime that you assess as an exponential distribution with an average of 1 hour. What is the shortest 90% confidence interval / probability interval for X, the time of guaranteed safe operation?
Frequentist 90% confidence interval: 12.1 hours - 13.8 hours
Bayesian 90% probability interval: 11.2 hours - 12.0 hours
Note: the frequentist interval has the strange property that we know for sure that the 90% confidence interval does not contain X (from the data we know that X <= 12). The Bayesian interval seems to match our common sense better.
My intuition would be that the interval should be bounded above by 12 - epsilon, since the probability that we got one component that failed at the theoretically fastest rate seems unlikely (probability zero?).
You can treat the interval as open at 12.0 if you like; it makes no difference.