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From my personal experience trying to learn basic statistics, I always got hung up
on the notion of a population, and of the standard deviation of the mean.
I found
the Bayesian approach to be both more intuitive, easier to apply to real data, and
more mathematically sound (there is a great article by E.T. Jaynes at
http://bayes.wustl.edu/etj/articles/confidence.pdf where he outlines several
pathologies in standard stats).
Bottom line: there is no population in the Bayesian approach. Probability is a
measure of ones state of knowledge, not a property of the system. In doing so, all
of the strained attempts at creating a fictitious population out of measurements
vanish (such as, say, analyzing measurements of the mass of the moon by imagining
many hypothetical universes of "identical" measurements). On instead is quantifying
your state of knowledge.
In almost all easy cases, the Bayesian approach yields the *exact same* numerical
result as the standard approach. The interpretation is a lot easier, and a lot
easier to communicate to students.