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I would refer everyone on this thread to the wonderful article "From
Laplace to Supernova SN 1987A: Bayesian Inference in Astrophysics"
by Tom Loredo, located at
http://www.astro.cornell.edu/staff/loredo/bayes/tjl.html
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.
Brian Blais