Re: [Phys-L] Bayesian statisitics

[John Denker <jsd@av8n.com>]

On 09/01/2013 12:47 PM, Dan Crowe wrote:
> I teach at a math-science magnet program for high school students.
> All of our juniors and seniors conduct research projects of their own
> design.  The majority of these projects are biological, but some of
> them are chemical, environmental, or engineering projects, and a few
> are physics projects.  Most of these projects have limited
> replication sample sizes (3 ≤ N ≤ 10) due to time constraints.
> Currently, we use frequentist statistics to analyze data.  I'm
> interested in learning more about Bayesian statistics to see if it
> would make sense for our students to use Bayesian analysis on their
> data.  I'm looking for recommendations for a good book, or possibly
> other material, that would help me better understand Bayesian
> statistics, including a comparison between the virtues of using
> Bayesian and frequentist statistics to analyze data.  If we decide
> that Bayesian analysis would be useful, we'd be interested in finding
> materials to help our students understand Bayesian analysis, but
> that's not my main focus right now.  All of our juniors and seniors
> study calculus: either AP Calculus AB followed by AP Calculus BC or
> AP Calculus BC followed by multivariable calculus.

Gaack, that's a complicated question.

For starters, there is endless confusion because the term "Bayesian" 
means different things to different people.
 ++ The Bayes inversion formula is a theorem.  Always was, 
  always will be.
 ++ The Bayes decision criterion is entirely reasonable and
  non-controversial.
 ++ Some folks who call themselves "Bayesians" are perfectly
  reasonable and sensible.
 -- On the other hand, some folks who call themselves "Bayesians"
  encumber the theory with layers of confusing restrictions and
  assumptions.  Crucially, it would be a treeemendously bad idea
  to assume there is any such thing as "The" Bayesian prior.

Secondly, the "frequentist" approach is unsophisticated, clumsy,
and 80 years out of date.

You can avoid both sets of problems by using the modern set-theoretic
definition of probability.  An excellent reference for this is
Apostol _Calculus_ volume II.  I strongly recommend the Apostol
books for this reason and eleventeen others.  They are sophisticated
and clever *and* down-to-earth, all at the same time.

Here is my writeup on the subject:
  http://www.av8n.com/physics/probability-intro.htm

The modern approach /includes/ the frequentist approach as a
clumsy, constraining special case:  It's what you get if you use
a Monte Carlo method for measuring the sets.  If you have some
other way of measuring the sets, which you usually do, you don't
need to mess with the frequentist approach ... but you still can
if/whenever you want to.
  http://www.av8n.com/physics/probability-intro.htm#sec-frequentist-def

The modern approach also /includes/ the good parts of the Bayesian
approach.  For instance, the Bayes inversion formula is an obvious
corollary:
  http://www.av8n.com/physics/probability-intro.htm#sec-bayes-inversion

At an even more fundamental level, rather than making some assumption
about "The" Bayesian prior, it is treeemendously better to realize
that there are many different probability measures (just as there
are many different non-Euclidean geometries) and omphaloskepsis is
not going to tell you which of them applies in any given physical
situation.

==============

Bottom line:  I strongly recommend re-formulating whatever you're
trying to do in terms of the set-theoretic definition of probability.
That's easy to do, and it washes away a whole lot of nonsense.

After doing that, if there are followup questions, we can discuss
them.