Theory of Statistics - George Mason University

Notes and Further Reading 377
An alternative approach to probabilistic reasoning is the Dempster-Shafer
theory of belief functions (see Shafer (1976) and Yager and Liu (2008)).
In some cases, especially in hypothesis, the Bayesian approach is fundamentally
different from the frequentist approach. The differences arise from
the definition of the problem that is addressed. The articles by Casella and Berger
(1987) (Roger) and (Jim) Berger and Sellke (1987) with accompanying discussion
by several authors identify some of the differences in perspectives.
Berger (1985) and Robert (2001) provide extensive coverage of statistical
inference from a Bayesian perspective. Both of these books compare the “frequentist”
and Bayesian approaches and argue that the Bayesian paradigm is
more solidly grounded.
Ghosh and Sen (1991) have considered Pitman closeness in the context of
a posterior distribution, and defined posterior Pitman closeness in terms of
probabilities evaluated with respect to the posterior distribution. Interestingly,
the posterior Pitman closeness is transitive, while as we have seen on page 215,
Pitman closeness does not have the transitive property.
Notation and Lingo
There are several instances in which the notation and terminology used in
Bayesian statistics differ from the classical statistics that had evolved with a
strong mathematical flavor.
I generally like to use uppercase letters to distinguish random variables
from realizations of those random variables, which I generally represent by
corresponding lowercase letters, but it is common in writing about a Bayesian
analysis not to distinguish a random variable from its realization.
People who work with simple Bayes procedures began calling the distribution
of the reciprocal of a chi-squared random variable an “inverse” chisquared
distribution. Because “inverse” is used in the names of distributions
in a different way (“inverse Gaussian”, for example), I prefer the term inverted
chi-squared, or inverted gamma.
What is often called a “simple hypothesis” by most statisticians is often
called a “sharp hypothesis in Bayesian analyses.
The Bayesian Religious Wars of the Mid-Twentieth Century
The analysis by Lindley and Phillips (1976) ********************.
in Example 3.12
Hartley (1963)
A rather humorous historical survey of the antithetical Bayesian and frequentist
approaches is given in McGrayne (2011).
Prior Distributions
Ghosh (2011) objective priors
Theory of Statistics c○2000–2013 James E. Gentle