15.4 Concluding Remarks
597
say that G is a family
of
conjugate priors
to
H.
Thus, the beta distribution
is
a
conjugate prior to the binomial, and the normal is self-conjugate. Conjugate priors
may not exist; when they do, selecting a member
of
the conjugate family
as
a prior is
done mostly for mathematical convenience, since the posterior can be evaluated very
simply. More generally, numerical methods
of
integration would have to be used to
evaluate the posterior.
~
ncluding
Remarks
This chapter has presented a brief introduction to two areas
of
mathematical statis-
tics, decision theory and Bayesian inference. The analysis has been somewhat more
abstract and theoretical than that
of
earlier chapters.
In order for decision theory to be relevant to a practical problem, the problem
must be such that a choice
is
to be made from a set
of
specified actions and a definite
measure
of
loss
is
associated with each possible action. Critics
of
decision theory
claim that most scientific investigations do not
fit
this paradigm; some have gone so
far as
to
call such a point
of
view totalitarian! In fact, experimental scientists very
rarely use decision theoretic methods in analyzing their data, but decision theory
has been used more often
in
business and economics, where it seems to be easier to
specify the relevant loss functions.
The subjectivist, or Bayesian, point
of
view toward statistics
is
somewhat contro-
versial. Arguments between frequentists and Bayesians have often been quite heated,
and it is probably fair
to
say that the Bayesian approach has not been adopted by
most practicing statisticians. Critics
of
the Bayesian paradigm look with disfavor on
the introduction
of
a prior distribution. Some dislike the subjective element thereby
introduced, maintaining that statistics should be "objective." Others, although not to-
tally unsympathetic, question how prior distributions can be determined
in
practice.
This book has been relatively unconcerned with philosophical or foundational
matters and has made no attempt to develop a consistent "theory"
of
statistics. The
subject matter under investigation and the role that statistics plays in the investigation
should effectively determine whether it is more appropriate for the user
of
statistics
to take a Bayesian, decision-theoretic, frequentist, or purely data-analytic point
of
view, or some combination
of
these.
)
)blems
The losses
of
actions
ai' a
2
,
and
a
3
,
depending on
t1
1
,
t1
2
,
and
t1
3
,
are given in the
following table:
1