The Bayesian paradigm provides an approach to statistical inference that is different from the methods we have considered thus far.

**BAYESIAN METHODS**

The Bayesian paradigm provides an approach to statistical
inference that is different from the methods we have considered thus far.
Although the topic is not commonly taught in introductory statistical courses,
we believe that Bayesian methods deserve coverage in this text. Despite the
fact that the basic idea goes back to Thomas Bayes’ treatise written more that
200 years ago, the use of the Bayesian idea as a tool of inference really took
place mostly in the 20th century. There are now many books on the subject, even
though it was not previously in favor among mainstream statisticians.

In the 1990s, Bayesian methods had a rebirth in
popularity with the advent of fast computational techniques (especially the
Markov chain Monte Carlo approach-es), which allowed computation of general
posterior probability distributions that had been difficult or impossible to
compute (or approximate) previously. Posterior distributions will be defined
shortly. Bayesian hierarchical methods now are being used in medical device
submissions to the FDA.

A good introductory text that provides the Bayesian
prospective was authored by Berry (1996). Bayesian hierarchical models also are
used as a method for doing meta-analyses (as described from the frequentist
approach in the previous section). An excellent treatment of use of
meta-analyses (Bayesian approaches) in many medical applications is given in
Stangl and Berry (2000), which we mentioned in the previous section.

Basically, in the Bayesian approach to inference,
the unknown parameters are treated as random quantities with probability
distributions to describe their uncer-tainty. Prior to collecting data, a
distribution called the prior distribution is chosen to describe our belief
about the possible values of the parameters.

Although Bayesian analysis is simple when there is
only one parameter, often we are interested in more than one parameter. In
addition, one or more nuisance pa-rameters may be involved, as is the case in
frequentist inference about a mean when the variance is unknown. In this
instance, the mean is the parameter of interest and the variance is a nuisance
parameter. In frequentist analysis, we estimate the vari-ance from the data and
use it to form a *t* statistic whose
frequency distribution does not depend on the nuisance parameter. In the
Bayesian approach, we determine a bivariate prior distribution for the mean and
variance; we use Bayes’ rule and the data to construct a bivariate posterior
distribution for the mean and variance; then we integrate over the values for
the variance to obtain a marginal posterior distribu-tion for the mean.

Bayes’ rule is simply a mathematical formula that
says that you find the posteri-or distribution for a parameter θ by taking the prior distribution for θ and
multiplying it by the likelihood for the data given a specified value of θ. For the mean, this likelihood can be regarded as the sample
distribution for when the population variance is assumed to be known and the population mean is
a specified μ. We know by the central limit theorem that this distribution is
approximately normal with mean and variance σ^{2}/*n*, where σ^{2} is the known variance and *n* is the sample size. The density function for this normal
distribution is the likelihood. We multiply the likelihood by the prior density
for to get the posterior density, called the posterior density of μ given the sample mean .

There is controversy among the schools of
statistical inference (Bayesian and fre-quentist). With respect to the Bayesian
approach, the controversy involves the treatment of μ as a random quantity with a prior distribution. In the discrete case,
it is a simple law of conditional probabilities that if *X* and *Y* are two random
quantities, then *P*[*X = x*|*Y = y*] =* P*[*X = x*,* Y = y*]/*P*[*Y = y*] =* P*[*Y = y*|*X = x*]*P*[*X = x*]/*P*[*Y
= y*]. Now,* P*[*Y = y*] =* **Σ*_{x}* P*[*Y = y*,* X *=*
x*]. This leads to Bayes’ rule, the uncontroversial mathematical result that
*P*[*X
= x*|*Y = y*] = *P*[*Y = y*|*X = x*]*P*[*X = x*]/ *Σ*_{x}*P*[*Y = y*, *X* = *x*].

In the problem of a population mean, the Bayesian
followers take *X* to be the population
mean and *Y* the sample estimate. The
left-hand side of the above equa-tion {*P*[*X = x*| *Y = y*]} is the posterior distribution (or density) for *X*, and the right-hand side is the
appropriately scaled likelihood for *Y*,
given *X* (*P*[*Y = y*|*X = x*]/ Σ_{x}*P*[*Y
= y*,* X *=* x*]) multiplied by the prior distribution (or density) for* X *at*
x *(namely,* P*[*X = x*]). The formula applies for continuous or discrete random
quantities but is* *derived more easily
in the discrete case. The mathematics cannot be disputed, but one can question
philosophically the existence of a prior distribution for *X* when *X* is an unknown
parameter of a probability distribution.

Point estimates of parameters usually are obtained
by taking the mode of the posterior distribution (but means or medians also can
be used). The analog to the confidence interval is called a credible region and
is obtained by finding points *a* and *b* such that the posterior probability
that the parameter μ falls in the interval [*a*, *b*]
is set at a value such as 0.95. Points* a *and* b *are not unique and generally are chosen
on grounds of symmetry. Sometimes the points are selected optimally in order to
make the width of the interval as short as possible.

For hypothesis testing, one constructs an odds
ratio for the alternative hypothesis relative to the null hypothesis as a prior
distribution and then applies Bayes’ rule to construct a posterior odds ratio
given the test data. That is, we have a distribution for the ratio of the
probability that the alternative is true to the probability that the null
hypothesis is true. Before collecting the data, one specifies how large this
ratio should be in order to reject the null hypothesis. See Berry (1996) for
more details and examples.

Markov chain Monte Carlo methods now have made it
computationally feasible to choose realistic prior distributions and solve
hierarchical Bayesian problems. This development has led to a great deal of
statistical research using the Bayesian approach to solve problems. Most
researchers are using the software Winbugs and associated diagnostics to solve
Bayesian problems. Developed in the United King-dom, this software is free of
charge. See Chapter 16 for details on Winbugs.

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