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536 Further Bayesian methods
resulting in useful treatments being discarded. Spiegelhalter et al. (1994) also
discuss several other kinds of priors that might be used in clinical trials.
Spiegelhalter et al. (1999) give a more recent perspective on these issues.
Non-informative priors
If it is decided to eschew specific prior knowledge, then a non-informative or
vague prior must be adopted. This is often the approach taken when the desire is
to have an `objective' Bayesian analysis in which the prior specification has
minimal effect. Many authors argue that the notion of vague prior knowledge
is unsatisfactory (see, for example, Bernardo & Smith, 1994, §5.4), for, while the
notion of an objective analysis might be superficially attractive, it is inherently
flawed. Attempts to reconcile these two views have led to the development of
reference priors, which attempt to be minimally informative in some well-defined
sense; a full discussion of this topic is beyond the scope of this chapter and the
interested reader is referred to §5.4 of Bernardo and Smith (1994) and to
Bernardo and RamoÂn (1998). Despite its shortcomings, several approaches to
the definition of vague priors are widely used and the rationale behind some of
these will be discussed below. An interesting account of the historical background
to this topic can be found in §5.6 of Bernardo and Smith (1994).
One of the most practical ways to represent prior ignorance is to choose a
prior distribution that is very diffuse. Typically, a prior distribution defined in
terms of parameters, h, is used and the elements of h that determine the dispersion
of the prior are chosen to be large. If possible, it will usually be convenient
to try to arrange for the prior to be conjugate (see §6.2) to the likelihood of the
data. For example, in a study of blood pressure, a normal prior with a standard
deviation of 1000 mmHg will convey virtually no useful information about the
location of the mean. With this standard deviation, it hardly matters what value
is ascribed to the prior mean.
Crudely speaking, a normal distribution becomes `flatter' as its standard
deviation increases. A natural limit to this is to take the standard deviation to
be infinite. This results in a prior that is flat and, as such, can be thought of as
representing an extreme version of prior ignorance, all values being equally
likely. Such a limit cannot be taken too literally, as the constant value achieved
is zero. Nevertheless, if the posterior is taken to be proportional to the likelihood,
i.e. p…ujx† ˆCp…xju†, then the prior has effectively been taken to be a
constant. Of course, a prior that is constant over the whole real line is not a
genuine probability distribution, because it does not integrate to 1, and is known
as an improper prior. Whether or not the posterior is improper depends on
whether or not the integral of the likelihood with respect to u is finite; in many
cases it is, so a proper posterior can emerge from an improper prior and this
approach is widely used. However, the analyst must be vigilant because, if an