Statistical Methods in Medical Research 4ed

6.1 Subjective and objective probability 167
tion should give equal probabilities, or probability densities, to all the possible
values of the parameter. However, that approach is ambiguous, because a uniform
distribution of probability across all values of a parameter would lead to a
non-uniform distribution on a transformed scale of measurement that might be
just as attractive as the original. For example, for a parameter u representing a
proportion of successes in an experiment, a uniform distribution of u between 0
and 1 would not lead to a uniform distribution of the logit of u ((14.5), p. 488)
between 1 and 1. This problem was one of the main objections to Bayesian
methods raised throughout the nineteenth century.
A convenient way out of the difficulty is to use the family of conjugate priors
appropriate for the situation under consideration, and to choose the extreme
member of that family to represent ignorance. This is called a non-informative or
vague prior. A further consideration is that the precise form of the prior distribution
is important only for small quantities of data. When the data are
extensive, the likelihood function is tightly concentrated around the maximum
likelihood value, and the only feature of the prior that has much influence in
Bayes' theorem is its behaviour in that same neighbourhood. Any prior distribution
will be rather flat in that region unless it is is very concentrated there or
elsewhere. Such a prior will lead to a posterior distribution very nearly proportional
to the likelihood, and thus almost independent of the prior. In other
words, as might be expected, large data sets almost completely determine the
posterior distribution unless the user has very strong prior evidence.
The main body of statistical methods described in this book was built on the
frequency view of probability, and we adhere mainly to this approach. Bayesian
methods based on suitable choices of non-informative priors (Lindley, 1965)
often correspond precisely to the more traditional methods, when appropriate
changes of wording are made. We shall indicate many of these points of correspondence
in the later sections of this chapter. Nevertheless, there are points at
which conflicts between the viewpoints necessarily arise, and it is wrong to
suggest that they are merely different ways of saying the same thing.
In our view both Bayesian and non-Bayesian methods have their proper
place in statistical methodology. If the purpose of an analysis is to express the
way in which a set of initial beliefs is modified by the evidence provided by the
data, then Bayesian methods are clearly appropriate. Formal introspection of
this sort is somewhat alien to the working practices of most scientists, but the
informal synthesis of prior beliefs and the assessment of evidence from data is
certainly commonplace. Any sensible use of statistical information must take
some account of prior knowledge and of prior assessments about the plausibility
of various hypotheses. In a card-guessing experiment to investigate extrasensory
perception, for example, a score in excess of chance expectation which was just
significant at the 1% level would be regarded by most people with some scepticism:
many would prefer to think that the excess had arisen by chance (to say