  This paper considers the problem of making statistical inferences about a
parameter when a narrow interval centred at a given value of the parameter is
considered special, which is interpreted as meaning that there is a substantial
degree of prior belief that the true value of the parameter lies in this
interval. A clear justification of the practical importance of this problem is
provided. The main difficulty with the standard Bayesian solution to this
problem is discussed and, as a result, a pseudo-Bayesian solution is put
forward based on determining lower limits for the posterior probability of the
parameter lying in the special interval by means of a sensitivity analysis.
Since it is not assumed that prior beliefs necessarily need to be expressed in
terms of prior probabilities, nor that post-data probabilities must be Bayesian
posterior probabilities, hybrid methods of inference are also proposed that are
based on specific ways of measuring and interpreting the classical concept of
significance. The various methods that are outlined are compared and contrasted
at both a foundational level, and from a practical viewpoint by applying them
to real data from meta-analyses that appeared in a well-known medical article.
