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

Bayesian, classical and hybrid methods of inference when one parameter value is special