  A fundamental class of inferential problems are those characterised by there
having been a substantial degree of pre-data (or prior) belief that the value
of a model parameter $\theta_j$ was equal or lay close to a specified value
$\theta^{*}_j$, which may, for example, be the value that indicates the absence
of a treatment effect or the lack of correlation between two variables. This
paper puts forward a generally applicable 'push-button' solution to problems of
this type that circumvents the severe difficulties that arise when attempting
to apply standard methods of inference, including the Bayesian method, to such
problems. Usually the only input of major note that is required from the user
in implementing this solution is the assignment of a pre-data or prior
probability to the hypothesis that the parameter $\theta_j$ lies in a narrow
interval $[\theta_{j0},\theta_{j1}]$ that is assumed to contain the value of
interest $\theta^{*}_j$. On the other hand, the end result that is achieved by
applying this method is, conveniently, a joint post-data distribution over all
the parameters $\theta_1,\theta_2,\ldots,\theta_k$ of the model concerned. The
proposed method is constructed by naturally combining a simple Bayesian
argument with an approach to inference called organic fiducial inference that
was developed in a number of earlier papers. To begin with, the main
theoretical arguments underlying this combined Bayesian and fiducial method are
presented and discussed in detail. Various applications and useful extensions
of this methodology are then outlined in the latter part of the paper. The
examples that are considered are made relevant to the analysis of clinical
trial data where appropriate.
