  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 was equal or lay close to a specified value, which may,
for example, be the value that indicates the absence of an effect. Standard
ways of tackling problems of this type, including the Bayesian method, are
often highly inadequate in practice. To address this issue, an inferential
framework called bispatial inference is put forward, which can be viewed as
both a generalisation and radical reinterpretation of existing approaches to
inference that are based on P values. It is shown that to obtain an appropriate
post-data density function for a given parameter, it is often convenient to
combine a special type of bispatial inference, which is constructed around
one-sided P values, with a previously outlined form of fiducial inference.
Finally, by using what are called post-data opinion curves, this
bispatial-fiducial theory is naturally extended to deal with the general
scenario in which any number of parameters may be unknown. The application of
the theory is illustrated in various examples, which are especially relevant to
the analysis of clinical trial data.
