  The power prior is a popular tool for constructing informative prior
distributions based on historical data. The method consists of raising the
likelihood to a discounting factor in order to control the amount of
information borrowed from the historical data. It is customary to perform a
sensitivity analysis reporting results for a range of values of the discounting
factor. However, one often wishes to assign it a prior distribution and
estimate it jointly with the parameters, which in turn necessitates the
computation of a normalising constant. In this paper we are concerned with how
to recycle computations from a sensitivity analysis in order to approximately
sample from joint posterior of the parameters and the discounting factor. We
first show a few important properties of the normalising constant and then use
these results to motivate a bisection-type algorithm for computing it on a
fixed budget of evaluations. We give a large array of illustrations and discuss
cases where the normalising constant is known in closed-form and where it is
not. We show that the proposed method produces approximate posteriors that are
very close to the exact distributions when those are available and also
produces posteriors that cover the data-generating parameters with higher
probability in the intractable case. Our results show that proper inclusion the
normalising constant is crucial to the correct quantification of uncertainty
and that the proposed method is an accurate and easy to implement technique to
include this normalisation, being applicable to a large class of models.
  Key-words: Doubly-intractable; elicitation; historical data; normalisation;
power prior; sensitivity analysis.
