  The Bayesian decision-theoretic approach to design of experiments involves
specifying a design (values of all controllable variables) to maximise the
expected utility function (expectation with respect to the distribution of
responses and parameters). For most common utility functions, the expected
utility is rarely available in closed form and requires a computationally
expensive approximation which then needs to be maximised over the space of all
possible designs. This hinders practical use of the Bayesian approach to find
experimental designs. However, recently, a new utility called Fisher
information gain has been proposed. The resulting expected Fisher information
gain reduces to the prior expectation of the trace of the Fisher information
matrix. Since the Fisher information is often available in closed form, this
significantly simplifies approximation and subsequent identification of optimal
designs. In this paper, it is shown that for exponential family models,
maximising the expected Fisher information gain is equivalent to maximising an
alternative objective function over a reduced-dimension space, simplifying even
further the identification of optimal designs. However, if this function does
not have enough global maxima, then designs that maximise the expected Fisher
information gain lead to non-identifiablility.
