Bayesian Design for the Normal Linear Model with Unknown
Most of the Bayesian theory of optimal experimental
design, for the normal linear model, has been developed
under the restrictive assumption that the variance is known.
In special cases, insensitivity of specific design criteria
to specific prior assumptions on the variance has been
demonstrated, but a general result to show the way in which
Bayesian optimal designs are affected by prior information
about the variance is lacking.
This paper stresses the important distinction between expected
utility functions and optimality criteria, examines a number of
expected utility functions -- some of which possess interesting
properties, and deserve wider use -- and derives the relevant
Bayesian optimality criteria under normal assumptions. This
unifying setup is useful for proving the main result of the
paper, that clarifies the issue of designing for the normal
linear model with unknown variance.
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