Title: Differences Among Noninformative Stopping Rules Are Often Relevant to Bayesian Decisions,
Abstract: L.J. Savage once hoped to show that "the superficially incompatible systems
of ideas associated on the one hand with [subjective Bayesianism] and on the
other hand with [classical statistics]...lend each other mutual support and
clarification." By 1972, however, he had largely "lost faith in the devices" of
classical statistics. One aspect of those "devices" that he found objectionable
is that differences among the "stopping rules" that are used to decide when to
end an experiment which are "noninformative" from a Bayesian perspective can
affect decisions made using a classical approach. Two experiments that produce
the same data using different stopping rules seem to differ only in the
intentions of the experimenters regarding whether or not they would have
carried on if the data had been different, which seem irrelevant to the
evidential import of the data and thus to facts about what actions the data
warrant.
I argue that classical and Bayesian ideas about stopping rules do in fact
"lend each other" the kind of "mutual support and clarification" that Savage
had originally hoped to find. They do so in a kind of case that is common in
scientific practice, in which those who design an experiment have different
interests from those who will make decisions in light of its results. I show
that, in cases of this kind, Bayesian principles provide qualified support for
the classical statistical practice of "penalizing" "biased" stopping rules.
However, they require this practice in a narrower range of circumstances than
classical principles do, and for different reasons. I argue that classical
arguments for this practice are compelling in precisely the class of cases in
which Bayesian principles also require it, and thus that we should regard
Bayesian principles as clarifying classical statistical ideas about stopping
rules rather than the reverse.