  Contemporary scientific research is a distributed, collaborative endeavor,
carried out by teams of researchers, regulatory institutions, funding agencies,
commercial partners, and scientific bodies, all interacting with each other and
facing different incentives. To maintain scientific rigor, statistical methods
should acknowledge this state of affairs. To this end, we study hypothesis
testing when there is an agent (e.g., a researcher or a pharmaceutical company)
with a private prior about an unknown parameter and a principal (e.g., a
policymaker or regulator) who wishes to make decisions based on the parameter
value. The agent chooses whether to run a statistical trial based on their
private prior and then the result of the trial is used by the principal to
reach a decision. We show how the principal can conduct statistical inference
that leverages the information that is revealed by an agent's strategic
behavior -- their choice to run a trial or not. In particular, we show how the
principal can design a policy to elucidate partial information about the
agent's private prior beliefs and use this to control the posterior probability
of the null. One implication is a simple guideline for the choice of
significance threshold in clinical trials: the type-I error level should be set
to be strictly less than the cost of the trial divided by the firm's profit if
the trial is successful.
