  I describe a framework for adaptive scientific exploration based on iterating
an Observation--Inference--Design cycle that allows adjustment of hypotheses
and observing protocols in response to the results of observation on-the-fly,
as data are gathered. The framework uses a unified Bayesian methodology for the
inference and design stages: Bayesian inference to quantify what we have
learned from the available data and predict future data, and Bayesian decision
theory to identify which new observations would teach us the most. When the
goal of the experiment is simply to make inferences, the framework identifies a
computationally efficient iterative ``maximum entropy sampling'' strategy as
the optimal strategy in settings where the noise statistics are independent of
signal properties. Results of applying the method to two ``toy'' problems with
simulated data--measuring the orbit of an extrasolar planet, and locating a
hidden one-dimensional object--show the approach can significantly improve
observational efficiency in settings that have well-defined nonlinear models. I
conclude with a list of open issues that must be addressed to make Bayesian
adaptive exploration a practical and reliable tool for optimizing scientific
exploration.
