  The process of discovery in the physical, biological and medical sciences can
be painstakingly slow. Most experiments fail, and the time from initiation of
research until a new advance reaches commercial production can span 20 years.
This tutorial is aimed to provide experimental scientists with a foundation in
the science of making decisions. Using numerical examples drawn from the
experiences of the authors, the article describes the fundamental elements of
any experimental learning problem. It emphasizes the important role of belief
models, which include not only the best estimate of relationships provided by
prior research, previous experiments and scientific expertise, but also the
uncertainty in these relationships. We introduce the concept of a learning
policy, and review the major categories of policies. We then introduce a
policy, known as the knowledge gradient, that maximizes the value of
information from each experiment. We bring out the importance of reducing
uncertainty, and illustrate this process for different belief models.
