Decision Theory

13Changes in utility as information

In previous chapters we discussed situations in which a decision maker, before making a decision, has the opportunity to observe data x. We now turn to the questions of whether this observation should be made and how worthwhile it is likely to be. Observing x could give information about the state of nature and, in this way, lead to a better decision; that is, a decision with higher expected utility. In this chapter we develop this idea more formally and present a general approach for assessing the value of information. Specifically, the value of information is quantified as the expected change in utility from observing x, compared to the “status quo” of not observing any additional data. This approach permits us to measure the information provided by an experiment on a metric that is tied to the decision problem at hand. Our discussion will follow Raiffa and Schlaifer (1961) and DeGroot (1984).

In many areas of science, data are collected to accumulate knowledge that will eventually contribute to many decisions. In that context the connection outlined above between information and a specific decision is not always useful. Motivated by this, we will also explore an idea of Lindley (1956) for measuring the information in a data set, which tries to capture, in a decision-theoretic way, “generic learning” rather than specific usefulness in a given problem.

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Lindley, D. V. (1956). On a measure of the information provided by an experiment, ...