The Value of Data for a Quadratic Decision Problem

A question that recurs in the decision analysis literature is: What is the value of additional data gathering? We use a very general definition of data that includes subjective probability distributions provided by experts, calculations from models, and traditional experimental data. General expressions for the expected value of data have little practical use because they involve probability distributions whose arguments are themselves probability distributions. The traditional approach to alleviate the problem of dimensionality is to introduce arbitrary parameterization of the data. This paper shows that for an interesting class of decision problems, arbitrary parameterization is not necessary. For the class of decision problems in which the value function over outcomes is a quadratic function of the state and decision variables, the value of data depends on the state variables only through the prior covariances of the posterior means. To use this result in the calculation of the approximate value of data for practical problems, it is necessary to simplify the value function. The paper suggests simplifying the value function by expanding it in a Taylor series with respect to the state and decision variables. The resulting expressions for the approximate value of information should be useful and inexpensive to evaluate for practical decision problems.