The value of information for correlated GLMs

- An approximation to the value of information is derived for GLM.
- The approximation is comparable to a Monte-Carlo alternative.
- Guidelines for assessing sensitivity to model parameters are given.
- The distribution of the gain in information is used to assess the risk of decisions.

We examine the situation where a decision maker is considering investing in a number of projects with uncertain revenues. Before making a decision, the investor has the option to purchase data which carry information about the outcomes from pertinent projects. When these projects are correlated, the data are informative about all the projects. The value of information is the maximum amount the investor would pay to acquire these data.The problem can be seen from a sampling design perspective where the sampling criterion is the maximisation of the value of information minus the sampling cost. The examples we have in mind are in the spatial setting where the sampling is performed at spatial coordinates or spatial regions.In this paper we discuss the case where the outcome of each project is modelled by a generalised linear mixed model. When the distribution is non-Gaussian, the value of information does not have a closed form expression. We use the Laplace approximation and matrix approximations to derive an analytical expression to the value of information, and examine its sensitivity under different parameter settings and distributions. In the Gaussian case the proposed technique is exact. Our analytical method is compared against the alternative Monte-Carlo method, and we show similarity of results for various sample sizes of the data. The closed form results are much faster to compute. Model weighting and bootstrap are used to measure the sensitivity of our analysis to model and parameter uncertainty. A general guidance on making decisions using our results is offered.Application of the method is presented in a spatial decision problem for treating the Bovine Tuberculosis in the United Kingdom, and for rock fall avoidance decisions in a Norwegian mine.