Sensor fusion maybe used to improve detection performance in applications. The idea is to make decisions locally, and then transmit them to a global fusion centre where the global decision is made. For global decision making, Bayes or Neyman-Pearson reasoning determines the optimal use of the local decision variables. However, the determination of the local decision variables that minimise global error probability is intractable. In this study, the authors design local decisions that maximise the mutual information between a binary decision variable and the underlying binary state. This serves as a benchmark against which globally optimum solutions maybe compared. Then, they use the theory of large deviations (LDs) to determine a local decision rule that minimises asymptotic global error probability. The use of LD produces a one-dimensional search on a receiver operating characteristic curve to equalise the error exponents for local false alarm and miss probabilities. Many interesting properties of the LD solution are proved. Numerical results illustrate the performance of the asymptotically optimum decision rule for finite collections of sensors.