A learning rule for inferring local distributions over space and time

A common feature of certain kinds of data is a high level of statistical dependence across space and time. This spatial and temporal dependence contains useful information that can be exploited to significantly reduce the uncertainty surrounding local distributions. This chapter develops a methodology for inferring local distributions that incorporates these dependencies. The approach accommodates active learning over space and time, and from aggregate data and distributions to disaggregate individual data and distributions. We combine data sets on Kansas winter wheat yields annual county-level yields over the period from 1947 through 2000 for all 105 counties in the state of Kansas, and 20,720 individual farm-level sample moments, based on ten years of the reported actual production histories for the winter wheat yields of farmers participating in the United States Department of Agriculture Federal Crop Insurance Corporation Multiple Peril Crop Insurance Program in each of the years 1991–2000. We derive a learning rule that combines state-wide, county, and local farm-level data using Bayes’ rule to estimate the moments of individual farm-level crop yield distributions. Information theory and the maximum entropy criterion are used to estimate farm-level crop yield densities from these moments. These posterior densities are found to substantially reduce the bias and volatility of crop insurance premium rates.