  We apply an empirical, data-driven approach for describing crop yield as a
function of monthly temperature and precipitation by employing generative
probabilistic models with parameters determined through Bayesian inference. Our
approach is applied to state-scale maize yield and meteorological data for the
US Corn Belt from 1981 to 2014 as an exemplar, but would be readily
transferable to other crops, locations and spatial scales. Experimentation with
a number of models shows that maize growth rates can be characterised by a
two-dimensional Gaussian function of temperature and precipitation with monthly
contributions accumulated over the growing period. This approach accounts for
non-linear growth responses to the individual meteorological variables, and
allows for interactions between them. Our models correctly identify that
temperature and precipitation have the largest impact on yield in the six
months prior to the harvest, in agreement with the typical growing season for
US maize (April to September). Maximal growth rates occur for monthly mean
temperature 18-19$^\circ$C, corresponding to a daily maximum temperature of
24-25$^\circ$C (in broad agreement with previous work) and monthly total
precipitation 115 mm. Our approach also provides a self-consistent way of
investigating climate change impacts on current US maize varieties in the
absence of adaptation measures. Keeping precipitation and growing area fixed, a
temperature increase of $2^\circ$C, relative to 1981-2014, results in the mean
yield decreasing by 8\%, while the yield variance increases by a factor of
around 3. We thus provide a flexible, data-driven framework for exploring the
impacts of natural climate variability and climate change on globally
significant crops based on their observed behaviour. In concert with other
approaches, this can help inform the development of adaptation strategies that
will ensure food security under a changing climate.
