As seen throughout this project, there are many different ways to apply Bayesian methods to spatial statistics problems. When doing a geospatial analysis, it is important to estimate the mean of the process. The traditional way is to build an OLS model, but Bayesians try to estimate these parameters by calculating a posterior distribution for the regression parameters. After doing this, Monte Carlo methods such as Metropolis-Hastings can be used to get multiple samples of parameters, which can then be used to create a posterior predictive distribution for the kriged values. Having a posterior predictive distribution is better than having one point estimate, but getting this by hand is complicated. It is almost impossible to know the likelihood of the data unless it is simulated, so although best guesses were made in this project, better results would probably achieved using the spBayes package.

This project really highlighted the utility of the spBayes package. It was able to perform 20,000 iterations of Metropolis-Hastings to estimate the posteriors of three different parameters in just a few seconds. Then when it came to prediction, it calculated the posterior predictive distribution for one point in less than 30 seconds. It would definitely take some time to complete this process for an entire grid of points, but there is almost no way to do this by hand with making many assumptions, which would not be feasible when working with non simulated data. In addition to computing a posterior predictive distribution, if a researcher cares about uncertainty, Bayesian kriging comes out on top. Not only is the credible interval interpreted in a way that is easier for a person without a mathematical background to understand, but it also is integrating out the uncertainty in the parameters. Even though traditional kriging takes less time and is easier to compute, only having one prediction for each point and not accounting for the uncertainty in the parameters is a big drawback.

During the semester in the spatial statistics course, the professor always mentioned ‘…everything always falls apart when doing spatial statistics’. This is due to the complicated nature in which everything is correlated with each other. This was definitely seen when doing Bayesian kriging by hand, as very specific assumptions had to be made to do it. Even with the simplest case, it took a very, very long time to compute. That being said, the spBayes package is very user friendly for someone who understands Bayesian statistics. Overall, this project scratched the surface by looking at ways to apply Bayesian methods to spatial problems. The results for some parts weren’t perfect, such as estimating the regression parameters, which goes to show how hard it is to estimate complex spatial data by hand. In the future, it would be interesting to experiment with the spBayes package using more complex data, different covariance functions, and including things like a nugget effect. There are also many other topics in spatial statistics not covered in this project such as conditional simulation, spectral methods, and spatial autoregressive models. Another project could be to see how Bayesian techniques could be applied to those topics as well.