Probability distribution over geographic coordinates I would like to model some quantity y that varies over different locations on the Earth using a Gaussian process. My concern with just using f(y|latitude, longitude), is that this assumes a Euclidean space over latitude and longitude, but clearly this is bad--I'd want something that works with a spherical system. Links to resources would be much appreciated.
 A: While you can of course use the squares exponential kernel as previously mentioned, there are in fact many kernels which decay monotonically with Euclidean distance. For example the matern kernels, the exponential kernel etc. The core difference between these is how smooth you expect the output function to look. These are all stationary kernels and use either absolute value or square (ie if you shift your input the relative similarity is preserved). As such you can switch $(x_*  - x)$ with, for example, the arc length about the sphere between the two points - this would be a valid kernel.
Another option, albeit less well known, are Gauss Markova Random Fields which have the same formulation as a Gaussian process but are parameterizes by the precision matrix and encode the fact that only neighboring nodes (or countries in your case) may interact with one another. Might be worth checking out! Good luck.
A: Lindgren et al 2011 addresses this by specifying the GP as a solution to a stochastic partial differential equation.  The stationary solutions have Matern covariance.
See Figure 4 for an example in the paper of estimation on the sphere.  The method is implemented in the R-INLA package, I'm not sure about implementations outside of R.
A: It might be useful to look into Riemannian distance metric [https://maths-people.anu.edu.au/~andrews/DG/DG_chap9.pdf] and geodesics [https://www.youtube.com/watch?v=8sVDceI70HM&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx&index=18]
