quadratures and cosmological inferences

At the end of a day not filled with research, I met up with Mike O'Neil (NYU) and Jon Wilkening (Berkeley) to talk about building basis functions in kernel space for kernels to use in Gaussian Process fitting of density fields in one to three ambient dimensions. Apparently there is a connection between this problem and quadratures. It has something to do with the fact that every matrix you make has a finite number of sample points but the positive-definite constraint on the kernel function is not just for every possible selection of spatial sample points but for the infinite dimensional limit. Quadratures relate integrals of functions to weighted sums of finite evaluations. As you can see from my vagueness, I don't really understand any of this yet, but if we could build a flexible basis for making covariance functions that are capable of representing (approximately) power laws with features, we could do a lot of interesting probabilistic cosmological inference.

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