As my reader knows, I have been working on a responsible and probabilistic use of the Jeans equations (which model velocity second moments) for inference. Today I re-built how I simultaneously model the density as a function of position and the velocity variance as a function of position to meet a few desiderata: (1) I want the model to be extremely flexible and data-driven, or non-parametric (meaning huge numbers of parameters). (2) I want the model to strictly enforce the Jeans equation given a parametric (or non-parametric) density or gravitational potential model. (3) I want the model to be parameterized so that I don't have to put harsh or ugly barriers in parameter space that will cause my optimizer to choke. (4) I want the parameterization to be relatively stable in the sense that I want small parameter changes to lead to small, smooth changes to the density and velocity moment models. I got all of this working with a very odd parameterization: I build a non-parametric model of the derivative with respect to position of the number density times the velocity second moment! This gets divided by a potential gradient to give the number density model, and it gets integrated and divided by the density model to get the velocity moment model. Crazy, but it works. I am sure there are much better solutions for my desiderata, but I found that I was much more willing to write code than go to the library!
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