one-dimensional velocity moment modeling

There is a standard technique for estimating gravitational potentials called Jeans modeling that uses the Jeans equations to relate the gravitational potential, the number density as a function of position, and the velocity dispersion. As my regular reader can imagine, I have many issues with it, some of them physical (what distributions are angle-mixed and integrable?) and some of them statistical (so you measure your data and then do arithmetic operations on those measurements?). But I spent my week of vacation (just ended) building a little sandbox for testing it out in one dimension and comparing it to better methods—methods that start from a likelihood function, or probability of getting the data given model parameters. I am sure the latter will win in every way, but I don't have my ducks in a row yet.

By the way, when I say likelihood function I don't mean I am going to do maximum likelihood, I mean I am going to transmit information from the data to the parameters of interest via a probability calculation! Just a reminder for those who hear maximum likelihood when all that is said is likelihood.