## simulation under zero measure constraints [a reply]

**F**ollowing my post of last Friday on simulating over zero measure sets, as, e.g., producing a sample with a given maximum likelihood estimator, Dennis Prangle pointed out the recent paper on the topic by Graham and Storkey, and a wee bit later, Matt Graham himself wrote an answer to my X Validated question detailing the resolution of the MLE problem for a Student’s t sample. Including the undoubtedly awesome picture of a 3 observation sample distribution over a non-linear manifold in **R**³. When reading this description I was then reminded of a discussion I had a few months ago with Gabriel Stolz about his free energy approach that managed the same goal through a Langevin process. Including the book Free Energy Computations he wrote in 2010 with Tony Lelièvre and Mathias Rousset. I now have to dig deeper in these papers, but in the meanwhile let me point out that there is a bounty of 200 points running on this X Validated question for another three days. Offered by Glen B., the #1 contributor to X Validated question for all times.

November 21, 2016 at 2:13 pm

On the question of your earlier post (production of a sample with given moments), I found another related X Validated question about ‘Constructing a continuous distribution to match m moments’ (http://stats.stackexchange.com/questions/141652/constructing-a-continuous-distribution-to-match-m-moments), to which I replied by using our momentify R package. It produces a density which approximately matches given moments and samples from it.

November 21, 2016 at 1:52 pm

Thanks for the kind words about the animation and comment on my answer! Just to add a couple more references – in addition to the Lelièvre, Rousset and Stoltz paper you mention also of interest may be ‘A constrained hybrid Monte-Carlo algorithm and the problem of calculating the free energy in several variables’, Hartmann and Schütte (2005) (https://page.mi.fu-berlin.de/chartman/ZAMM.pdf) and ‘An ergodic sampling scheme for constrained Hamiltonian systems with applications to molecular dynamics’, Hartmann (2007) (https://page.mi.fu-berlin.de/chartman/hmc.pdf).