tomography, detailed balance, exoplanet non-parametrics

After I started to become convinced that we might be able to make progress on the Cryo-EM problem, Greengard described to me an easier (in some ways) problem: cryo-electron tomography. This is similar to Cryo-EM, but the experimenter controls the angles! In principle this should make it easier (and it does), but according to mathematical standards the problem is still ill-posed: The device can't be turned to all possible angles, or not even enough to fill out the tomographic information needed for general reconstructions. Of course this doesn't phase me!.

Useful conversations with Foreman-Mackey included two interesting subjects. One is that even if you have K different MCMC moves, each of which satisfies detailed balance, it is not guaranteed that a deterministic sequence of them will satisfy detailed balance! That blew me away for a few minutes but then started to make sense. Ish. According to Foreman-Mackey, there is a connection between this issue and the point that a product of symmetric matrices will not necessarily be symmetric!

The other interesting subject with Foreman-Mackey was on exoplanet system modeling. We want to explore some non-parametrics: That is, instead of considering the exoplanet population as a mixture of one-planet, two-planet, three-planet (and so-on) systems, model it just with K-planet systems, where K is very large (or infinite). This model would require having a significant amount of the planet-mass pdf at very low masses (or sizes). Not only might this have many technical advantages, it also accords with our intuitions: After all, I think we (almost) all think that if you have one or two major planets, you probably have many, many minor ones. Like way many.

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