I enjoyed this recent paper by Schneider & Dawson concerning a Bayesian algorithm for estimating orbital elements of Geosynchronous Earth Orbit objects (GEOs) from the direction, length, and width of their streaks in ground-based telescope images. The authors detail a number of pragmatic modelling assumptions made to allow for fast computational approximation of the likelihood function contributed by each single image from amongst a set of images containing the same object. The trick then is to make use of their partial posterior samples of orbital elements constrained by each single image in order to efficiently sample the full joint posterior. Their solution here is to use a multiple importance sampling approach (Elvira et al. 2015; a la Veach & Guibas 1995). This requires new evaluations of the approximate likelihood function for each partial posterior—although a KDE-based version is also noted to have been effective which leads me to point out a connection with the wider statistical literature on combining partial posteriors from data partitioning, especially Neiswanger et al.‘s ’embarrassingly parallel MCMC’.
Of course, KDE approaches suffer problems due to poor scaling statistically with the dimension of the parameter space and poor scaling computationally with the number of particles in the estimator, and with limitations when inferences are made far from the positions of the available particles. The other common strategy (‘consensus MCMC‘) for combing partial posteriors is to use a Normal approximation to each (for which the product is also Normal) but other course this is limited by the suitability of the Normal for representing non-Normal (perhaps even multimodal) partial posteriors. Ignoring multi-modality for now, I would speculate that astronomers might be well placed to devise a neat higher order approximation scheme to improve on the consensus MCMC approach given our familiarity with expansions about Gaussian potentials (e.g. Blinnikov & Moessner 1997; although I seem to remember their naming of this as an Edgeworth expansion is non-canonical?).