@andrewgelman and I just uploaded a new case study: https://mc-stan.org/users/documentation/case-studies/planetary_motion/planetary_motion.html. Comments and questions are welcomed!

Our initial intent was to construct a simple textbook example for an ODE-based model, but the example (planetary motion using Hamiltonian mechanics) turned out to be a bit more complicated. There was unexpected multimodality, and in the tail of the posterior, the ODE became very difficult to solve. The case study illustrates some advanced use of posterior predictive checks to understand how modes arise, even when there is no identifiability problem. And itās a good reminder that the behavior of your ODE can vary greatly depending on where you are in the parameter space. There is also a question being raised on what constitutes acceptable initial starting points for MCMC; this gets less attention (at least from what I can tell in the Stan community) but itās another tuning parameter to worry about.

A summary of this case study can be found in Section 11 of our recent paper on the Bayesian workflow: https://arxiv.org/pdf/2011.01808.pdf.