One has to be careful with concepts like â€śrejectâ€ť â€“ in particular we have to differentiate between what is *model* and what is *computation*.

Right now many dynamical systems specified via ODEs contain extreme behavior that manifests as certain parameter values. This stresses the ODE solvers when theyâ€™re not configured appropriately, leading to large numerical errors in the final states and their sensitivities which then induces problems with the computation. In other words the computational problems are a manifestation of extreme behavior in the model.

The computational solution is to use, well, better computation. By putting more work into each ODE solve we can better resolve the extreme dynamics and accurately quantify their contributions to the posterior.

The modeling solution is to remove the extreme behavior from the model. The most direct approach is to use more informative prior models that suppress the extreme parameter configurations; this has critical benefit of yielding a smooth target distribution that facilities accurate computation of the resulting model.

Jury-rigging the â€śrejectâ€ť function based on observed numerical solver behavior, on the other hand, only implicitly changes the model. Because the rejection is based on solver output it can account only for extreme behavior that manifests in particular numerical behavior; for example if some collision events integrated fine but some didnâ€™t then the reject approach wouldnâ€™t be removing all collision events just some particular collision events. This selection can have unintended effects on the final inferences. On top of that the rejection introduces a discontinuity â€“ the target density is non-zero on one side of the reject condition and zero on the other â€“ which complicates the computation.

Ultimately removing extreme behavior can be extremely useful, but one has to do it in a principled way lest the final inferences be compromised. Almost always this involves careful prior modeling motivated by numerical pathologies instead of trying to define implicit models based on those numerical pathologies.