Diagnosting posterior geometry with divergences

[nhuurre]

@jsocolar

I wonder whether in some of these cases the divergences actually do concentrate in a particular region of parameter space, but HMC is working well enough so that when Stan samples along the trajectory that terminated in a divergence, it frequently returns a location in parameter space that is far away from where the divergence occurred.

Yes, that does sound plausible to me.

@jsocolar

forgo standard sampling along the trajectory and instead return a point that is at most some fixed smallish number of leapfrog steps from where the divergence actually got flagged

It’s a Markov chain, the next transition starts from the last sample. If you pick a diverging sample instead of a proper sample then the chain must continue from the difficult region and is more likely to get stuck. I think the divergent samples must go into a separate diagnostic output stream.
The other question is, what is a good “fixed smallish number of leapfrog steps?” The divergence detection heuristic is somewhat arbitrary and even in theory a divergence does not have a precise location. I guess the easiest solution would be to just pick a random sample from the rejected half of the trajectory. The sampler already records candidate samples as it build the trajectory; when the trajectory blows up you could print the latest candidate instead of just discarding it.

These sort of ideas have been discussed before:

Getting the location + gradients of divergences (not of iteration starting points)

When I was writing the non-identifiability blog post, I really tried to think hard where divergences are expected to occur. And while the divergences somewhat gathered in the areas I expected them too, they were also often marked quite far from those area. I noticed that the Stan manual says: ShinyStan provides visualizations that highlight the starting point of divergent transitions to diagnose where the divergences arise in parameter space. So this means that if an iteration is divergent, …

Sampler Diagnostics

When encountering divergences, I often have the problem that these divergences are spread out all over the posterior distribution (or at least the part of it that was already sampled). This makes it difficult to pin down the cause of these divergences. Only in the minority of cases I encountered divergences which are thightly grouped in a certain area, thus giving a hint about what causes them. I think that these divergences are so often spread out is because of two things: The location where…

It’s a very interesting problem.