Prior for negative binomial dispersion moderator

Questions version agnostic
1

I’m defining my negative binomial like this currently:

dispersion = pm.HalfNormal(‘dispersion’, 1)
sigma = mean_rate * (1 + mean_rate / dispersion)
counts = pm.NegativeBinomial(“counts”, mu=rates, alpha=sigma, observed=event_count)

So the dispersion is moderating the over-dispersion of the negative binomial. However, I’d like to reparameterize the prior on dispersion so that 1/sqrt(dispersion) is from a half-normal, rather than just dispersion. In Stan I’d just do this:

1/sqrt(dispersion) ~ half-N(0, 1)

But I can’t work out how to do it in PYMC. Any help would be gratefully received.

2

PyMC does not allow for that kind of syntax. In the future it would allow for something defined in terms of dispersion, that is:

with pm.Model() as m:
  dispersion = 1 / HalfNormal.dist()**2
  m.register_rv(dispersion, name="dispersion")

But that relies on Aeppl, where we are implementing this sort of functionality, and we dont yet have support for squaring transformations. You can follow progres in GitHub - aesara-devs/aeppl: Tools for an Aesara-based PPL.

3

Seems like the easiest solution would be to use a deterministic?

disp_base = pm.HalfNormal("disp_base", 0, 1)
dispersion = pm.Deterministic("dispersion", pm.math.sqr(1 / disp_base))
4

That’s the easiest but it’s not exactly the same. You are not putting a prior directly on the transformation. It can be perfectly adequate though.

5

Thanks for the replies - much appreciated.