⁂ George Ho

Understanding NUTS and HMC

“Bayesian modeling is harder than deep learning” is a sentiment I’ve been hearing a lot lately. While I’m skeptical of sweeping statements like that, I agree when it comes to the central inference algorithm — how MCMC samplers work (especially the de facto standard samplers, NUTS and HMC) is one of the most difficult concepts I’ve tried to learn, and is certainly harder than autodifferentiation or backpropagation.

So I thought I’d share what worked for me when I tried to teach myself NUTS and HMC. In chronological order of publication, these are the three resources that I’d recommend reading to grok NUTS/HMC:

  1. Radford Neal’s chapter in the MCMC handbook
  2. Matthew Hoffman’s The No-U-Turn Sampler (a.k.a. the original NUTS paper)
  3. Michael Betancourt’s Conceptual Introduction to Hamiltonian Monte Carlo

Not only did I find it useful to read these papers several times (as one would read any sequence of “important” papers), but also to read them in both chronological and reverse-chronological order. Reading both forwards and backwards gave me multiple expositions of important ideas and also let me mentally “diff” the papers to see the progression of ideas over time. For example, Neal’s chapter was written before NUTS was discovered, which gives you a sense of what the MCMC world looked like prior to Hoffman’s work: making progress in fits and starts, but in need of a real leap forward.

In terms of reading code, I’d recommend looking through Colin Carroll’s minimc for a minimal working example of NUTS in Python, written for pedagogy rather than actual sampling. For a “real world” implementation of NUTS/HMC, I’d recommend looking through my littlemcmc for a standalone version of PyMC3’s NUTS/HMC samplers.

Finally, for anyone who wants to read around computational methods for Bayesian inference more generally (i.e. not restricted to HMC, for example), I’d (unashamedly) point to my blog post on this.