Recall: simple binomial model

This is the same as an intercept-only logistic distribution with a logit-transformed uniform for the prior on

Bayes' rule

Recall: simple binomial model

This is the same as an intercept-only logistic distribution with a logit-transformed uniform for the prior on

Bayes' rule

Simulate a
physical system

“Energy” at any point in the parameter space is proportional to the negative log posterior probability:

Get random draws
by ‘perturbing’ a particle in that field

Place a particle in that system, give it a push in a random direction, and use Hamiltonian dynamics to simulate its motion.


Wherever the particle ends up after a fixed amount of time is the next candidate draw from the posterior.

Takes advantage
of gradient

Gradient (slope) information helps HMC adjust to the shape of the posterior.

Reduces autocorrelation

HMC tends to explore the plausible areas of the parameter space much more quickly than ‘standard’ MCMC like Metropolis–Hastings. It is not likely to spend too much time in one small area.

“No-U-Turn sampler”
(NUTS)

A version of HMC that automatically optimizes some of the meta-parameters of the algorithm.

• HMC can ‘overshoot’ if the posterior density has sharp edges
• Common in multilevel models
• Stan (and brms) will warn you about these
• Address with re-specification of model or finer-grained steps