Hamiltonian Monte Carlo (HMC) is a hybrid method that leverages the first-order derivative information of the gradient of the likelihood to propose new states for exploration and overcome some of the challenges of MCMC. In addition, it incorporates momentum to efficiently jump around the posterior. As a result, it converges faster to a high-dimensional target distribution than simpler random-walk Metropolis or Gibbs sampling.

The No-U-Turn sampler is a self-tuning HMC extension that adaptively regulates the size and number of moves around the posterior before selecting a proposal. It works well on high-dimensional and complex posterior distributions and allows many complex models to be fit without specialized knowledge about the fitting algorithm itself. As we will see in the next section, it is the default sampler in PyMC3.