Suppose the target distribution to sample is and a chain of samples is required. The Hamilton's equations are and where and are the th component of the position and momentum vector respectively and is the Hamiltonian. Let be a mass matrix which is symmetric and positive definite, then the Hamiltonian is where is the potential energy. The potential energy for a target is given as which comes from the Boltzmann's factor. The algorithm requires a positive integer for number of leap frog steps and a positive number for the step size. Suppose the chain is at. Let. First, a randomGaussianmomentum is drawn from. Next, the particle will run under Hamiltonian dynamics for time, this is done by solving the Hamilton's equations numerically using the leap frog algorithm. The position and momentum vectors after time using the leap frog algorithm are These equations are to be applied to and times to obtain and. Because the leap frog algorithm is a numerical method and does not solve the Hamilton's equations exactly, a Metropolis–Hastings step is used. The transition from to is where This is repeated to obtain.
No U-Turn Sampler
The No U-Turn Sampler is an extension by controlling automatically. Tuning is critical. For example in the one dimensional case, the potential is which corresponds to the potential of a simple harmonic oscillator. For too large, the particle will oscillate and this waste computational time. For too small, the particle will behave like a random walk. Loosely, NUTS runs the Hamiltonian dynamics both forwards and backwards in time randomly until a U-Turn condition is satisfied. When that happens, a random point from the path is chosen for the MCMC sample and the process is repeated from that new point. In detail, a binary tree is constructed to trace the path of the leap frog steps. To produce a MCMC sample, an iterative procedure is conducted. A slice variable is sampled. Let and be the position and momentum of the forward particle respectively. Similarly, and for the backward particle. In each iteration, the binary tree selects at random uniformly to move the forward particle forwards in time or the backward particle backwards in time. Also for each iteration, the number of leap frog steps increase by a factor of 2. For example, in the first iteration, the forward particle moves forwards in time using 1 leap frog step. In the next iteration, the backward particle moves backwards in time using 2 leap frog steps. The iterative procedure continues until the U-Turn condition is met, that is or when the Hamiltonian becomes inaccurate or where, for example,. Once the U-Turn condition is met, the next MCMC sample,, is obtained by sampling uniformly the leap frog path traced out by the binary tree which satisfies This is usually satisfied if the remaining HMC parameters are sensible.
