# MCMC: Hamiltonian Monte Carlo (a.k.a. Hybrid Monte Carlo) The Clever Machine

The random-walk behavior of many Markov Chain Monte Carlo (MCMC) algorithms makes Markov chain convergence to a target stationary distribution \$latex p(x)&s=-1\$ inefficient, resulting in slow mixing. Hamiltonian/Hybrid Monte Carlo (HMC), is a MCMC method that adopts physical system dynamics rather than a probability distribution to propose future states in the Markov chain. This allows the Markov chain to explore the target distribution much more efficiently, resulting in faster convergence. Here we introduce basic analytic and numerical concepts for simulation of Hamiltonian dynamics. We then show how Hamiltonian dynamics can be used as the Markov chain proposal function for an MCMC sampling algorithm (HMC).

## First off, a brief physics lesson in Hamiltonian dynamics

Before we can develop Hamiltonian Monte Carlo, we need to become familiar with the concept of Hamiltonian dynamics. Hamiltonian dynamics is one way that physicists describe how objects move throughout a system. Hamiltonian dynamics describe an object’s…

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