# Why are Hamiltonian dynamics used in MCMC?

In Hamiltonian Monte Carlo, Hamiltonian dynamics are used to generate new proposals from the current state. I understand why these dynamics are used as opposed to random walk behavior to generate proposals, but I don't understand why Hamiltonian dynamics are used as opposed to any other dynamics. Could someone please explain this to me? Also, this seems to be a difficult thing to search for as well, so any pointers to references would be appreciated too.

• Does, e.g., sections 2.2 and 3 of cs.utoronto.ca/~radford/ftp/ham-mcmc.pdf help, or maybe not? – Kirill Nov 8 '16 at 15:20
• What do you mean by "why Hamiltonian dynamics are used as opposed to any other dynamics"? Normally, people consider Hamiltonian, Lagrangian or Vector mechanics equivalent formulations of a problem. But with some operative advantage. – nicoguaro Nov 8 '16 at 15:26
• @Kirill, section 2.2 is what I was looking for, thanks. – ugabooga Nov 8 '16 at 15:58
• @nicoguaro, could you give a reference for or briefly describe these operative advantages? Section 2.2 of Kirill's reference doesn't provide a comparative analysis. Also, I'm not a physicist, so I'm not sure what operative advantages you're talking about. – ugabooga Nov 8 '16 at 16:00
• You might be interested in the Wikipedia's article on Analytical mechanics. – nicoguaro Nov 8 '16 at 16:56

I'm a little late to reply but I found this review to be really informative. One particularly nice feature of Hamiltonian flows is that they preserve volume in phase space. If we take an infinitesimal box of size $(\delta q, \delta p)$ around a point $(q, p)$ in phase space, then propagate every point inside the box forward in time by $\delta t$ under the Hamiltonian flow, the volume of the transformed box will stay the same. If you were to generate new proposals using some arbitrary transformation $\Psi$ that does not necessarily preserve volume in phase space, you'd also have to keep track of Jacobian factors $|\det\nabla\Psi|$ that may or may not be feasible to compute. Hamiltonian flows are volume-conserving, so this Jacobian factor is always equal to 1.