Not fully sure what the point is. Supposedly covers evenly random space better than random walks by some measure.

Related to simulating the states of molecules.

MCMC: Markov Chain Monte Carlo

Momentum: mass times velocity

Bunch of proofs as to why Hamiltonian dynamics are useful for stuff.

"Metropolis updates."

Hamiltonian MCMC

Current Markov state is inferred from the positiong of a weighted puck moving through 2D space.

One update step pulls a new random momentum from Gaussian distributions.

Another update step simulates the ball rolling around and proposes a new position

The new position is probablistically accepted or rejected as the new state for the markov machine.

Hamiltonian Dynamics

In a 2D system you have a two dimensional position vector, a two dimensional momentum vector, a potential energy function U(q) based on the surface being traversed, and kinetic energy function K(p).

The puck has a mass measured by 'm'.

Kinetic energy is equal to |p|^2/(2m)

Potential energy is proportional to the height of the surface at the current position.

The ball moves proportionate to how much energy it has and this energy is consumed at some rate; moving down a hill builds energy in to the ball (gravity) while moving up a hill consumes the momentum. When the ball lacks enough momentum to resist gravity it will be pulled down slopes.