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This is a bit of a curious situation. I have an energy function $E=S+N$ which is the sum of a smooth differentiable function $S$ and a piecewise constant "noise" function $N$. This means that on average the gradient does point in the right direction, but even a tiny step in the gradient's direction will fall on a different piece.

Is there any way to adapt HMC to handle this situation? For all practical purposes, the "noise" function is a black-box.

So far I'm thinking of using HMC updates based only on the gradient of $S$, with a MH acceptance step at the end. The only difference with regular HMC being that the acceptance ratio may not be 1 (or close to 1). Is this correct or am I missing something?

P.S. I put "noise" in quotes because it does capture the look of the function (low spatial correlation), but it is not noise in the sense that I do not want to ignore its contribution to the energy function.

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  • $\begingroup$ Hamiltonian can be defined on a symplectic manifold. A symplectic manifold must be differentiable (and smooth) to be classed as such. Therein lies the problem that you are most likely aware of. This is an interesting problem though. I would expect that one can solve the Equations of Motion (EOM) in a piecewise fashion and bootstrap them together though. I will look into this in more detail over the summer when I start my thesis on HMC. $\endgroup$ Apr 26 '16 at 16:24
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This paper is likely relevant. Abstract:

Hamiltonian Monte Carlo (HMC) is a successful approach for sampling from continuous densities. However, it has difficulty simulating Hamiltonian dynamics with non-smooth functions, leading to poor performance. This paper is motivated by the behavior of Hamiltonian dynamics in physical systems like optics. We introduce a modification of the Leapfrog discretization of Hamiltonian dynamics on piecewise continuous energies, where intersections of the trajectory with discontinuities are detected, and the momentum is reflected or refracted to compensate for the change in energy.

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