# Sampling states of an "unnatural" Hamiltonian System

I would like to sample from a Gibbs distribution given by

$$f(p, q) = \frac{1}{\mathcal{Z}}e^{-H(p, q; \omega, J)}$$

where $$H$$ is the Hamiltonian on generalized coordinates $$(p,q)\in \mathbb{R}^{2n}$$ defined by

\begin{align} H(p,q; \omega, J)& = \sum_{\ell=1}^n \frac{\omega_{\ell}}{2}(q_{\ell}^2 + p_{\ell}^2) \\ &+ \frac{1}{4}\sum_{\ell, m = 1}^n J_{\ell, m}(q_{\ell}p_{m} - q_m p_{\ell}) \cdot (q_m^2 + p_m^2 - q_{\ell}^2 - p_{\ell}^2) \end{align}



explored in this paper. We take $$J$$ to be symmetric. Notice that $$H$$ is not a natural" Hamiltonian expessible as $$H(p,q) = K(q) + U(p)$$ so that, for instance, it is not easy to calculate a marginal on $$p$$ or $$q$$ and then use Hamilton's equations to propose new samples for the remaining coordinates in the manner of Hamiltonian MCMC. I'm not sure I can use Gibbs sampling either, since I don't see any tractable conditionals.

Any help with this or references to "unnatural" Hamiltonian systems would be greatly appreciated.

Edit: "Eq. 4" in the comments is the action-angle reparameterization,

$$H(I, \theta; \omega, J) = \sum_{\ell}^n \omega_{\ell} I_{\ell} - \sum_{\ell, m = 1}^n J_{\ell, m}\sqrt{I_{\ell}I_m} \cdot (I_m - I_{\ell})\sin(\theta_m - \theta_{\ell}).$$

• is it clear that this Gibbs distribution can be normalised? e.g. won't the Hamiltonian diverge to $-\infty$ in some nontrivial part of the tails?
– πr8
Feb 9, 2020 at 23:23
• Thanks for your comment, @πr8. I'd certainly believe that it diverges. I suppose I don't know the details concerning which Hamiltonian systems admit a Gibbsian description. But how about in the case that, for instance, the version of the Hamiltonian given by Eq. 4 in the linked paper is used except that each $I_{\ell} \leq 1$? Then $H$ is clearly bounded. This would still be useful for my purposes. Feb 10, 2020 at 4:06
• okay, that would make things normalise. is it essential that you use the Hamiltonian structure to do sampling here? there's nothing stopping you from just doing some form of Metropolis-Hastings on $f(p, q)$
– πr8
Feb 10, 2020 at 10:29
• The Hamiltonian structure isn't essential, except inasmuch as it allows for the use of Hamiltonian MCMC, which, to my understanding, has some nice properties. Is there anything about the structure of Eq. 4 in the paper that makes any particular form of MH seem best? From my inspection, it seems like both HMC and Gibbs sampling are difficult owing to the intractable marginals and conditionals. Feb 10, 2020 at 15:21
• Interesting. Perhaps another way to write it is to consider each $(p_{\ell}, q_{\ell})$ as a vector, $z_{\ell}$, on the unit disk and then do HMC by including a "kinetic energy" term defined on two-dimensional momenta, say $w_{\ell}$, which are marginally distributed according to bivariate gaussians. Then do HMC on $(z, w)$. I will give this a shot. Thanks, @πr8! Feb 10, 2020 at 17:26