Microcanonical Hamiltonian Monte Carlo relates to thermodynamics and new sampling methods
Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem
Artificial Intelligence
Summary
Some scientists studied a special method called Microcanonical Hamiltonian Monte Carlo, which is used to explore complex probabilities. They showed that this method follows ideas from thermodynamics, the science of heat and energy, by proving it meets a key principle called the Helmholtz theorem. They also created a new version of the method that works better for simpler problems with fewer variables. Finally, they suggest that other common methods called canonical Markov Chain Monte Carlo may be easier to understand from energy and information perspectives.
Microcanonical Hamiltonian Monte CarlothermodynamicsHelmholtz theoremmicrocanonical ensembleMarkov Chain Monte Carlosampling algorithmsthermodynamic potentialsfirst law of thermodynamicsinference problems
Authors
Heinrich von Campe, Bjoern Malte Schaefer
Abstract
The recently proposed Microcanonical Hamiltonian Monte Carlo algorithm has not yet been studied in detail from a thermodynamic point of view; this work aims to fill that gap. We demonstrate how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble. In particular, we demonstrate (analytically and numerically) that the algorithm fulfils the Helmholtz theorem, an alternative formulation of the first law of thermodynamics. Furthermore, we construct a new sampling algorithm that extends the original to lower-dimensional inference problems. Finally, we argue that canonical Markov Chain Monte Carlo algorithms are more natural than Microcanonical Hamiltonian Monte Carlo from the thermodynamic and information-theoretic point of view.