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I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo.

In this paper The geometric foundations of Hamiltonian Monte Carlo it is mentioned that a good reference is John Lee's Introduction to smooth manifolds and here is my question:

  • What are the core concepts that I should know from smooth manifolds theory in order to understand Hamiltonian Monte Carlo from a mathematics perspective?

Reading Lee's book from cover to back seems a daunting task, so I'd like to have more guidance over what sections (or topics) I should definitely read.

P.S-1: I'm not constrained to Lee's book(s), I'm just looking for a kind of "syllabus" of the core topics to understand HMC.

P.S-2: If it is helpful, I have a background in measure-theoretic probability, real analysis and introductory topology.

Thanks for your answers!

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    $\begingroup$ I see there is a vote to close. I see this as unfortunate. I think this is a carefully constructed question at an appropriate level whose answers have the potential to be useful to others. $\endgroup$
    – Anthony Quas
    Commented Sep 14, 2021 at 22:36

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Core concepts: i) Co-tangent bundle of a given manifold, ii) canonical one form $\theta$ on any co-tangent bundle, associated symplectic form $d \theta$, associated canonical Liouville (phase-space) volume form (=measure) $d \theta \wedge \ldots \wedge d \theta$. The latter equals $dq dp$ in any coordinate system, iii) conservation of this symplectic form hence Liouville measure by any Hamiltonian flow (see also Poisson brackets formalism), iv) Lagrange/Hamiltonian formalism of classical mechanics.

Basically, HMC constructs a Markov Chain with prescribed invariant distribution using i) the conservation of Liouville by Hamiltonian flows, ii) the conservation of the Hamiltonian function. The reversibility of such flows when the Hamiltonian is invariant under momentum reversal is also useful for Metropolis / reversibility considerations.

So all you need is basics of exterior calculus and classical mechanics, arguably main ref is this book.

So in Lee's book this would be covered by chapters: Cotangent bundle, differential forms, symplectic manifolds.

For a numerical view point, see also this book.

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