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A Conceptual Introduction to Hamiltonian Monte Carlo

Michael Betancourt

2017 · arXiv

Statistics & probabilityMathematics
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From the abstract

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

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3 selected excerpts · 3 total mentions

crystal_revenge
Neutral mention · rule
I think you're confusing the sampling process and the convergence of those samples with the warmup process (also called 'burn-in') in HMC. When doing HMC MCMC we typically don't start sampling right away (or, more precisely we throw out those samples) because we may be initializing the sampler in a part of the distribution that involves pretty low probability density. After the chain has run awhile it tends to end up sampling from the typical set which, especially in high dimensional distribution, tends to more correctly represent the distribution we actually want to integrate over. So for language when I say "Bob has three apples, Jane gives him four and Judy takes two…
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how many apples does Bob have" we're actually pretty far from the part of the linguistic manifold where the correct answer is likely to be. As the chain wanders this space it's getting closer until it finally statistically follow the path "this answer is..." and when it's sampling from this path it's in a much more likely neighborhood of the correct answer. That is, after wandering a bit, more and more of the possible paths are closer to where the actual answer lies than they would be if we had just forced the model to choose early. edit: Michael Betancourt has great introduction to HMC which covers warm-up and the typical set https://arxiv.org/pdf/1701.02434 (he has a ton more content that dives much more deeply into the specifics)
Original comment What I don’t like about chains of thoughts (2023)
winwang
Neutral mention · rule
Also did a bit of QC research. Since I'm tipsy, I skimmed the Twitter post. Here's a pdf on arxiv about HMC (2017) which I found in 20 seconds: https://arxiv.org/abs/1701.02434 Seems like Claude3 is decently logical in paraphrasing key parts of text in a good order, but with repetition scattered everywhere. Also Monte Carlo is something which sounds cool but is literally just conceptual dart-throwing. I've always hated the name.
Original comment Claude 3 Opus reinvented this quantum algorithm from scratch in just 2 prompts
tgb
Neutral mention · rule
I won't dispute the main point of the article but a couple minor errors bug me. First, he kept referring to a Gaussian distribution as being the unit sphere, when of course the radius depends upon the parameters of the Gaussian (the standard deviation). If not, then it wouldn't be invariant under which units you chose. A bizarre mistake to repeat many times throughout the article. Less importantly, the last paragraph says that the probability that two samples are orthogonal is "very high". Being precisely orthogonal is technically a probability zero event. There author means "very close to orthogonal." There was a good discussion about this problem in the context of Monte Carlo simulations in (1). (1) https://arxiv.org/abs/1701.02434
Original comment Gaussian Distributions Are Soap Bubbles

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