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The approach, referred to as eHMC, where\n                    <jats:italic>e<\/jats:italic>\n                    stands for empirical, leverages importance sampling to construct an empirical distribution on discretization parameters, thereby eliminating the need for manual burn-in diagnostics and online adaptation. The proposal distribution used in the calibration stage is obtained via a Population Monte Carlo scheme with tempering and relies on flexible parametric variational families such as normalizing flows. Once the calibration stage complete, the resulting algorithm defines a homogeneous Markov chain via a mixture of HMC kernels with a fixed mixing distribution, and hence preserves the target distribution. Numerical experiments indicate that eHMC can achieve competitive or improved sampling efficiency compared to the No-U-Turn Sampler (NUTS) in the case useful integration times can be summarized by the offline distribution. The comparison is assessed by standard efficiency metrics normalized by the number of leapfrog steps during the post-calibration sampling phase.\n                  <\/jats:p>","DOI":"10.1007\/s11222-026-10972-z","type":"journal-article","created":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T05:51:07Z","timestamp":1790142667000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Faster Hamiltonian Monte Carlo by learning leapfrog scale: an offline randomized solution"],"prefix":"10.1007","volume":"36","author":[{"given":"Changye","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Pudlo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian P.","family":"Robert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julien","family":"Stoehr","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,23]]},"reference":[{"key":"10972_CR1","unstructured":"Aufort, G., Pudlo, P., Burgarella, D.: Tempered, anti-trunctated, multiple importance sampling (2022). https:\/\/arxiv.org\/abs\/2205.01501"},{"issue":"3","key":"10972_CR2","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1214\/17-STS611","volume":"32","author":"S Agapiou","year":"2017","unstructured":"Agapiou, S., Papaspiliopoulos, O., Sanz-Alonso, D., Stuart, A.M.: Importance sampling: intrinsic dimension and computational cost. 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