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  1. We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of e^{−f(x)} on a convex body M ⊂ R^n. We show that for distributions in the form of e−^{a x} on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of ∥a∥_2 and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of mn^3 to achieve ϵ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form e^{−f(x)} in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of our old result, which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice. 
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    Free, publicly-accessible full text available June 12, 2024
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    Background: Healthcare workers are at the forefront of the COVID-19 pandemic and it is essential to monitor the relative incidence rate of this group, as compared to workers in other occupations. This study aimed to produce estimates of the relative incidence ratio between healthcare workers and workers in non-healthcare occupations. Methods: Analysis of cross-sectional data from a daily, web-based survey of 1,822,662 Facebook users from September 8, 2020 to October 20, 2020. Participants were Facebook users in the United States aged 18 and above who were tested for COVID-19 because of an employer or school requirement in the past 14 days. The exposure variable was a self-reported history of working in healthcare in the past four weeks and the main outcome was a self-reported positive test for COVID-19. Results: On October 20, 2020, in the United States, there was a relative COVID-19 incidence ratio of 0.73 (95% UI 0.68 to 0.80) between healthcare workers and workers in non-healthcare occupations. Conclusions: In fall of 2020, in the United States, healthcare workers likely had a lower COVID-19 incidence rate than workers in non-healthcare occupations. 
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