Sub d to Stat The Geometric Foundations of Hamiltonian Monte Carlo Michael Betancourt,Simon Byrne,Sam Livingstone,and Mark Girolami Abstract.Although Hamiltonian Monte Carlo has proven an empirical success,the lack of a rigorous theoretical understanding of the algo- rithm has in many ways impeded both principled developments of the method and use of the algorithm in practice.In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds,and demonstrate how the theory natu- rally identifies efficient implementations and motivates promising gen- 回 eralizations. Key words and phrases:Markoy Chain monte Carlo.Hamiltonian monte Carlo,Disintegration,Differential Geometry,Smooth Manifold,Fiber Bundle,Riemannian Geometry,Symplectic Geometry. Th quires algorithms capable of fitting mod- wit if not thousand ia oft n hierarchica Carlo (Duane er na- proven tremen suc scien and S 2013 alakhutdin V,2013 )ecology(Sch ,0 Terada,Inoue and Nishihara, epidemio (e( ing et a 2012), (Husain, Vasishth and Srinivasan 14),pnarma asche et al.,2010;Porter and Carre Sanders,Betancourt and Soderberg,2014;Wang et al.,2014),and political science(Ghitza and Gelman, 2014,t name a few.Despite such widesprea empirical success,however,there remains an alr or mys oncerning the efficacy of the algorithm This la f understanding not only limits the adoption of Hamiltonian Monte Carlo but may also foster unprincipled and,ultimately,fragile implementations that restrict the Michael Betancourt is a Postdoctoral Research As ociate at the University of Warwick Coventry CV4 7AL,UK (e-mail:betanalphaugmail.com).Simon Byrne is an EPSRC Postdoctoral Research Fellow at University College London,Gower Street,London WCIE 6B Sam Livingstone is a PhD candidate at University Co London,Gower Street,London,WC1E 6BT Mark Girolami is an ESPRC Established Career Research Fellow at the University of Warwick.Coventry CVA TAL.UK. 1 arXiv:1410.5110v1 [stat.ME] 19 Oct 2014 Submitted to Statistical Science The Geometric Foundations of Hamiltonian Monte Carlo Michael Betancourt, Simon Byrne, Sam Livingstone, and Mark Girolami Abstract. Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algo￾rithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory natu￾rally identifies efficient implementations and motivates promising gen￾eralizations. Key words and phrases: Markov Chain Monte Carlo, Hamiltonian Monte Carlo, Disintegration, Differential Geometry, Smooth Manifold, Fiber Bundle, Riemannian Geometry, Symplectic Geometry. The frontier of Bayesian inference requires algorithms capable of fitting complex mod￾els with hundreds, if not thousands of parameters, intricately bound together with non￾linear and often hierarchical correlations. Hamiltonian Monte Carlo (Duane et al., 1987; Neal, 2011) has proven tremendously successful at extracting inferences from these mod￾els, with applications spanning computer science (Sutherland, P´oczos and Schneider, 2013; Tang, Srivastava and Salakhutdinov, 2013), ecology (Schofield et al., 2014; Terada, Inoue and Nishihara, 2013), epidemiology (Cowling et al., 2012), linguistics (Husain, Vasishth and Srinivasan, 2014), pharmacokinetics (Weber et al., 2014), physics (Jasche et al., 2010; Porter and Carr´e, 2014; Sanders, Betancourt and Soderberg, 2014; Wang et al., 2014), and political science (Ghitza and Gelman, 2014), to name a few. Despite such widespread empirical success, however, there remains an air of mystery concerning the efficacy of the algorithm. This lack of understanding not only limits the adoption of Hamiltonian Monte Carlo but may also foster unprincipled and, ultimately, fragile implementations that restrict the Michael Betancourt is a Postdoctoral Research Associate at the University of Warwick, Coventry CV4 7AL, UK (e-mail: betanalpha@gmail.com). Simon Byrne is an EPSRC Postdoctoral Research Fellow at University College London, Gower Street, London, WC1E 6BT. Sam Livingstone is a PhD candidate at University College London, Gower Street, London, WC1E 6BT. Mark Girolami is an ESPRC Established Career Research Fellow at the University of Warwick, Coventry CV4 7AL, UK. 1