2 BETANCOURT ET AL of Fang,San tation in ical Monte Carlo (Lan 20121.1m effort to r len +1 rical inte to Ha niltonia Monte Carlo.Alth gh this lea ads dels th ce rapidly in pgimg 2012)in ndard Hami ehow critical to manc but why? paper we he theoretical fou on of Hamiltonian Monte Carlo wer que ions like We demo ritical to the of the algorith hence i several gene 1 ementa the success miltonian a We begin esPoeartiofg efficient Markov kernels and po sibl tho on mot ates the ols in differen e try, theory of probabilistic olds the penult at tha a sh the ion.and analysis of H tonian Monte this spective directs generalizations of the algorithm familiarity with differential geometry a complete understanding of this work will be a challe and we recommend that readers without a backgroun only scan through Section 2 to develop some intuition for the probabilistic interpretation bundles,Hemannian metrics,and symplectic forms,as wel I as the utility of Hamiltonia For those readers interesting in developing new implementations c Hamiltonian Monte Carlo we recommend a more careful reading of these sections and suggest introductory literature on the mathematics necessary to do so in the introduction of Section 2. 1 CONSTRUCTING EFFICIENT MARKOV KERNELS Bayesian inference is conceptually straightforward:the information about a system is first modeled with the construction of a posterior distribution,and then statistical questions can be answered by computing expectations with respect to that distribution.Many of the limitations of Bayesian inference arise not in the modeling of a posterior distribution but rather in computing the subsequent expectations.Because it provides a generic means of estimating these expectations markoy chain monte carlo has been critical to the success of the Bayesian methodology in practice. In this section we first review the Markov kernels intrinsic to Markov Chain Monte Carlo and then consider the dynamic systems perspective to motivate a strategy for constructing Markov kernels that yield computationally efficient inferences.2 BETANCOURT ET AL. scalability of the algorithm. Consider, for example, the Compressible Generalized Hybrid Monte Carlo scheme of Fang, Sanz-Serna and Skeel (2014) and the particular implemen￾tation in Lagrangian Dynamical Monte Carlo (Lan et al., 2012). In an effort to reduce the computational burden of the algorithm, the authors sacrifice the costly volume-preserving numerical integrators typical to Hamiltonian Monte Carlo. Although this leads to improved performance in some low-dimensional models, the performance rapidly diminishes with in￾creasing model dimension (Lan et al., 2012) in sharp contrast to standard Hamiltonian Monte Carlo. Clearly, the volume-preserving numerical integrator is somehow critical to scalable performance; but why? In this paper we develop the theoretical foundation of Hamiltonian Monte Carlo in order to answer questions like these. We demonstrate how a formal understanding naturally identifies the properties critical to the success of the algorithm, hence immediately providing a framework for robust implementations. Moreover, we discuss how the theory motivates several generalizations that may extend the success of Hamiltonian Monte Carlo to an even broader array of applications. We begin by considering the properties of efficient Markov kernels and possible strategies for constructing those kernels. This construction motivates the use of tools in differential geometry, and we continue by curating a coherent theory of probabilistic measures on smooth manifolds. In the penultimate section we show how that theory provides a skeleton for the development, implementation, and formal analysis of Hamiltonian Monte Carlo. Finally, we discuss how this formal perspective directs generalizations of the algorithm. Without a familiarity with differential geometry a complete understanding of this work will be a challenge, and we recommend that readers without a background in the subject only scan through Section 2 to develop some intuition for the probabilistic interpretation of forms, fiber bundles, Riemannian metrics, and symplectic forms, as well as the utility of Hamiltonian flows. For those readers interesting in developing new implementations of Hamiltonian Monte Carlo we recommend a more careful reading of these sections and suggest introductory literature on the mathematics necessary to do so in the introduction of Section 2. 1. CONSTRUCTING EFFICIENT MARKOV KERNELS Bayesian inference is conceptually straightforward: the information about a system is first modeled with the construction of a posterior distribution, and then statistical questions can be answered by computing expectations with respect to that distribution. Many of the limitations of Bayesian inference arise not in the modeling of a posterior distribution but rather in computing the subsequent expectations. Because it provides a generic means of estimating these expectations, Markov Chain Monte Carlo has been critical to the success of the Bayesian methodology in practice. In this section we first review the Markov kernels intrinsic to Markov Chain Monte Carlo and then consider the dynamic systems perspective to motivate a strategy for constructing Markov kernels that yield computationally efficient inferences