We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm 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 naturally identifies efficient implementations and motivates promising generalizations.

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm 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 naturally identifies efficient implementations and motivates promising generalizations.

Scoring functions are used to evaluate and compare partially probabilistic forecasts. We investigate the use of rank-sum functions such as empirical Area Under the Curve (AUC), a widely used measure of classification performance, as a scoring function for the prediction of probabilities of a set of binary outcomes. It is shown that the AUC is not generally a proper scoring function, that is, under certain circumstances it is possible to improve on the expected AUC by modifying the quoted probabilities from their true values. However with some restrictions, or with certain modifications, it can be made proper.

This paper considers the problem of defining distributions over graphical structures. We propose an extension of the hyper Markov properties of Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272-1317], which we term structural Markov properties, for both undirected decomposable and directed acyclic graphs, which requires that the structure of distinct components of the graph be conditionally independent given the existence of a separating component. This allows the analysis and comparison of multiple graphical structures, while being able to take advantage of the common conditional independence constraints. Moreover, we show that these properties characterise exponential families, which form conjugate priors under sampling from compatible Markov distributions.

We describe a Langevin diffusion with a target stationary density with respect to Lebesgue measure, as opposed to the volume measure of a previously-proposed diffusion. The two are sometimes equivalent but in general distinct and lead to different Metropolis-adjusted Langevin algorithms, which we compare.

Scandinavian Journal of StatisticsVolume 41, Issue 1 p. 19-21 Original Article Rejoinder: Geodesic Monte Carlo on Embedded Manifolds Simon Byrne, Corresponding Author Simon Byrne Department of Statistical Science University College London Simon Byrne, Department of Statistical Science University College London. simon.byrne@ucl.ac.ukSearch for more papers by this authorMark Girolami, Mark Girolami Department of Statistical Science University College LondonSearch for more papers by this author Simon Byrne, Corresponding Author Simon Byrne Department of Statistical Science University College London Simon Byrne, Department of Statistical Science University College London. simon.byrne@ucl.ac.ukSearch for more papers by this authorMark Girolami, Mark Girolami Department of Statistical Science University College LondonSearch for more papers by this author First published: 06 January 2014 https://doi.org/10.1111/sjos.12058Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume41, Issue1March 2014Pages 19-21 RelatedInformation

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064)

Hamiltonian Monte Carlo can provide powerful inference in complex statistical problems, but ultimately its performance is sensitive to various tuning parameters. In this paper we use the underlying geometry of Hamiltonian Monte Carlo to construct a universal optimization criteria for tuning the step size of the symplectic integrator crucial to any implementation of the algorithm as well as diagnostics to monitor for any signs of invalidity. An immediate outcome of this result is that the suggested target average acceptance probability of 0.651 can be relaxed to $0.6 \lesssim a \lesssim 0.9$ with larger values more robust in practice.

Prentice & Pyke (1979) established that the maximum likelihood estimate of an odds-ratio in a case-control study is the same as would be found by running a logistic regression: in other words, for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model. Similar results have been obtained for other models, and conditions have also been identified under which the corresponding Bayesian property holds, namely that the posterior distribution of the odds-ratio be the same, whether computed using the prospective or the retrospective likelihood. Here we demonstrate how these results follow directly from certain parameter independence properties of the models and priors, and identify prior laws that support such reverse analysis, for both standard and stratified designs.

Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton–Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices.