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.

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.

We provide a clarification of the description of Langevin diffusions on Riemannian manifolds and of the measure underlying the invariant density. As a result we propose a new position-dependent Metropolis-adjusted Langevin algorithm (MALA) based upon a Langevin diffusion in $\mathbb{R}^d$ which has the required invariant density with respect to Lebesgue measure. We show that our diffusion and the diffusion upon which a previously-proposed position-dependent MALA is based are equivalent in some cases but are distinct in general. A simulation study illustrates the gain in efficiency provided by the new position-dependent MALA

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 positionindependent 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.MSC 2010 subject classifications: Primary 60J05; secondary 60J20, 60J22, 65C05, 65C40, 62F15, 60H30, 37A50.

Parental reflective functioning (PRF) describes a parent's capacity for considering both their own their child's thoughts, feelings, and behaviors, which can help parents to guide interactions with children. Given the cognitive demands of keeping infants in mind whilst caregiving, we examined the association between aspects of executive function (i.e., working memory and set-shifting) and PRF (employing the Parental Reflective Functioning Questionnaire) in recent mothers. In Study 1 (N=50), we found that better working memory was associated with higher levels of maternal interest and curiosity in their child's feelings. In Study 2 (N=68), we found that visual working memory and set-shifting capacity were also associated with higher levels of maternal interest and curiosity in their child's thoughts and feelings. Our results provide preliminary support for the association between executive processes and maternal reflective functioning. The implications of these findings and important future directions are discussed, including advancing our understanding of executive processes and PRF to support the broader family system.