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Data assimilation of observational data into full atmospheric states is essential for weather forecast model initialization. Recently, methods for deep generative data assimilation have been proposed which allow for using new input data without retraining the model. They could also dramatically accelerate the costly data assimilation process used in operational regional weather models. Here, in a central US testbed, we demonstrate the viability of score-based data assimilation in the context of realistically complex km-scale weather. We train an unconditional diffusion model to generate snapshots of a state-of-the-art km-scale analysis product, the High Resolution Rapid Refresh. Then, using score-based data assimilation to incorporate sparse weather station data, the model produces maps of precipitation and surface winds. The generated fields display physically plausible structures, such as gust fronts, and sensitivity tests confirm learnt physics through multivariate relationships. Preliminary skill analysis shows the approach already outperforms a naive baseline of the High-Resolution Rapid Refresh system itself. By incorporating observations from 40 weather stations, 10\% lower RMSEs on left-out stations are attained. Despite some lingering imperfections such as insufficiently disperse ensemble DA estimates, we find the results overall an encouraging proof of concept, and the first at km-scale. It is a ripe time to explore extensions that combine increasingly ambitious regional state generators with an increasing set of in situ, ground-based, and satellite remote sensing data streams., Comment: 18 pages, 7 figures

MPI is the most widely used interface for high-performance computing (HPC) workloads. Its success lies in its embrace of libraries and ability to evolve while maintaining backward compatibility for older codes, enabling them to run on new architectures for many years. In this paper, we propose a new level of MPI compatibility: a standard Application Binary Interface (ABI). We review the history of MPI implementation ABIs, identify the constraints from the MPI standard and ISO C, and summarize recent efforts to develop a standard ABI for MPI. We provide the current proposal from the MPI Forum's ABI working group, which has been prototyped both within MPICH and as an independent abstraction layer called Mukautuva. We also list several use cases that would benefit from the definition of an ABI while outlining the remaining constraints.

We introduce ClimateMachine, a new open-source atmosphere modeling framework using the Julia language to be performance portable on central processing units (CPUs) and graphics processing units (GPUs). ClimateMachine uses a common framework both for coarser-resolution global simulations and for high-resolution, limited-area large-eddy simulations (LES). Here, we demonstrate the LES configuration of the atmosphere model in canonical benchmark cases and atmospheric flows, using an energy-conserving nodal discontinuous-Galerkin (DG) discretization of the governing equations. Resolution dependence, conservation characteristics and scaling metrics are examined in comparison with existing LES codes. They demonstrate the utility of ClimateMachine as a modelling tool for limited-area LES flow configurations., Comment: 40 pages, 15 figures

Random variables and their distributions are a central part in many areas of statistical methods. The Distributions.jl package provides Julia users and developers tools for working with probability distributions, leveraging Julia features for their intuitive and flexible manipulation, while remaining highly efficient through zero-cost abstractions.

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., Comment: 29 pages + supplement (included in arXival as Appendix), 1 figure

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., Comment: 15 pages

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., Comment: 36 pages, 5 figures

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., Comment: 45 pages, 13 figures

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., Comment: Published at http://dx.doi.org/10.1214/15-AOS1319 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064), Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal of Statistics. 18 pages

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.

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.

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., Comment: 16 pages, to appear in Biometrika