A variational Gaussian approximation of the posterior distribution can be an excellent way to infer posterior quantities. However, to capture all posterior correlations the parametrization of the full covariance is required, which scales quadratic with the problem size. This scaling prohibits full-covariance approximations for large-scale problems. As a solution to this limitation we propose Metric Gaussian Variational Inference (MGVI). This procedure approximates the variational covariance such that it requires no parameters on its own and still provides reliable posterior correlations and uncertainties for all model parameters. We approximate the variational covariance with the inverse Fisher metric, a local estimate of the true posterior uncertainty. This covariance is only stored implicitly and all necessary quantities can be extracted from it by independent samples drawn from the approximating Gaussian. MGVI requires the minimization of a stochastic estimate of the Kullback-Leibler divergence only with respect to the mean of the variational Gaussian, a quantity that scales linearly with the problem size. We motivate the choice of this covariance from an information geometric perspective. We validate the method against established approaches, demonstrate its scalability into the regime over a million parameters and capability to capture posterior distributions over complex models with multiple components and strongly non-Gaussian prior distributions.

The immediate vicinity of an active supermassive black hole—with its event horizon, photon ring, accretion disk and relativistic jets—is an appropriate place to study physics under extreme conditions, particularly general relativity and magnetohydrodynamics. Observing the dynamics of such compact astrophysical objects provides insights into their inner workings, and the recent observations of M87* by the Event Horizon Telescope1–6 using very-long-baseline interferometry techniques allows us to investigate the dynamical processes of M87* on timescales of days. Compared with most radio interferometers, very-long-baseline interferometry networks typically have fewer antennas and low signal-to-noise ratios. Furthermore, the source is variable, prohibiting integration over time to improve signal-to-noise ratio. Here, we present an imaging algorithm7,8 that copes with the data scarcity and temporal evolution, while providing an uncertainty quantification. Our algorithm views the imaging task as a Bayesian inference problem of a time-varying brightness, exploits the correlation structure in time and reconstructs (2 + 1 + 1)-dimensional time-variable and spectrally resolved images. We apply this method to the Event Horizon Telescope observations of M87*9 and validate our approach on synthetic data. The time- and frequency-resolved reconstruction of M87* confirms variable structures on the emission ring and indicates extended and time-variable emission structures outside the ring itself.

The next-generation Event Horizon Telescope (ngEHT) will be a significant enhancement of the Event Horizon Telescope (EHT) array, with ∼10 new antennas and instrumental upgrades of existing antennas. The increased uv-coverage, sensitivity, and frequency coverage allow a wide range of new science opportunities to be explored. The ngEHT Analysis Challenges have been launched to inform the development of the ngEHT array design, science objectives, and analysis pathways. For each challenge, synthetic EHT and ngEHT datasets are generated from theoretical source models and released to the challenge participants, who analyze the datasets using image reconstruction and other methods. The submitted analysis results are evaluated with quantitative metrics. In this work, we report on the first two ngEHT Analysis Challenges. These have focused on static and dynamical models of M87* and Sgr A* and shown that high-quality movies of the extended jet structure of M87* and near-horizon hourly timescale variability of Sgr A* can be reconstructed by the reference ngEHT array in realistic observing conditions using current analysis algorithms. We identify areas where there is still room for improvement of these algorithms and analysis strategies. Other science cases and arrays will be explored in future challenges.