Cosmological and astrophysical data are now sufficiently large and complex to become almost intractable by traditional statistical techniques. Unravelling the twin mysteries of dark energy and dark matter will require new data analysis methods capable of extracting knowledge from upcoming data streams, including the Euclid satellite, the Nancy Grace Roman space telescope, LSST/Vera Rubin observatory and LISA. While machine learning offers great promise, it suffers from difficulties in interpretability as well as from poor generalisation when the training set is not representative of the target data (known as “covariate shift”) - an almost ubiquitous problem in astronomy due to selection effects.  I will motivate and present some of the latest machine learning tools being developed for fast and scalable inference, focusing on simulation based inference and particularly on neural ratio estimation for amortised marginal posterior inference as the natural heir to classical Bayesian inference techniques. I will then discuss a general, statistically principled solution to covariate shift in supervised learning, which achieves gold standard performance in a variety of important test cases. I will conclude with example applications in supernova type Ia cosmology, dark matter direct detection and gravitational waves astronomy. Those unable to attend the colloquium in person are invited to participate online through Zoom (Meeting ID: 942 0262 2849, passcode 792771) using the link: https://eu02web.zoom-x.de/j/94202622849?pwd=dGlPQXBiUytzY1M2UE5oUDRhbzNOZz09 During his visit to Heidelberg Professor Trotta will be available for meetings by arrangement with his host, Ivelina Momcheva (momcheva@mpia.de).