To constrain physic in distant systems, astrophysics heavily relies on parametric models which are constrained and compared with data. Bayesian inference combined with Monte Carlo methods is the most common choice for practical computation of parameter probability distributions. However, we still lack a comprehensive and principled understanding of this inference process. Through a survey of inference problems across sub-fields, this talk presents a characterization of parametric inference spaces, defining a space of inference spaces. I will present new developments in nested sampling which scale the data analysis to current and upcoming large surveys, including systematic analyses of exoplanet candidates and high-energy surveys of accreting black holes. The research is enabling and vitalizing interdisciplinary collaborations of researchers from cosmology, astrophysics, particle physics and beyond.