A sample of parametric Bayesian inference applications from astronomy, cosmology and particle physics is studied, augmented by mock data sets and toy problems. The parameter spaces of these parametric physical models and their posterior distributions from analysing specific data are characterized by (1) the number of model parameters, (2) whether the posterior shape is similar to a Gaussian, (3) whether the posterior has light or heavy tails, (4) how small the posterior is compared to the prior, i.e., how informative the data are, (5) whether some parameters remain unconstrained while others are highly constrained, (6) whether the posterior has multiple, disconnected modes, and (7) whether the inference undergoes phase transitions. These axis define a parameter space of inference problems. We characterize each of the inference problems and observe that inference in astrophysics spans the entire parameter space, from low to high dimensionality, mono- to multi-modal, and a variety of complex distributions that range from uninformative to highly informative. Furthermore, the computational cost of the physical models can range from milliseconds to dozens of seconds. The collated sample of inference problems is proposed as a standard test bed for new samplers. For reproducibility and ease of use, a Docker compute image is provided.
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