  Physical theories that depend on many parameters or are tested against data
from many different experiments pose unique challenges to statistical
inference. Many models in particle physics, astrophysics and cosmology fall
into one or both of these categories. These issues are often sidestepped with
statistically unsound ad hoc methods, involving intersection of parameter
intervals estimated by multiple experiments, and random or grid sampling of
model parameters. Whilst these methods are easy to apply, they exhibit
pathologies even in low-dimensional parameter spaces, and quickly become
problematic to use and interpret in higher dimensions. In this article we give
clear guidance for going beyond these procedures, suggesting where possible
simple methods for performing statistically sound inference, and
recommendations of readily-available software tools and standards that can
assist in doing so. Our aim is to provide any physicists lacking comprehensive
statistical training with recommendations for reaching correct scientific
conclusions, with only a modest increase in analysis burden. Our examples can
be reproduced with the code publicly available at
https://doi.org/10.5281/zenodo.4322283.
