Optimal prior for Bayesian inference in a constrained parameter space

Abstract: Bayesian parameter inference depends on a choice of prior probability distribution for the parameters in question. The prior which makes the posterior distribution maximally sensitive to data is called the Jeffreys prior, and it is completely determined by the response of the likelihood to changes in parameters. Under the assumption that the likelihood is a Gaussian distribution, the Jeffreys prior is a constant, i.e. flat. However, if one parameter is constrained by physical considerations, the Gaussian approximation fails and the flat prior is no longer the Jeffreys prior. In this paper we compute the correct Jeffreys prior for a multivariate normal distribution constrained in one dimension, and we apply it to the sum of neutrino masses $\Sigma m_\nu$ and the tensor-to-scalar ratio $r$. We find that one-dimensional marginalised posteriors for these two parameters change considerably and that the 68% and 95% Bayesian upper limits increase by 9% and 4% respectively for $\Sigma m_\nu$ and 22% and 3% for $r$. Adding the prior to an existing chain can be done as a trivial importance sampling in the final step of the analysis proces.