Optimal prior for Bayesian inference in a constrained parameter space. (arXiv:1710.08899v2 [astro-ph.CO] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Hannestad_S/0/1/0/all/0/1">Steen Hannestad</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Tram_T/0/1/0/all/0/1">Thomas Tram</a>
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.
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.
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