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Optimal prior for Bayesian inference in a constrained parameter space · arXivDesk

1710.08899Oct 24, 2017As it was pointed out in 1802.09450 and later in 1902.07667, the Jeffreys prior is unchanged by restricting a parameter based on physical considerations. This breaks our assumption of a truncated Gaussian in Eq. 2.10 and thus invalidates our conclusions. (In situations where Eq. 2.10 is a good model of the data, the calculation may still be useful.)

Optimal prior for Bayesian inference in a constrained parameter space

Steen Hannestad, Thomas Tram

Cosmology and Nongalactic Astrophysics

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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 $Σm_ν$ 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 $Σm_ν$

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Researchers working in astro-ph.CO and adjacent areas.

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The paper addresses: Optimal prior for Bayesian inference in a constrained parameter space.
Bayesian parameter inference depends on a choice of prior probability distribution for the parameters in question.

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Author comment: As it was pointed out in 1802.09450 and later in 1902.07667, the Jeffreys prior is unchanged by restricting a parameter based on physical considerations. This breaks our assumption of a truncated Gaussian in Eq. 2.10 and thus invalidates our conclusions. (In situations where Eq. 2.10 is a good model of the data, the calculation may still be useful.)